1.
Free surface
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Unlike liquids, gases cannot form a free surface on their own. Fluidized/liquified solids, including slurries, granular materials, and powders may form a free surface, a liquid in a gravitational field will form a free surface if unconfined from above. Under mechanical equilibrium this free surface must be perpendicular to the acting on the liquid, if not there would be a force along the surface. Thus, on the surface of the Earth, all surfaces of liquids are horizontal unless disturbed. In a free liquid that is not affected by forces such as a gravitational field. Its free surface will assume the shape with the least surface area for its volume, such behaviour can be expressed in terms of surface tension. It can be demonstrated experimentally by observing a large globule of oil placed below the surface of a mixture of water, if the free surface of a liquid is disturbed, waves are produced on the surface. These waves are not elastic waves due to any elastic force, momentum causes the wave to overshoot, thus oscillating and spreading the disturbance to the neighboring portions of the surface. The velocity of the surface varies as the square root of the wavelength if the liquid is deep. Capillary ripples are damped both by sub-surface viscosity and by surface rheology. The free surface at each point is at an angle to the force acting at it, which is the resultant of the force of gravity. Since the main mirror in a telescope must be parabolic, this principle is used to create liquid mirror telescopes. In hydrodynamics, the surface is defined mathematically by the free-surface condition, that is. In fluid dynamics, a vortex, also known as a potential vortex or whirlpool, forms in an irrotational flow. In naval architecture and marine safety, the surface effect occurs when liquids or granular materials under a free surface in partially filled tanks or holds shift when the vessel heels. In hydraulic engineering a free-surface jet is one where the entrainment of the fluid outside the jet is minimal, a liquid jet in air approximates a free surface jet. In fluid mechanics a free flow, also called open channel flow, is the gravity driven flow of a fluid under a free surface. Free surface effect Surface tension Laser-heated pedestal growth Liquid level Splash Slosh dynamics Riabouchinsky solid
2.
Hydrostatics
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Fluid statics or hydrostatics is the branch of fluid mechanics that studies incompressible fluids at rest. It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamics, hydrostatics are categorized as a part of the fluid statics, which is the study of all fluids, incompressible or not, at rest. Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids and it is also relevant to geophysics and astrophysics, to meteorology, to medicine, and many other fields. Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisterns, aqueducts and fountains. Archimedes is credited with the discovery of Archimedes Principle, which relates the force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The fair cup or Pythagorean cup, which dates from about the 6th century BC, is a technology whose invention is credited to the Greek mathematician. It was used as a learning tool, the cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom. The height of this pipe is the same as the line carved into the interior of the cup, the cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, due to the drag that molecules exert on one another, the cup will be emptied. Herons fountain is a device invented by Heron of Alexandria that consists of a jet of fluid being fed by a reservoir of fluid. The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, the device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir. Pascal made contributions to developments in both hydrostatics and hydrodynamics, due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface, if a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force, thus, the pressure on a fluid at rest is isotropic, i. e. it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes, i. e. a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in an extended form, by Blaise Pascal. In a fluid at rest, all frictional and inertial stresses vanish, when this condition of V =0 is applied to the Navier-Stokes equation, the gradient of pressure becomes a function of body forces only
3.
Coriolis force
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In physics, the Coriolis force is an inertial force that acts on objects that are in motion relative to a rotating reference frame. In a reference frame with clockwise rotation, the acts to the left of the motion of the object. In one with anticlockwise rotation, the acts to the right. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology, deflection of an object due to the Coriolis force is called the Coriolis effect. Newtons laws of motion describe the motion of an object in a frame of reference. When Newtons laws are transformed to a frame of reference. Both forces are proportional to the mass of the object, the Coriolis force is proportional to the rotation rate and the centrifugal force is proportional to its square. The Coriolis force acts in a perpendicular to the rotation axis. The centrifugal force acts outwards in the direction and is proportional to the distance of the body from the axis of the rotating frame. These additional forces are termed inertial forces, fictitious forces or pseudo forces and they allow the application of Newtons laws to a rotating system. They are correction factors that do not exist in a non-accelerating or inertial reference frame, a commonly encountered rotating reference frame is the Earth. The Coriolis effect is caused by the rotation of the Earth, such motions are constrained by the surface of the Earth, so only the horizontal component of the Coriolis force is generally important. This force causes moving objects on the surface of the Earth to be deflected to the right in the Northern Hemisphere, the horizontal deflection effect is greater near the poles, since the effective rotation rate about a local vertical axis is largest there, and smallest at the equator. This effect is responsible for the rotation of large cyclones, riccioli, Grimaldi, and Dechales all described the effect as part of an argument against the heliocentric system of Copernicus. In other words, they argued that the Earths rotation should create the effect, the effect was described in the tidal equations of Pierre-Simon Laplace in 1778. Gaspard-Gustave Coriolis published a paper in 1835 on the yield of machines with rotating parts. That paper considered the forces that are detected in a rotating frame of reference. Coriolis divided these forces into two categories
4.
Conservation of mass
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Hence, the quantity of mass is conserved over time. Thus, during any chemical reaction, nuclear reaction, or radioactive decay in an isolated system, the concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. e. Those completely isolated from all exchanges with the environment, in this circumstance, the mass–energy equivalence theorem states that mass conservation is equivalent to total energy conservation, which is the first law of thermodynamics. By contrast, for a closed system mass is only approximately conserved. Certain types of matter may be created or destroyed, but in all of these processes, for a discussion, see mass in general relativity. An important idea in ancient Greek philosophy was that Nothing comes from nothing, so that what exists now has always existed, no new matter can come into existence where there was none before. A further principle of conservation was stated by Epicurus who, describing the nature of the Universe, wrote that the totality of things was always such as it is now, and always will be. Jain philosophy, a non-creationist philosophy based on the teachings of Mahavira, states that the universe, the Jain text Tattvarthasutra states that a substance is permanent, but its modes are characterised by creation and destruction. A principle of the conservation of matter was also stated by Nasīr al-Dīn al-Tūsī and he wrote that A body of matter cannot disappear completely. It only changes its form, condition, composition, color and other properties, the principle of conservation of mass was first outlined by Mikhail Lomonosov in 1748. He proved it by experiments—though this is sometimes challenged, antoine Lavoisier had expressed these ideas in 1774. Others whose ideas pre-dated the work of Lavoisier include Joseph Black, Henry Cavendish, the conservation of mass was obscure for millennia because of the buoyancy effect of the Earths atmosphere on the weight of gases. For example, a piece of wood weighs less after burning, the vacuum pump also enabled the weighing of gases using scales. Once understood, the conservation of mass was of importance in progressing from alchemy to modern chemistry. His research indicated that in certain reactions the loss or gain could not have more than from 2 to 4 parts in 100,000. The difference in the accuracy aimed at and attained by Lavoisier on the one hand, in special relativity, the conservation of mass does not apply if the system is open and energy escapes. However, it continue to apply to totally closed systems. If energy cannot escape a system, its mass cannot decrease, in relativity theory, so long as any type of energy is retained within a system, this energy exhibits mass
5.
Momentum
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In classical mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object, quantified in kilogram-meters per second. It is dimensionally equivalent to impulse, the product of force and time, Newtons second law of motion states that the change in linear momentum of a body is equal to the net impulse acting on it. If the truck were lighter, or moving slowly, then it would have less momentum. Linear momentum is also a quantity, meaning that if a closed system is not affected by external forces. In classical mechanics, conservation of momentum is implied by Newtons laws. It also holds in special relativity and, with definitions, a linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory. It is ultimately an expression of one of the symmetries of space and time. Linear momentum depends on frame of reference, observers in different frames would find different values of linear momentum of a system. But each would observe that the value of linear momentum does not change with time, momentum has a direction as well as magnitude. Quantities that have both a magnitude and a direction are known as vector quantities, because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Below, the properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations, the momentum of a particle is traditionally represented by the letter p. It is the product of two quantities, the mass and velocity, p = m v, the units of momentum are the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity in meters per second then the momentum is in kilogram meters/second, in cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters/second. Being a vector, momentum has magnitude and direction, for example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg m/s due north measured from the ground. The momentum of a system of particles is the sum of their momenta, if two particles have masses m1 and m2, and velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2. If all the particles are moving, the center of mass will generally be moving as well, if the center of mass is moving at velocity vcm, the momentum is, p = m v cm. This is known as Eulers first law, if a force F is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t
6.
Hydraulic jump
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A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flow such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, in an open channel flow, this manifests as the fast flow rapidly slowing and piling up on top of itself similar to how a shockwave forms. The phenomenon is dependent upon the fluid speed. If the initial speed of the fluid is below the critical speed, for initial flow speeds which are not significantly above the critical speed, the transition appears as an undulating wave. As the initial flow speed increases further, the transition becomes more abrupt, until at high enough speeds, when this happens, the jump can be accompanied by violent turbulence, eddying, air entrainment, and surface undulations, or waves. There are two main manifestations of hydraulic jumps and historically different terminology has been used for each, the different manifestations are, The stationary hydraulic jump – rapidly flowing water transitions in a stationary jump to slowly moving water as shown in Figures 1 and 2. The tidal bore – a wall or undulating wave of water moves upstream against water flowing downstream as shown in Figures 3 and 4. If considered from a frame of reference which moves with the wave front, a related case is a cascade – a wall or undulating wave of water moves downstream overtaking a shallower downstream flow of water as shown in Figure 5. If considered from a frame of reference which moves with the wave front and these phenomena are addressed in an extensive literature from a number of technical viewpoints. Hydraulic jumps can be seen in both a form, which is known as a hydraulic jump, and a dynamic or moving form. They can be described using the same approaches and are simply variants of a single phenomenon. A tidal bore is a jump which occurs when the incoming tide forms a wave of water that travel up a river or narrow bay against the direction of the current. Figure 3 shows a tidal bore with the common to shallow upstream water – a large elevation difference is observed. Figure 4 shows a tidal bore with the common to deep upstream water – a small elevation difference is observed. In both cases the tidal wave moves at the characteristic of waves in water of the depth found immediately behind the wave front. A key feature of tidal bores and positive surges is the turbulent mixing induced by the passage of the bore front. Another variation of the hydraulic jump is the cascade. In the cascade, a series of waves or undulating waves of water moves downstream overtaking a shallower downstream flow of water
7.
Stream bed
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A stream bed is the channel bottom of a stream, river or creek, the physical confine of the normal water flow. The lateral confines or channel margins are known as the banks or river banks, during all. Under certain conditions a river can branch from one bed to multiple stream beds. A flood occurs when a stream overflows its banks and flows onto its flood plain, as a general rule, the bed is the part of the channel up to the normal water line, and the banks are that part above the normal water line. However, because water flow varies, this differentiation is subject to local interpretation, the nature of any stream bed is always a function of the flow dynamics and the local geologic materials, influenced by that flow. With small streams in mesophytic regions, the nature of the bed is strongly responsive to conditions of precipitation runoff. Where natural conditions of either grassland or forest ameliorate peak flows, stream beds are stable, possibly rich, with organic matter and these streams support a rich biota. Where conditions produce unnatural levels of runoff, such as occurs below roads and this process greatly increases watershed erosion and results in thinner soils, upslope from the stream bed, as the channel adjusts to the increase in flow. The stream bed is very complex in terms of erosion, sediment is transported, eroded and deposited on the stream bed. With global warming there is a fear that the size and shape of riverbeds will change due to increased flood magnitude and this shows that the stream bed is left mostly unchanged in size and shape. Dry stream beds are also subject to becoming underground water pockets and flooding by heavy rains and water rising from the ground and may sometimes be part of the rejuvenation of the stream
8.
Friction
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Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction, Dry friction resists relative lateral motion of two surfaces in contact. Dry friction is subdivided into static friction between non-moving surfaces, and kinetic friction between moving surfaces, fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces, skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body. Internal friction is the force resisting motion between the making up a solid material while it undergoes deformation. When surfaces in contact move relative to other, the friction between the two surfaces converts kinetic energy into thermal energy. This property can have consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to thermal energy whenever motion with friction occurs, another important consequence of many types of friction can be wear, which may lead to performance degradation and/or damage to components. Friction is a component of the science of tribology, Friction is not itself a fundamental force. Dry friction arises from a combination of adhesion, surface roughness, surface deformation. The complexity of interactions makes the calculation of friction from first principles impractical and necessitates the use of empirical methods for analysis. Friction is a non-conservative force - work done against friction is path dependent, in the presence of friction, some energy is always lost in the form of heat. Thus mechanical energy is not conserved, the Greeks, including Aristotle, Vitruvius, and Pliny the Elder, were interested in the cause and mitigation of friction. They were aware of differences between static and kinetic friction with Themistius stating in 350 A. D. that it is easier to further the motion of a moving body than to move a body at rest. The classic laws of sliding friction were discovered by Leonardo da Vinci in 1493, a pioneer in tribology and these laws were rediscovered by Guillaume Amontons in 1699. Amontons presented the nature of friction in terms of surface irregularities, the understanding of friction was further developed by Charles-Augustin de Coulomb. Coulomb further considered the influence of sliding velocity, temperature and humidity, the distinction between static and dynamic friction is made in Coulombs friction law, although this distinction was already drawn by Johann Andreas von Segner in 1758. Leslie was equally skeptical about the role of adhesion proposed by Desaguliers, in Leslies view, friction should be seen as a time-dependent process of flattening, pressing down asperities, which creates new obstacles in what were cavities before
9.
Viscosity
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The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. For liquids, it corresponds to the concept of thickness, for example. Viscosity is a property of the fluid which opposes the motion between the two surfaces of the fluid in a fluid that are moving at different velocities. For a given velocity pattern, the stress required is proportional to the fluids viscosity, a fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at low temperatures in superfluids. Otherwise, all fluids have positive viscosity, and are said to be viscous or viscid. A fluid with a high viscosity, such as pitch. The word viscosity is derived from the Latin viscum, meaning mistletoe, the dynamic viscosity of a fluid expresses its resistance to shearing flows, where adjacent layers move parallel to each other with different speeds. It can be defined through the situation known as a Couette flow. This fluid has to be homogeneous in the layer and at different shear stresses, if the speed of the top plate is small enough, the fluid particles will move parallel to it, and their speed will vary linearly from zero at the bottom to u at the top. Each layer of fluid will move faster than the one just below it, in particular, the fluid will apply on the top plate a force in the direction opposite to its motion, and an equal but opposite one to the bottom plate. An external force is required in order to keep the top plate moving at constant speed. The magnitude F of this force is found to be proportional to the u and the area A of each plate. The proportionality factor μ in this formula is the viscosity of the fluid, the ratio u/y is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction perpendicular to the plates. Isaac Newton expressed the forces by the differential equation τ = μ ∂ u ∂ y, where τ = F/A. This formula assumes that the flow is moving along parallel lines and this equation can be used where the velocity does not vary linearly with y, such as in fluid flowing through a pipe. Use of the Greek letter mu for the dynamic viscosity is common among mechanical and chemical engineers. However, the Greek letter eta is used by chemists, physicists
10.
Density
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The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume. The symbol most often used for density is ρ, although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume, ρ = m V, where ρ is the density, m is the mass, and V is the volume. In some cases, density is defined as its weight per unit volume. For a pure substance the density has the numerical value as its mass concentration. Different materials usually have different densities, and density may be relevant to buoyancy, purity, osmium and iridium are the densest known elements at standard conditions for temperature and pressure but certain chemical compounds may be denser. Thus a relative density less than one means that the floats in water. The density of a material varies with temperature and pressure and this variation is typically small for solids and liquids but much greater for gases. Increasing the pressure on an object decreases the volume of the object, increasing the temperature of a substance decreases its density by increasing its volume. In most materials, heating the bottom of a results in convection of the heat from the bottom to the top. This causes it to rise relative to more dense unheated material, the reciprocal of the density of a substance is occasionally called its specific volume, a term sometimes used in thermodynamics. Density is a property in that increasing the amount of a substance does not increase its density. Archimedes knew that the irregularly shaped wreath could be crushed into a cube whose volume could be calculated easily and compared with the mass, upon this discovery, he leapt from his bath and ran naked through the streets shouting, Eureka. As a result, the term eureka entered common parlance and is used today to indicate a moment of enlightenment, the story first appeared in written form in Vitruvius books of architecture, two centuries after it supposedly took place. Some scholars have doubted the accuracy of this tale, saying among other things that the method would have required precise measurements that would have been difficult to make at the time, from the equation for density, mass density has units of mass divided by volume. As there are units of mass and volume covering many different magnitudes there are a large number of units for mass density in use. The SI unit of kilogram per metre and the cgs unit of gram per cubic centimetre are probably the most commonly used units for density.1,000 kg/m3 equals 1 g/cm3. In industry, other larger or smaller units of mass and or volume are often more practical, see below for a list of some of the most common units of density
11.
Product rule
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In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated as ′ = f ′ ⋅ g + f ⋅ g ′ or in the Leibniz notation d d x = u ⋅ d v d x + v ⋅ d u d x. In differentials notation, this can be written as d = u d v + v d u, discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. Here is Leibnizs argument, Let u and v be two functions of x. Then the differential of uv is d = ⋅ − u ⋅ v = u ⋅ d v + v ⋅ d u + d u ⋅ d v. Since the term du·dv is negligible, Leibniz concluded that d = v ⋅ d u + u ⋅ d v, suppose we want to differentiate ƒ = x2 sin. By using the rule, one gets the derivative ƒ = 2x sin + x2cos. This follows from the rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear, the rule for integration by parts is derived from the product rule, as is the quotient rule. Let h = f g, and suppose that f and g are each differentiable at x and we want to prove that h is differentiable at x and that its derivative h is given by f g + f g. To do this f g − f g is added to the numerator to permit its factoring, a rigorous proof of the product rule can be given using the definition of the derivative as a limit, and the basic properties of limits. Let h = f g, and suppose that f and g are each differentiable at x0 and we want to prove that h is differentiable at x0 and that its derivative h′ is given by f′ g + f g′. Let Δh = h − h, note that although x0 is fixed, Δh depends on the value of Δx, which is thought of as being small. The function h is differentiable at x0 if the limit lim Δ x →0 Δ h Δ x exists, as with Δh, let Δf = f − f and Δg = g − g which, like Δh, also depends on Δx. Then f = f + Δf and g = g + Δg, using the basic properties of limits and the definition of the derivative, we can tackle this term-by term. First, lim Δ x →0 = f ′ g, similarly, lim Δ x →0 = f g ′. The third term, corresponding to the small rectangle, winds up being negligible because Δf Δg vanishes to second order. Then, f g − f g = − f g = f ′ g h + f g ′ h + O Taking the limit for small h gives the result, Let f = uv and suppose u and v are positive functions of x
12.
Zonal and meridional
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The terms zonal and meridional are used to describe directions on a globe. Zonal means along a circle or in the west–east direction. These terms are used in the atmospheric and earth sciences to describe global phenomena, such as meridional wind flow. Meridional is also used to describe the close to the chain orientation in a polymer fiber. For vector fields, the component is denoted as u. The word comes from Latin meri dies, meaning the position of the Sun at that time. As the original Latin territory was in the Northern Hemisphere, this is used with that sense in some Romance languages such as Portuguese, Spanish, French. Meridional flow Zonal flow Zonal and poloidal Zonal flow Meridione
13.
Acceleration
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Acceleration, in physics, is the rate of change of velocity of an object with respect to time. An objects acceleration is the net result of any and all forces acting on the object, the SI unit for acceleration is metre per second squared. Accelerations are vector quantities and add according to the parallelogram law, as a vector, the calculated net force is equal to the product of the objects mass and its acceleration. For example, when a car starts from a standstill and travels in a line at increasing speeds. If the car turns, there is an acceleration toward the new direction, in this example, we can call the forward acceleration of the car a linear acceleration, which passengers in the car might experience as a force pushing them back into their seats. When changing direction, we call this non-linear acceleration, which passengers might experience as a sideways force. If the speed of the car decreases, this is an acceleration in the direction from the direction of the vehicle. Passengers may experience deceleration as a force lifting them forwards, mathematically, there is no separate formula for deceleration, both are changes in velocity. Each of these accelerations might be felt by passengers until their velocity matches that of the car, an objects average acceleration over a period of time is its change in velocity divided by the duration of the period. Mathematically, a ¯ = Δ v Δ t, instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time. The SI unit of acceleration is the metre per second squared, or metre per second per second, as the velocity in metres per second changes by the acceleration value, every second. An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, in this case it is said to be undergoing centripetal acceleration. Proper acceleration, the acceleration of a relative to a free-fall condition, is measured by an instrument called an accelerometer. As speeds approach the speed of light, relativistic effects become increasingly large and these components are called the tangential acceleration and the normal or radial acceleration. Geometrical analysis of space curves, which explains tangent, normal and binormal, is described by the Frenet–Serret formulas. Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an amount in every equal time period. A frequently cited example of uniform acceleration is that of an object in free fall in a gravitational field. The acceleration of a body in the absence of resistances to motion is dependent only on the gravitational field strength g
14.
Gravity
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Gravity, or gravitation, is a natural phenomenon by which all things with mass are brought toward one another, including planets, stars and galaxies. Since energy and mass are equivalent, all forms of energy, including light, on Earth, gravity gives weight to physical objects and causes the ocean tides. Gravity has a range, although its effects become increasingly weaker on farther objects. The most extreme example of this curvature of spacetime is a hole, from which nothing can escape once past its event horizon. More gravity results in time dilation, where time lapses more slowly at a lower gravitational potential. Gravity is the weakest of the four fundamental interactions of nature, the gravitational attraction is approximately 1038 times weaker than the strong force,1036 times weaker than the electromagnetic force and 1029 times weaker than the weak force. As a consequence, gravity has an influence on the behavior of subatomic particles. On the other hand, gravity is the dominant interaction at the macroscopic scale, for this reason, in part, pursuit of a theory of everything, the merging of the general theory of relativity and quantum mechanics into quantum gravity, has become an area of research. While the modern European thinkers are credited with development of gravitational theory, some of the earliest descriptions came from early mathematician-astronomers, such as Aryabhata, who had identified the force of gravity to explain why objects do not fall out when the Earth rotates. Later, the works of Brahmagupta referred to the presence of force, described it as an attractive force. Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th and this was a major departure from Aristotles belief that heavier objects have a higher gravitational acceleration. Galileo postulated air resistance as the reason that objects with less mass may fall slower in an atmosphere, galileos work set the stage for the formulation of Newtons theory of gravity. In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. Newtons theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted for by the actions of the other planets. Calculations by both John Couch Adams and Urbain Le Verrier predicted the position of the planet. A discrepancy in Mercurys orbit pointed out flaws in Newtons theory, the issue was resolved in 1915 by Albert Einsteins new theory of general relativity, which accounted for the small discrepancy in Mercurys orbit. The simplest way to test the equivalence principle is to drop two objects of different masses or compositions in a vacuum and see whether they hit the ground at the same time. Such experiments demonstrate that all objects fall at the rate when other forces are negligible
15.
Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
16.
Drag (physics)
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In fluid dynamics, drag is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist between two layers or a fluid and a solid surface. Unlike other resistive forces, such as dry friction, which are independent of velocity. Drag force is proportional to the velocity for a laminar flow, even though the ultimate cause of a drag is viscous friction, the turbulent drag is independent of viscosity. Drag forces always decrease fluid velocity relative to the object in the fluids path. In the case of viscous drag of fluid in a pipe, in physics of sports, the drag force is necessary to explain the performance of runners, particularly of sprinters. Types of drag are generally divided into the categories, parasitic drag, consisting of form drag, skin friction, interference drag, lift-induced drag. The phrase parasitic drag is used in aerodynamics, since for lifting wings drag it is in general small compared to lift. For flow around bluff bodies, form and interference drags often dominate, further, lift-induced drag is only relevant when wings or a lifting body are present, and is therefore usually discussed either in aviation or in the design of semi-planing or planing hulls. Wave drag occurs either when an object is moving through a fluid at or near the speed of sound or when a solid object is moving along a fluid boundary. Drag depends on the properties of the fluid and on the size, shape, at low R e, C D is asymptotically proportional to R e −1, which means that the drag is linearly proportional to the speed. At high R e, C D is more or less constant, the graph to the right shows how C D varies with R e for the case of a sphere. As mentioned, the equation with a constant drag coefficient gives the force experienced by an object moving through a fluid at relatively large velocity. This is also called quadratic drag, the equation is attributed to Lord Rayleigh, who originally used L2 in place of A. Sometimes a body is a composite of different parts, each with a different reference areas, in the case of a wing the reference areas are the same and the drag force is in the same ratio to the lift force as the ratio of drag coefficient to lift coefficient. Therefore, the reference for a wing is often the area rather than the frontal area. For an object with a surface, and non-fixed separation points—like a sphere or circular cylinder—the drag coefficient may vary with Reynolds number Re. For an object with well-defined fixed separation points, like a disk with its plane normal to the flow direction
17.
Kinematic viscosity
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The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. For liquids, it corresponds to the concept of thickness, for example. Viscosity is a property of the fluid which opposes the motion between the two surfaces of the fluid in a fluid that are moving at different velocities. For a given velocity pattern, the stress required is proportional to the fluids viscosity, a fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at low temperatures in superfluids. Otherwise, all fluids have positive viscosity, and are said to be viscous or viscid. A fluid with a high viscosity, such as pitch. The word viscosity is derived from the Latin viscum, meaning mistletoe, the dynamic viscosity of a fluid expresses its resistance to shearing flows, where adjacent layers move parallel to each other with different speeds. It can be defined through the situation known as a Couette flow. This fluid has to be homogeneous in the layer and at different shear stresses, if the speed of the top plate is small enough, the fluid particles will move parallel to it, and their speed will vary linearly from zero at the bottom to u at the top. Each layer of fluid will move faster than the one just below it, in particular, the fluid will apply on the top plate a force in the direction opposite to its motion, and an equal but opposite one to the bottom plate. An external force is required in order to keep the top plate moving at constant speed. The magnitude F of this force is found to be proportional to the u and the area A of each plate. The proportionality factor μ in this formula is the viscosity of the fluid, the ratio u/y is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction perpendicular to the plates. Isaac Newton expressed the forces by the differential equation τ = μ ∂ u ∂ y, where τ = F/A. This formula assumes that the flow is moving along parallel lines and this equation can be used where the velocity does not vary linearly with y, such as in fluid flowing through a pipe. Use of the Greek letter mu for the dynamic viscosity is common among mechanical and chemical engineers. However, the Greek letter eta is used by chemists, physicists
18.
Rotational symmetry
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Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
19.
Surface runoff
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Surface runoff is the flow of water that occurs when excess stormwater, meltwater, or other sources flows over the Earths surface. Surface runoff is a component of the water cycle. It is the agent in soil erosion by water. Runoff that occurs on the surface before reaching a channel is also called a nonpoint source. If a nonpoint source contains man-made contaminants, or natural forms of pollution the runoff is called nonpoint source pollution, a land area which produces runoff that drains to a common point is called a drainage basin. When runoff flows along the ground, it can pick up soil contaminants including petroleum, pesticides, surface runoff can be generated either by rainfall, snowfall or by the melting of snow, or glaciers. Snow and glacier melt occur only in areas cold enough for these to form permanently, typically snowmelt will peak in the spring and glacier melt in the summer, leading to pronounced flow maxima in rivers affected by them. The determining factor of the rate of melting of snow or glaciers is both air temperature and the duration of sunlight, in high mountain regions, streams frequently rise on sunny days and fall on cloudy ones for this reason. In areas where there is no snow, runoff will come from rainfall, however, not all rainfall will produce runoff because storage from soils can absorb light showers. This occurs when the rate of rainfall on a surface exceeds the rate at which water can infiltrate the ground and this is called flooding excess overland flow, Hortonian overland flow, or unsaturated overland flow. This more commonly occurs in arid and semi-arid regions, where rainfall intensities are high and this occurs largely in city areas where pavements prevent water from flooding. When the soil is saturated and the depression storage filled, and rain continues to fall, the level of antecedent soil moisture is one factor affecting the time until soil becomes saturated. This runoff is called saturation excess overland flow or saturated overland flow, soil retains a degree of moisture after a rainfall. This residual water moisture affects the soils infiltration capacity, during the next rainfall event, the infiltration capacity will cause the soil to be saturated at a different rate. The higher the level of antecedent soil moisture, the more quickly the soil becomes saturated, once the soil is saturated, runoff occurs. After water infiltrates the soil on a portion of a hill, the water may flow laterally through the soil. This is called subsurface return flow or throughflow, any remaining surface water eventually flows into a receiving water body such as a river, lake, estuary or ocean. Urbanization increases surface runoff by creating more impervious surfaces such as pavement and it is instead forced directly into streams or storm water runoff drains, where erosion and siltation can be major problems, even when flooding is not
20.
Cross section (geometry)
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In geometry and science, a cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections, conic sections – circles, ellipses, parabolas, and hyperbolas – are formed by cross-sections of a cone at various different angles, as seen in the diagram at left. Any planar cross-section passing through the center of an ellipsoid forms an ellipse on its surface, a cross-section of a cylinder is a circle if the cross-section is parallel to the cylinders base, or an ellipse with non-zero eccentricity if it is neither parallel nor perpendicular to the base. If the cross-section is perpendicular to the base it consists of two line segments unless it is just tangent to the cylinder, in which case it is a single line segment. A cross section of a polyhedron is a polygon, if instead the cross section is taken for a fixed value of the density, the result is an iso-density contour. For the normal distribution, these contours are ellipses, a cross section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown at left. In economics, a function f specifies the output that can be produced by various quantities x and y of inputs, typically labor. The production function of a firm or a society can be plotted in three-dimensional space, also in economics, a cardinal or ordinal utility function u gives the degree of satisfaction of a consumer obtained by consuming quantities w and v of two goods. Cross sections are used in anatomy to illustrate the inner structure of an organ. A cross section of a trunk, as shown at left, reveals growth rings that can be used to find the age of the tree. Cavalieris principle states that solids with corresponding sections of equal areas have equal volumes. The cross-sectional area of an object when viewed from an angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has A ′ = π r 2 when viewed along its central axis, a sphere of radius r has A ′ = π r 2 when viewed from any angle. For a convex body, each ray through the object from the viewers perspective crosses just two surfaces, descriptive geometry Exploded view drawing Graphical projection Plans
21.
Computer simulation
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Computer simulations reproduce the behavior of a system using a model. Simulation of a system is represented as the running of the systems model and it can be used to explore and gain new insights into new technology and to estimate the performance of systems too complex for analytical solutions. The scale of events being simulated by computer simulations has far exceeded anything possible using traditional paper-and-pencil mathematical modeling, other examples include a 1-billion-atom model of material deformation, a 2. Because of the computational cost of simulation, computer experiments are used to perform such as uncertainty quantification. A computer model is the algorithms and equations used to capture the behavior of the system being modeled, by contrast, computer simulation is the actual running of the program that contains these equations or algorithms. Simulation, therefore, is the process of running a model, thus one would not build a simulation, instead, one would build a model, and then either run the model or equivalently run a simulation. It was a simulation of 12 hard spheres using a Monte Carlo algorithm, Computer simulation is often used as an adjunct to, or substitute for, modeling systems for which simple closed form analytic solutions are not possible. The external data requirements of simulations and models vary widely, for some, the input might be just a few numbers, while others might require terabytes of information. Because of this variety, and because diverse simulation systems have common elements. Systems that accept data from external sources must be careful in knowing what they are receiving. While it is easy for computers to read in values from text or binary files, often they are expressed as error bars, a minimum and maximum deviation from the value range within which the true value lie. Even small errors in the data can accumulate into substantial error later in the simulation. While all computer analysis is subject to the GIGO restriction, this is true of digital simulation. Indeed, observation of this inherent, cumulative error in digital systems was the main catalyst for the development of chaos theory, another way of categorizing models is to look at the underlying data structures. For time-stepped simulations, there are two classes, Simulations which store their data in regular grids and require only next-neighbor access are called stencil codes. Many CFD applications belong to this category, if the underlying graph is not a regular grid, the model may belong to the meshfree method class. Equations define the relationships between elements of the system and attempt to find a state in which the system is in equilibrium. Such models are used in simulating physical systems, as a simpler modeling case before dynamic simulation is attempted
22.
HEC-RAS
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HEC-RAS is a computer program that models the hydraulics of water flow through natural rivers and other channels. The release of Version 5.0 introduced two-dimensional modeling of flow as well as sediment transfer modeling capabilities, the Hydrologic Engineering Center in Davis, California developed the River Analysis System to aid hydraulic engineers in channel flow analysis and floodplain determination. It includes numerous data entry capabilities, hydraulic components, data storage and management capabilities. The basic computational procedure of HEC-RAS for steady flow is based on the solution of the energy equation. Energy losses are evaluated by friction and contraction / expansion, the momentum equation may be used in situations where the water surface profile is rapidly varied. These situations include hydraulic jumps, hydraulics of bridges, and evaluating profiles at river confluences, for unsteady flow, HEC-RAS solves the full, dynamic, 1-D Saint Venant Equation using an implicit, finite difference method. The unsteady flow equation solver was adapted from Dr. Robert L. Barkau’s UNET package, HEC-RAS is equipped to model a network of channels, a dendritic system or a single river reach. Certain simplifications must be made in order to some complex flow situations using the HEC-RAS one-dimensional approach. It is capable of modeling subcritical, supercritical, and mixed flow regime flow along with the effects of bridges, culverts, weirs, HEC-RAS is a computer program for modeling water flowing through systems of open channels and computing water surface profiles. HEC-RAS finds particular commercial application in floodplain management and flood insurance studies to evaluate floodway encroachments, some of the additional uses are, bridge and culvert design and analysis, levee studies, and channel modification studies. It can be used for dam breach analysis, though other modeling methods are more widely accepted for this purpose. HEC-RAS has merits, notably its support by the US Army Corps of Engineers, the enhancements in progress. It is in the domain and peer-reviewed, and available to download free of charge from HECs web site. Various private companies are registered as vendors and offer consulting services. Some also distribute the software in countries that are not permitted to access US Army web sites, however, the direct download from HEC includes extensive documentation, and scientists and engineers versed in hydraulic analysis should have little difficulty utilizing the software. Users may find numerical instability problems during unsteady analyses, especially in steep and/or highly dynamic rivers and it is often possible to use HEC-RAS to overcome instability issues on river problems. A is a software for canals and river engineering works design. Seamless integration in CAD environment makes it easy to learn
23.
Partial differential equation
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In mathematics, a partial differential equation is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs, just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations, Partial differential equations are equations that involve rates of change with respect to continuous variables. The dynamics for the body take place in a finite-dimensional configuration space. This distinction usually makes PDEs much harder to solve ordinary differential equations. Classic domains where PDEs are used include acoustics, fluid dynamics, electrodynamics, a partial differential equation for the function u is an equation of the form f =0. If f is a function of u and its derivatives. Common examples of linear PDEs include the equation, the wave equation, Laplaces equation, Helmholtz equation, Klein–Gordon equation. A relatively simple PDE is ∂ u ∂ x =0 and this relation implies that the function u is independent of x. However, the equation gives no information on the dependence on the variable y. Hence the general solution of equation is u = f. The analogous ordinary differential equation is d u d x =0, which has the solution u = c and these two examples illustrate that general solutions of ordinary differential equations involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A solution of a PDE is generally not unique, additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the example above, the function f can be determined if u is specified on the line x =0. Even if the solution of a differential equation exists and is unique. The mathematical study of questions is usually in the more powerful context of weak solutions. The derivative of u with respect to y approaches 0 uniformly in x as n increases and this solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y
24.
Area
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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
25.
Shear stress
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A shear stress, often denoted τ, is defined as the component of stress coplanar with a material cross section. Shear stress arises from the vector component parallel to the cross section. Normal stress, on the hand, arises from the force vector component perpendicular to the material cross section on which it acts. The formula to calculate average shear stress is force per unit area, τ = F A, where, τ = the shear stress, F = the force applied, A = the cross-sectional area of material with area parallel to the applied force vector. Pure shear stress is related to shear strain, denoted γ, by the following equation, τ = γ G where G is the shear modulus of the isotropic material. Here E is Youngs modulus and ν is Poissons ratio, beam shear is defined as the internal shear stress of a beam caused by the shear force applied to the beam. The beam shear formula is known as Zhuravskii shear stress formula after Dmitrii Ivanovich Zhuravskii who derived it in 1855. Shear stresses within a structure may be calculated by idealizing the cross-section of the structure into a set of stringers. Dividing the shear flow by the thickness of a portion of the semi-monocoque structure yields the shear stress. Any real fluids moving along solid boundary will incur a shear stress on that boundary, the no-slip condition dictates that the speed of the fluid at the boundary is zero, but at some height from the boundary the flow speed must equal that of the fluid. The region between two points is aptly named the boundary layer. For all Newtonian fluids in laminar flow the shear stress is proportional to the rate in the fluid where the viscosity is the constant of proportionality. However, for non-Newtonian fluids, this is no longer the case as for these fluids the viscosity is not constant, the shear stress is imparted onto the boundary as a result of this loss of velocity. Specifically, the shear stress is defined as, τ w ≡ τ = μ ∂ u ∂ y | y =0. For an isotropic Newtonian flow it is a scalar, while for anisotropic Newtonian flows it can be a second-order tensor too. On the other hand, given a shear stress as function of the flow velocity, the constant one finds in this case is the dynamic viscosity of the flow. On the other hand, a flow in which the viscosity were and this nonnewtonian flow is isotropic, so the viscosity is simply a scalar, μ =1 u. This relationship can be exploited to measure the shear stress
26.
Wetted perimeter
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The wetted perimeter is the perimeter of the cross sectional area that is wet. Engineers commonly cite the cross sectional area of a river, the wetted perimeter can be defined mathematically as P = ∑ i =0 ∞ l i where li is the length of each surface in contact with the aqueous body. In open channel flow, the perimeter is defined as the surface of the channel bottom. Friction losses typically increase with an increasing wetted perimeter, resulting in a decrease in heat, in a practical experiment, one is able to measure the wetted perimeter with a tape measure weighted down to the river bed to get a more accurate measurement. When a channel is wider than it is deep, the wetted perimeter approximates the channel width. Hydrological transport model Manning formula Hydraulic radius
27.
Gravity of Earth
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The gravity of Earth, which is denoted by g, refers to the acceleration that is imparted to objects due to the distribution of mass within the Earth. In SI units this acceleration is measured in metres per second squared or equivalently in newtons per kilogram and this quantity is sometimes referred to informally as little g. The precise strength of Earths gravity varies depending on location, the nominal average value at the Earths surface, known as standard gravity is, by definition,9.80665 m/s2. This quantity is denoted variously as gn, ge, g0, gee, the weight of an object on the Earths surface is the downwards force on that object, given by Newtons second law of motion, or F = ma. Gravitational acceleration contributes to the acceleration, but other factors, such as the rotation of the Earth, also contribute. The Earth is not spherically symmetric, but is slightly flatter at the poles while bulging at the Equator, there are consequently slight deviations in both the magnitude and direction of gravity across its surface. The net force as measured by a scale and plumb bob is called effective gravity or apparent gravity, effective gravity includes other factors that affect the net force. These factors vary and include such as centrifugal force at the surface from the Earths rotation. Effective gravity on the Earths surface varies by around 0. 7%, in large cities, it ranges from 9.766 in Kuala Lumpur, Mexico City, and Singapore to 9.825 in Oslo and Helsinki. The surface of the Earth is rotating, so it is not a frame of reference. At latitudes nearer the Equator, the centrifugal force produced by Earths rotation is larger than at polar latitudes. This counteracts the Earths gravity to a small degree – up to a maximum of 0. 3% at the Equator –, the same two factors influence the direction of the effective gravity. Gravity decreases with altitude as one rises above the Earths surface because greater altitude means greater distance from the Earths centre, all other things being equal, an increase in altitude from sea level to 9,000 metres causes a weight decrease of about 0. 29%. It is a misconception that astronauts in orbit are weightless because they have flown high enough to escape the Earths gravity. In fact, at an altitude of 400 kilometres, equivalent to an orbit of the Space Shuttle. Weightlessness actually occurs because orbiting objects are in free-fall, the effect of ground elevation depends on the density of the ground. A person flying at 30000 ft above sea level over mountains will feel more gravity than someone at the same elevation, however, a person standing on the earths surface feels less gravity when the elevation is higher. The following formula approximates the Earths gravity variation with altitude, g h = g 02 Where gh is the acceleration at height h above sea level
28.
Geodetic datum
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A geodetic datum or geodetic system is a coordinate system, and a set of reference points, used to locate places on the Earth. An approximate definition of sea level is the datum WGS84, other datums are defined for other areas or at other times, ED50 was defined in 1950 over Europe and differs from WGS84 by a few hundred meters depending on where in Europe you look. Mars has no oceans and so no sea level, but at least two martian datums have been used to locate places there. Datums are used in geodesy, navigation, and surveying by cartographers, each starts with an ellipsoid, and then defines latitude, longitude and altitude coordinates. One or more locations on the Earths surface are chosen as anchor base-points, the difference in co-ordinates between datums is commonly referred to as datum shift. The datum shift between two particular datums can vary from one place to another within one country or region, the North Pole, South Pole and Equator will be in different positions on different datums, so True North will be slightly different. Different datums use different interpolations for the shape and size of the Earth. Because the Earth is an ellipsoid, localised datums can give a more accurate representation of the area of coverage than WGS84. OSGB36, for example, is an approximation to the geoid covering the British Isles than the global WGS84 ellipsoid. However, as the benefits of a global system outweigh the greater accuracy, horizontal datums are used for describing a point on the Earths surface, in latitude and longitude or another coordinate system. Vertical datums measure elevations or depths, in surveying and geodesy, a datum is a reference system or an approximation of the Earths surface against which positional measurements are made for computing locations. Horizontal datums are used for describing a point on the Earths surface, vertical datums are used to measure elevations or underwater depths. The horizontal datum is the used to measure positions on the Earth. A specific point on the Earth can have different coordinates. There are hundreds of local horizontal datums around the world, usually referenced to some convenient local reference point, contemporary datums, based on increasingly accurate measurements of the shape of the Earth, are intended to cover larger areas. The WGS84 datum, which is almost identical to the NAD83 datum used in North America, a vertical datum is used as a reference point for elevations of surfaces and features on the Earth including terrain, bathymetry, water levels, and man-made structures. Vertical datums are either, tidal, based on sea levels, gravimetric, based on a geoid, or geodetic, for the purpose of measuring the height of objects on land, the usual datum used is mean sea level. This is a datum which is described as the arithmetic mean of the hourly water elevation taken over a specific 19 years cycle
29.
Continuity equation
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A continuity equation in physics is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, continuity equations are a stronger, local form of conservation laws. For example, a version of the law of conservation of energy states that energy can neither be created nor destroyed—i. e. The total amount of energy is fixed and this statement does not immediately rule out the possibility that energy could disappear from a field in Canada while simultaneously appearing in a room in Indonesia. A stronger statement is that energy is conserved, Energy can neither be created nor destroyed. A continuity equation is the way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of charge at any point can only change by the amount of electric current flowing into or out of that point. In an everyday example, there is a continuity equation for the number of people alive, it has a term to account for people being born. Any continuity equation can be expressed in a form, which applies to any finite region. Continuity equations underlie more specific transport equations such as the equation, Boltzmann transport equation. Before we can write down the continuity equation, we must first define flux, the continuity equation is useful when there is some quantity q which can flow or move, such as mass, energy, electric charge, momentum, number of molecules, etc. Let ρ be the density of this property, i. e. the amount of q per unit volume. The way that this quantity q is flowing is described by its flux, the flux of q is a vector field, which we denote as j. Here are some examples and properties of flux, The dimension of flux is amount of q flowing per unit time, outside the pipe, where there is no water, the flux is zero. In a well-known example, the flux of electric charge is the current density. In a simple example, V could be a building, and q could be the number of people in the building, the surface S would consist of the walls, doors, roof, and foundation of the building. Terms that generate or remove q are referred to as a sources and this general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation, mathematically it is an automatic consequence of Maxwells equations, although charge conservation is more fundamental than Maxwells equations