The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity can be conceptualized as quantifying the frictional force that arises between adjacent layers of fluid that are in relative motion. For instance, when a fluid is forced through a tube, it flows more near the tube's axis than near its walls. In such a case, experiments show; this is because a force is required to overcome the friction between the layers of the fluid which are in relative motion: the strength of this force is proportional to the viscosity. A fluid that has no resistance to shear stress is known as an inviscid fluid. Zero viscosity is observed only at low temperatures in superfluids. Otherwise, the second law of thermodynamics requires all fluids to have positive viscosity. A fluid with a high viscosity, such as pitch, may appear to be a solid; the word "viscosity" is derived from the Latin "viscum", meaning mistletoe and a viscous glue made from mistletoe berries.
In materials science and engineering, one is interested in understanding the forces, or stresses, involved in the deformation of a material. For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called elastic stresses. In other materials, stresses are present which can be attributed to the rate of change of the deformation over time; these are called. For instance, in a fluid such as water the stresses which arise from shearing the fluid do not depend on the distance the fluid has been sheared. Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation. Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow. In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed u.
If the speed of the top plate is low enough in steady state the fluid particles move parallel to it, their speed varies from 0 at the bottom to u at the top. Each layer of fluid moves faster than the one just below it, friction between them gives rise to a force resisting their relative motion. In particular, the fluid applies on the top plate a force in the direction opposite to its motion, an equal but opposite force on the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed. In many fluids, the flow velocity is observed to vary linearly from zero at the bottom to u at the top. Moreover, the magnitude F of the force acting on the top plate is found to be proportional to the speed u and the area A of each plate, inversely proportional to their separation y: F = μ A u y; the proportionality factor μ is the viscosity of the fluid, with units of Pa ⋅ s. The ratio u / y is called the rate of shear deformation or shear velocity, is the derivative of the fluid speed in the direction perpendicular to the plates.
If the velocity does not vary linearly with y the appropriate generalization is τ = μ ∂ u ∂ y, where τ = F / A, ∂ u / ∂ y is the local shear velocity. This expression is referred to as Newton's law of viscosity. In shearing flows with planar symmetry, it is what defines μ, it is a special case of the general definition of viscosity, which can be expressed in coordinate-free form. Use of the Greek letter mu for the viscosity is common among mechanical and chemical engineers, as well as physicists. However, the Greek letter eta is used by chemists and the IUPAC; the viscosity μ is sometimes referred to as the shear viscosity. However, at least one author discourages the use of this terminology, noting that μ can appear in nonshearing flows in addition to shearing flows. In general terms, the viscous stresses in a fluid are defined as those resulting from the relative velocity of different fluid particles; as such, the viscous stresses. If the velocity gradients are small to a first approximation the v
The velocity of an object is the rate of change of its position with respect to a frame of reference, is a function of time. Velocity is equivalent to a specification of an object's direction of motion. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity; the scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI as metres per second or as the SI base unit of. For example, "5 metres per second" is a scalar. If there is a change in speed, direction or both the object has a changing velocity and is said to be undergoing an acceleration. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes.
Hence, the car is considered to be undergoing an acceleration. Speed describes only how fast an object is moving, whereas velocity gives both how fast it is and in which direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified; the big difference can be noticed. When something moves in a circular path and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle; this is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Velocity is defined as the rate of change of position with respect to time, which may be referred to as the instantaneous velocity to emphasize the distinction from the average velocity.
In some applications the "average velocity" of an object might be needed, to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v, over some time period Δt. Average velocity can be calculated as: v ¯ = Δ x Δ t; the average velocity is always equal to the average speed of an object. This can be seen by realizing that while distance is always increasing, displacement can increase or decrease in magnitude as well as change direction. In terms of a displacement-time graph, the instantaneous velocity can be thought of as the slope of the tangent line to the curve at any point, the average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity; the average velocity is the same as the velocity averaged over time –, to say, its time-weighted average, which may be calculated as the time integral of the velocity: v ¯ = 1 t 1 − t 0 ∫ t 0 t 1 v d t, where we may identify Δ x = ∫ t 0 t 1 v d t and Δ t = t 1 − t 0.
If we consider v as velocity and x as the displacement vector we can express the velocity of a particle or object, at any particular time t, as the derivative of the position with respect to time: v = lim Δ t → 0 Δ x Δ t = d x d t. From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time is the displacement, x. In calculus terms, the integral of the velocity function v is the displacement function x. In the figure, this corresponds to the yellow area under the curve labeled s. X = ∫ v d t. Since the derivative of the position with respect to time gives the change in position divided by the change in time, velocity is measured in metres per second. Although the concept of an instantaneous velocity might at first seem counter-intuitive, it
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences and engineering disciplines, as well as in the social sciences. A model may help to explain a system and to study the effects of different components, to make predictions about behaviour. Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models; these and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements leads to important advances as better theories are developed.
In the physical sciences, a traditional mathematical model contains most of the following elements: Governing equations Supplementary sub-models Defining equations Constitutive equations Assumptions and constraints Initial and boundary conditions Classical constraints and kinematic equations Mathematical models are composed of relationships and variables. Relationships can be described by operators, such as algebraic operators, differential operators, etc. Variables are abstractions of system parameters of interest. Several classification criteria can be used for mathematical models according to their structure: Linear vs. nonlinear: If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise; the definition of linearity and nonlinearity is dependent on context, linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables.
A differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented by linear equations the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation the model is known as a nonlinear model. Nonlinearity in simple systems, is associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are tied to nonlinearity. Static vs. dynamic: A dynamic model accounts for time-dependent changes in the state of the system, while a static model calculates the system in equilibrium, thus is time-invariant.
Dynamic models are represented by differential equations or difference equations. Explicit vs. implicit: If all of the input parameters of the overall model are known, the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, the corresponding inputs must be solved for by an iterative procedure, such as Newton's method or Broyden's method. In such a case the model is said to be implicit. For example, a jet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties. Discrete vs. continuous: A discrete model treats objects as discrete, such as the particles in a molecular model or the states in a statistical model.
Deterministic vs. probabilistic: A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, variable states are not described by unique values, but rather by probability distributions. Deductive, inductive, or floating: A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them; the floating model rests on neither theory nor observation, but is the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models. Application of catastrophe theory in science has been characterized as a floating model. Mathematical models are of great importance in the natural sciences in physics. Physical theories are invariably expressed using mathematic
In fluid dynamics, a vortex is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, may be observed in smoke rings, whirlpools in the wake of a boat, the winds surrounding a tropical cyclone, tornado or dust devil. Vortices are a major component of turbulent flow; the distribution of velocity, vorticity, as well as the concept of circulation are used to characterize vortices. In most vortices, the fluid flow velocity is greatest next to its axis and decreases in inverse proportion to the distance from the axis. In the absence of external forces, viscous friction within the fluid tends to organize the flow into a collection of irrotational vortices superimposed to larger-scale flows, including larger-scale vortices. Once formed, vortices can move, stretch and interact in complex ways. A moving vortex carries with it some angular and linear momentum and mass. A key concept in the dynamics of vortices is the vorticity, a vector that describes the local rotary motion at a point in the fluid, as would be perceived by an observer that moves along with it.
Conceptually, the vorticity could be observed by placing a tiny rough ball at the point in question, free to move with the fluid, observing how it rotates about its center. The direction of the vorticity vector is defined to be the direction of the axis of rotation of this imaginary ball while its length is twice the ball's angular velocity. Mathematically, the vorticity is defined as the curl of the velocity field of the fluid denoted by ω → and expressed by the vector analysis formula ∇ × u →, where ∇ is the nabla operator and u → is the local flow velocity; the local rotation measured by the vorticity ω → must not be confused with the angular velocity vector of that portion of the fluid with respect to the external environment or to any fixed axis. In a vortex, in particular, ω → may be opposite to the mean angular velocity vector of the fluid relative to the vortex's axis. In theory, the speed u of the particles in a vortex may vary with the distance r from the axis in many ways. There are two important special cases, however: If the fluid rotates like a rigid body – that is, if the angular rotational velocity Ω is uniform, so that u increases proportionally to the distance r from the axis – a tiny ball carried by the flow would rotate about its center as if it were part of that rigid body.
In such a flow, the vorticity is the same everywhere: its direction is parallel to the rotation axis, its magnitude is equal to twice the uniform angular velocity Ω of the fluid around the center of rotation. Ω → =, r → =, u → = Ω → × r → =, ω → = ∇ × u → = = 2 Ω →. If the particle speed u is inversely proportional to the distance r from the axis the imaginary test ball would not rotate over itself. In this case the vorticity ω → is zero at any point not on that axis, the flow is said to be irrotational. Ω → =, r → =, u → = Ω → × r → =, ω → = ∇ × u → = 0. In the absence of external forces, a vortex evolves quickly toward the irrotational flow pattern, where the flow velocity u is inversely proportional to the distance r. Irrotational vortices are called free vortices. For an irrotational vortex, the circulation is zero along any closed contour that does not encl
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university in Cambridge, Massachusetts. Founded in 1861 in response to the increasing industrialization of the United States, MIT adopted a European polytechnic university model and stressed laboratory instruction in applied science and engineering; the Institute is a land-grant, sea-grant, space-grant university, with a campus that extends more than a mile alongside the Charles River. Its influence in the physical sciences and architecture, more in biology, linguistics and social science and art, has made it one of the most prestigious universities in the world. MIT is ranked among the world's top universities; as of March 2019, 93 Nobel laureates, 26 Turing Award winners, 8 Fields Medalists have been affiliated with MIT as alumni, faculty members, or researchers. In addition, 58 National Medal of Science recipients, 29 National Medals of Technology and Innovation recipients, 50 MacArthur Fellows, 73 Marshall Scholars, 45 Rhodes Scholars, 41 astronauts, 16 Chief Scientists of the US Air Force have been affiliated with MIT.
The school has a strong entrepreneurial culture, the aggregated annual revenues of companies founded by MIT alumni would rank as the tenth-largest economy in the world. MIT is a member of the Association of American Universities. In 1859, a proposal was submitted to the Massachusetts General Court to use newly filled lands in Back Bay, Boston for a "Conservatory of Art and Science", but the proposal failed. A charter for the incorporation of the Massachusetts Institute of Technology, proposed by William Barton Rogers, was signed by the governor of Massachusetts on April 10, 1861. Rogers, a professor from the University of Virginia, wanted to establish an institution to address rapid scientific and technological advances, he did not wish to found a professional school, but a combination with elements of both professional and liberal education, proposing that: The true and only practicable object of a polytechnic school is, as I conceive, the teaching, not of the minute details and manipulations of the arts, which can be done only in the workshop, but the inculcation of those scientific principles which form the basis and explanation of them, along with this, a full and methodical review of all their leading processes and operations in connection with physical laws.
The Rogers Plan reflected the German research university model, emphasizing an independent faculty engaged in research, as well as instruction oriented around seminars and laboratories. Two days after MIT was chartered, the first battle of the Civil War broke out. After a long delay through the war years, MIT's first classes were held in the Mercantile Building in Boston in 1865; the new institute was founded as part of the Morrill Land-Grant Colleges Act to fund institutions "to promote the liberal and practical education of the industrial classes" and was a land-grant school. In 1863 under the same act, the Commonwealth of Massachusetts founded the Massachusetts Agricultural College, which developed as the University of Massachusetts Amherst. In 1866, the proceeds from land sales went toward new buildings in the Back Bay. MIT was informally called "Boston Tech"; the institute adopted the European polytechnic university model and emphasized laboratory instruction from an early date. Despite chronic financial problems, the institute saw growth in the last two decades of the 19th century under President Francis Amasa Walker.
Programs in electrical, chemical and sanitary engineering were introduced, new buildings were built, the size of the student body increased to more than one thousand. The curriculum drifted with less focus on theoretical science; the fledgling school still suffered from chronic financial shortages which diverted the attention of the MIT leadership. During these "Boston Tech" years, MIT faculty and alumni rebuffed Harvard University president Charles W. Eliot's repeated attempts to merge MIT with Harvard College's Lawrence Scientific School. There would be at least six attempts to absorb MIT into Harvard. In its cramped Back Bay location, MIT could not afford to expand its overcrowded facilities, driving a desperate search for a new campus and funding; the MIT Corporation approved a formal agreement to merge with Harvard, over the vehement objections of MIT faculty and alumni. However, a 1917 decision by the Massachusetts Supreme Judicial Court put an end to the merger scheme. In 1916, the MIT administration and the MIT charter crossed the Charles River on the ceremonial barge Bucentaur built for the occasion, to signify MIT's move to a spacious new campus consisting of filled land on a mile-long tract along the Cambridge side of the Charles River.
The neoclassical "New Technology" campus was designed by William W. Bosworth and had been funded by anonymous donations from a mysterious "Mr. Smith", starting in 1912. In January 1920, the donor was revealed to be the industrialist George Eastman of Rochester, New York, who had invented methods of film production and processing, founded Eastman Kodak. Between 1912 and 1920, Eastman donated $20 million in cash and Kodak stock to MIT. In the 1930s, President Karl Taylor Compton and Vice-President Vannevar Bush emphasized the importance of pure sciences like physics and chemistry and reduced the vocational practice required in shops and drafting studios; the Compton reforms "renewed confidence in the ability of the Institute to develop leadership in science as well as in engineering". Unlike Ivy League schools, MIT catered more to middle-class families, depended more on tuition than on endow