In statistical mechanics, Maxwell–Boltzmann statistics describes the average distribution of non-interacting material particles over various energy states in thermal equilibrium, is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible. The expected number of particles with energy ε i for Maxwell–Boltzmann statistics is ⟨ N i ⟩ N = g i e / k T = 1 Z g i e − ε i / k T, where: ε i is the i-th energy level, ⟨ N i ⟩ is the average number of particles in the set of states with energy ε i, g i is the degeneracy of energy level i, that is, the number of states with energy ε i which may be distinguished from each other by some other means, μ is the chemical potential, k is Boltzmann's constant, T is absolute temperature, N is the total number of particles: N = ∑ i N i,Z is the partition function: Z = ∑ i g i e − ε i / k T, e is the exponential function. Equivalently, the number of particles is sometimes expressed as ⟨ N i ⟩ N = 1 e / k T = 1 Z e − ε i / k T, where the index i now specifies a particular state rather than the set of all states with energy ε i, Z = ∑ i e − ε i / k T. Maxwell–Boltzmann statistics may be used to derive the Maxwell–Boltzmann distribution.
However, they apply to other situations as well. Maxwell–Boltzmann statistics can be used to extend that distribution to particles with a different energy–momentum relation, such as relativistic particles. In addition, hypothetical situations can be considered, such as particles in a box with different numbers of dimensions. Maxwell–Boltzmann statistics are described as the statistics of "distinguishable" classical particles. In other words, the configuration of particle A in state 1 and particle B in state 2 is different from the case in which particle B is in state 1 and particle A is in state 2; this assumption leads to the proper statistics of particles in the energy states, but yields non-physical results for the entropy, as embodied in the Gibbs paradox. At the same time, there are no real particles which have the characteristics required by Maxwell–Boltzmann statistics. Indeed, the Gibbs paradox is resolved if we treat all particles of a certain type as indistinguishable, this assumption can be justified in the context of quantum mechanics.
Once this assumption is made, the particle statistics change. Quantum particles fermions. Both of these quantum statistics approach the Maxwell–Boltzmann statistics in the limit of high temperature and low particle density, without the need for any ad hoc assumptions; the Fermi–Dirac and Bose–Einstein statistics give the energy level occupation as: ⟨ N i ⟩ = g i e / k T ± 1. It can be seen that the condition under which the Maxwell–Boltzmann statistics are valid is when e / k T ≫ 1, where ε m
Einstein coefficients are mathematical quantities which are a measure of the probability of absorption or emission of light by an atom or molecule. The Einstein A coefficient is related to the rate of spontaneous emission of light and the Einstein B coefficients are related to the absorption and stimulated emission of light. In physics, one thinks of a spectral line from two viewpoints. An emission line is formed when an atom or molecule makes a transition from a particular discrete energy level E2 of an atom, to a lower energy level E1, emitting a photon of a particular energy and wavelength. A spectrum of many such photons will show an emission spike at the wavelength associated with these photons. An absorption line is formed when an atom or molecule makes a transition from a lower, E1, to a higher discrete energy state, E2, with a photon being absorbed in the process; these absorbed photons come from background continuum radiation and a spectrum will show a drop in the continuum radiation at the wavelength associated with the absorbed photons.
The two states must be bound states in which the electron is bound to the atom or molecule, so the transition is sometimes referred to as a "bound–bound" transition, as opposed to a transition in which the electron is ejected out of the atom into a continuum state, leaving an ionized atom, generating continuum radiation. A photon with an energy equal to the difference E2 - E1 between the energy levels is released or absorbed in the process; the frequency ν at which the spectral line occurs is related to the photon energy by Bohr's frequency condition E2 - E1 = hν where h denotes Planck's constant. An atomic spectral line refers to emission and absorption events in a gas in which n 2 is the density of atoms in the upper energy state for the line, n 1 is the density of atoms in the lower energy state for the line; the emission of atomic line radiation at frequency ν may be described by an emission coefficient ϵ with units of energy/time/volume/solid angle. Ε dt dV dΩ is the energy emitted by a volume element d V in time d t into solid angle d Ω.
For atomic line radiation: ϵ = h ν 4 π n 2 A 21 where A 21 is the Einstein coefficient for spontaneous emission, fixed by the intrinsic properties of the relevant atom for the two relevant energy levels. The absorption of atomic line radiation may be described by an absorption coefficient κ with units of 1/length; the expression κ' dx gives the fraction of intensity absorbed for a light beam at frequency ν while traveling distance dx. The absorption coefficient is given by: κ ′ = h ν 4 π where B 12 and B 21 are the Einstein coefficients for photo absorption and induced emission respectively. Like the coefficient A 21, these are fixed by the intrinsic properties of the relevant atom for the two relevant energy levels. For thermodynamics and for the application of Kirchhoff's law, it is necessary that the total absorption be expressed as the algebraic sum of two components, described by B 12 and B 21, which may be regarded as positive and negative absorption, which are the direct photon absorption, what is called stimulated or induced emission.
The above equations have ignored the influence of the spectroscopic line shape. To be accurate, the above equations need to be multiplied by the spectral line shape, in which case the units will change to include a 1/Hz term. For conditions of thermodynamic equilibrium, together the number densities n 2 and n 1, the Einstein coefficients, the spectral energy density provide sufficient information to determine the absorption and emission rates; the number densities n 2 and n 1 are set by the physical state of the gas in which the spectral line occurs, including the local spectral radiance. When that state is either one of strict thermodynamic equilibrium, or one of so-called "local thermodynamic equilibrium" the distribution of atomic states of excitation determines the rates of atomic emissions and absorptions to be such that Kirchhoff's law of equality of radiative absorptivity and emissivity holds. In strict thermodynamic equilibrium, the radiation field is said to be black-body radiation and is described by Planck's law.
For local thermodynamic equilibrium, the radiation fi
William Sutherland (physicist)
William Sutherland was a Scottish-born Australian theoretical physicist, physical chemist and writer for The Age newspaper. Sutherland was born in Glasgow, son of George Sutherland, a woodcarver, his wife Jane, née Smith. William had George Sutherland and Jane Sutherland; the family emigrated to Australia in 1864, staying in Sydney for six years and moving to Melbourne in 1870. Sutherland graduated from Wesley College; the headmaster was Martin Howy Irving, the second professor of classics at the University of Melbourne, but the influence of the second master, H. M. Andrew, afterwards professor of natural philosophy at the same university, was of more importance to Sutherland. From Wesley Sutherland enrolled at the University of Melbourne in February 1876, graduating B. A. in 1879 with first-class final honours and the scholarship in natural science, third-class honours in engineering. Sutherland was nominated by the Melbourne University council for the Gilchrist scholarship at University College, London, in England, awarded to him and he left for England in July 1879.
Entering as a science student at University College London, Sutherland came under the influence of Professor Carey Foster, in the final examination for the BSc degree took first place and first class honours in experimental physics and the clothworkers scholarship of £50 for two years. Sutherland had not enjoyed his time in England and arrived back in Melbourne in February 1882. Sutherland's home life meant a lot to him, it was a home of affection and culture, every member of it excelled in either literature, music or art. In July 1882 Sutherland was offered the position of superintendent of the School of Mines, but it was too far from his home and the public library, the offer was declined. For many years he earned just enough to pay his way by acting as an examiner and contributing articles to the press. In 1884 he applied without success for the chair of chemistry at the University of Adelaide, in 1888 when the professor of natural philosophy Henry Martyn Andrew died Sutherland was appointed lecturer at the University of Melbourne until the chair could be filled.
Sutherland had applied for this position through the Victorian agent-general in London, but the application was mis-filed and was not considered. Professor Thomas Ranken Lyle was appointed and in 1897, when he was away on leave, Sutherland was again made lecturer. Sutherland had begun contributing to the Philosophical Magazine in 1885, on an average about two articles a year front his pen appeared in it for the next 25 years. For the last 10 years of his life he was a regular contributor and leader writer on the Melbourne Age on scientific subjects. Sutherland declined an offer of an appointment on the staff of the paper. Sutherland wrote on such topics as the surface tension of liquids, the rigidity of solids, the properties of solutions, the origin of spectra and the source of the Earth's magnetic field. Sutherland devoted most of his time to scientific research. A list of 69 of his contributions to scientific magazines appears in W. A. Osborne's, William Sutherland a Biography. Sutherland died in his sleep on 5 October 1911 from a ruptured heart.
Sutherland was a well-built man of under medium height quiet in manner. He could have been a painter if he had been able to give the time. One of the earlier papers to bring Sutherland into notice was on the viscosity of gases which appeared in the Philosophical Magazine in December 1893. Other important papers dealt with the constitution of water, the viscosity of water, molecular attractions and ionization, ionic velocities and atomic sizes; the ordinary reader may refer to a discussion of his scientific work in chapter VI of Osborne's biography of Sutherland, but the full value of it could only be computed by a physicist willing to collate his papers with the state of knowledge at the time each was written. It was well known and valued in England and America. Professor T. R. Lyle said at the time of Sutherland's death that he was "the greatest authority living in molecular physics". Modest and selfless, Sutherland was content to add to the sum of human knowledge and to hope that another person would carry the work further.
Sutherland never married. Sutherland wrote an equation describing Brownian motion and diffusion, published in a 1904 paper, which he presented at a Dunedin ANZAAS conference. Albert Einstein's first published work on the same topic was published in 1905. Essay on William Sutherland by Prof. Roderick Weir Home 2005 http://williamsutherland.wordpress.com/ Sutherland potential equation and description at SklogWiki.org
Albert Einstein was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics. His work is known for its influence on the philosophy of science, he is best known to the general public for his mass–energy equivalence formula E = mc2, dubbed "the world's most famous equation". He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, for his discovery of the law of the photoelectric effect", a pivotal step in the development of quantum theory. Near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field; this led him to develop his special theory of relativity during his time at the Swiss Patent Office in Bern. However, he realized that the principle of relativity could be extended to gravitational fields, he published a paper on general relativity in 1916 with his theory of gravitation.
He continued to deal with problems of statistical mechanics and quantum theory, which led to his explanations of particle theory and the motion of molecules. He investigated the thermal properties of light which laid the foundation of the photon theory of light. In 1917, he applied the general theory of relativity to model the structure of the universe. Except for one year in Prague, Einstein lived in Switzerland between 1895 and 1914, during which time he renounced his German citizenship in 1896 received his academic diploma from the Swiss federal polytechnic school in Zürich in 1900. After being stateless for more than five years, he acquired Swiss citizenship in 1901, which he kept for the rest of his life. In 1905, he was awarded a PhD by the University of Zurich; the same year, he published four groundbreaking papers during his renowned annus mirabilis which brought him to the notice of the academic world at the age of 26. Einstein taught theoretical physics at Zurich between 1912 and 1914 before he left for Berlin, where he was elected to the Prussian Academy of Sciences.
In 1933, while Einstein was visiting the United States, Adolf Hitler came to power. Because of his Jewish background, Einstein did not return to Germany, he settled in the United States and became an American citizen in 1940. On the eve of World War II, he endorsed a letter to President Franklin D. Roosevelt alerting him to the potential development of "extremely powerful bombs of a new type" and recommending that the US begin similar research; this led to the Manhattan Project. Einstein supported the Allies, but he denounced the idea of using nuclear fission as a weapon, he signed the Russell–Einstein Manifesto with British philosopher Bertrand Russell, which highlighted the danger of nuclear weapons. He was affiliated with the Institute for Advanced Study in Princeton, New Jersey, until his death in 1955. Einstein published more than 150 non-scientific works, his intellectual achievements and originality have made the word "Einstein" synonymous with "genius". Albert Einstein was born in Ulm, in the Kingdom of Württemberg in the German Empire, on 14 March 1879.
His parents were Hermann Einstein, a salesman and engineer, Pauline Koch. In 1880, the family moved to Munich, where Einstein's father and his uncle Jakob founded Elektrotechnische Fabrik J. Einstein & Cie, a company that manufactured electrical equipment based on direct current; the Einsteins were non-observant Ashkenazi Jews, Albert attended a Catholic elementary school in Munich, from the age of 5, for three years. At the age of 8, he was transferred to the Luitpold Gymnasium, where he received advanced primary and secondary school education until he left the German Empire seven years later. In 1894, Hermann and Jakob's company lost a bid to supply the city of Munich with electrical lighting because they lacked the capital to convert their equipment from the direct current standard to the more efficient alternating current standard; the loss forced the sale of the Munich factory. In search of business, the Einstein family moved to Italy, first to Milan and a few months to Pavia; when the family moved to Pavia, Einstein 15, stayed in Munich to finish his studies at the Luitpold Gymnasium.
His father intended for him to pursue electrical engineering, but Einstein clashed with authorities and resented the school's regimen and teaching method. He wrote that the spirit of learning and creative thought was lost in strict rote learning. At the end of December 1894, he travelled to Italy to join his family in Pavia, convincing the school to let him go by using a doctor's note. During his time in Italy he wrote a short essay with the title "On the Investigation of the State of the Ether in a Magnetic Field". Einstein always excelled at math and physics from a young age, reaching a mathematical level years ahead of his peers; the twelve year old Einstein taught himself algebra and Euclidean geometry over a single summer. Einstein independently discovered his own original proof of the Pythagorean theorem at age 12. A family tutor Max Talmud says that after he had given the 12 year old Einstein a geometry textbook, after a short time " had worked through the whole book, he thereupon devoted himself to higher mathematics...
Soon the flight of his mathematical genius was so high I could not follow." His passion for geometry and algebra led the twelve year old to become convinced that nature could be understood as a "mathematical structure". Einstein started teaching himself calculus at
Classical physics refers to theories of physics that predate modern, more complete, or more applicable theories. If a accepted theory is considered to be modern, its introduction represented a major paradigm shift the previous theories, or new theories based on the older paradigm, will be referred to as belonging to the realm of "classical physics"; as such, the definition of a classical theory depends on context. Classical physical concepts are used when modern theories are unnecessarily complex for a particular situation. Most classical physics refers to pre-1900 physics, while modern physics refers to post-1900 physics which incorporates elements of quantum mechanics and relativity. Classical theory has at least two distinct meanings in physics. In the context of quantum mechanics, classical theory refers to theories of physics that do not use the quantisation paradigm, which includes classical mechanics and relativity. Classical field theories, such as general relativity and classical electromagnetism, are those that do not use quantum mechanics.
In the context of general and special relativity, classical theories are those that obey Galilean relativity. Depending on point of view, among the branches of theory sometimes included in classical physics are variably: Classical mechanics Newton's laws of motion Classical Lagrangian and Hamiltonian formalisms Classical electrodynamics Classical thermodynamics Special relativity and general relativity Classical chaos theory and nonlinear dynamics In contrast to classical physics, "modern physics" is a looser term which may refer to just quantum physics or to 20th and 21st century physics in general. Modern physics includes quantum relativity, when applicable. A physical system can be described by classical physics when it satisfies conditions such that the laws of classical physics are valid. In practice, physical objects ranging from those larger than atoms and molecules, to objects in the macroscopic and astronomical realm, can be well-described with classical mechanics. Beginning at the atomic level and lower, the laws of classical physics break down and do not provide a correct description of nature.
Electromagnetic fields and forces can be described well by classical electrodynamics at length scales and field strengths large enough that quantum mechanical effects are negligible. Unlike quantum physics, classical physics is characterized by the principle of complete determinism, although deterministic interpretations of quantum mechanics do exist. From the point of view of classical physics as being non-relativistic physics, the predictions of general and special relativity are different than those of classical theories concerning the passage of time, the geometry of space, the motion of bodies in free fall, the propagation of light. Traditionally, light was reconciled with classical mechanics by assuming the existence of a stationary medium through which light propagated, the luminiferous aether, shown not to exist. Mathematically, classical physics equations are those. According to the correspondence principle and Ehrenfest's theorem, as a system becomes larger or more massive the classical dynamics tends to emerge, with some exceptions, such as superfluidity.
This is why we can ignore quantum mechanics when dealing with everyday objects and the classical description will suffice. However, one of the most vigorous on-going fields of research in physics is classical-quantum correspondence; this field of research is concerned with the discovery of how the laws of quantum physics give rise to classical physics found at the limit of the large scales of the classical level. Today a computer performs millions of arithmetic operations in seconds to solve a classical differential equation, while Newton would take hours to solve the same equation by manual calculation if he were the discoverer of that particular equation. Computer modeling is essential for relativistic physics. Classic physics is considered the limit of quantum mechanics for large number of particles. On the other hand, classic mechanics is derived from relativistic mechanics. For example, in many formulations from special relativity, a correction factor 2 appears, where v is the velocity of the object and c is the speed of light.
For velocities much smaller than that of light, one can neglect the terms with c2 and higher that appear. These formulas reduce to the standard definitions of Newtonian kinetic energy and momentum; this is as it should be, for special relativity must agree with Newtonian mechanics at low velocities. Computer modeling has to be as real as possible. Classical physics would introduce an error as in the superfluidity case. In order to produce reliable models of the world, we can not use classic physics, it is true that quantum theories consume time and computer resources, the equations of classical physics could be resorted to provide a quick solution, but such a solution would lack reliability. Computer modeling would use only the energy criteria to determine which theory to use: relativity or quantum theory, when attempting to describe the behavior of an object. A physicist would use a classical model to provide an approximation before more exacting models are applied and those calculations proceed.
In a computer model, there is no need to use the speed of the object if classical physics is excluded. Low energy objects would be handled by quantum theory and high energy objects by relativity theory. Glossary of classical physics Semiclassical physics
A semiconductor material has an electrical conductivity value falling between that of a metal, like copper, etc. and an insulator, such as glass. Their resistance decreases as their temperature increases, behaviour opposite to that of a metal, their conducting properties may be altered in useful ways by the deliberate, controlled introduction of impurities into the crystal structure. Where two differently-doped regions exist in the same crystal, a semiconductor junction is created; the behavior of charge carriers which include electrons and electron holes at these junctions is the basis of diodes and all modern electronics. Some examples of semiconductors are silicon and gallium arsenide. After silicon, gallium arsenide is the second most common semiconductor used in laser diodes, solar cells, microwave frequency integrated circuits, others. Silicon is a critical element for fabricating most electronic circuits. Semiconductor devices can display a range of useful properties such as passing current more in one direction than the other, showing variable resistance, sensitivity to light or heat.
Because the electrical properties of a semiconductor material can be modified by doping, or by the application of electrical fields or light, devices made from semiconductors can be used for amplification and energy conversion. The conductivity of silicon is increased by adding a small amount of trivalent atoms; this process is known as doping and resulting semiconductors are known as doped or extrinsic semiconductors. Apart from doping, the conductivity of a semiconductor can be improved by increasing its temperature; this is contrary to the behaviour of a metal in which conductivity decreases with increase in temperature. The modern understanding of the properties of a semiconductor relies on quantum physics to explain the movement of charge carriers in a crystal lattice. Doping increases the number of charge carriers within the crystal; when a doped semiconductor contains free holes it is called "p-type", when it contains free electrons it is known as "n-type". The semiconductor materials used in electronic devices are doped under precise conditions to control the concentration and regions of p- and n-type dopants.
A single semiconductor crystal can have many p- and n-type regions. Although some pure elements and many compounds display semiconductor properties, silicon and compounds of gallium are the most used in electronic devices. Elements near the so-called "metalloid staircase", where the metalloids are located on the periodic table, are used as semiconductors; some of the properties of semiconductor materials were observed throughout the mid 19th and first decades of the 20th century. The first practical application of semiconductors in electronics was the 1904 development of the cat's-whisker detector, a primitive semiconductor diode used in early radio receivers. Developments in quantum physics in turn allowed the development of the transistor in 1947 and the integrated circuit in 1958. Variable electrical conductivity Semiconductors in their natural state are poor conductors because a current requires the flow of electrons, semiconductors have their valence bands filled, preventing the entry flow of new electrons.
There are several developed techniques that allow semiconducting materials to behave like conducting materials, such as doping or gating. These modifications have two outcomes: p-type; these refer to the shortage of electrons, respectively. An unbalanced number of electrons would cause a current to flow through the material. Heterojunctions Heterojunctions occur when two differently doped semiconducting materials are joined together. For example, a configuration could consist of n-doped germanium; this results in an exchange of electrons and holes between the differently doped semiconducting materials. The n-doped germanium would have an excess of electrons, the p-doped germanium would have an excess of holes; the transfer occurs until equilibrium is reached by a process called recombination, which causes the migrating electrons from the n-type to come in contact with the migrating holes from the p-type. A product of this process is charged ions. Excited electrons A difference in electric potential on a semiconducting material would cause it to leave thermal equilibrium and create a non-equilibrium situation.
This introduces electrons and holes to the system, which interact via a process called ambipolar diffusion. Whenever thermal equilibrium is disturbed in a semiconducting material, the number of holes and electrons changes; such disruptions can occur as a result of a temperature difference or photons, which can enter the system and create electrons and holes. The process that creates and annihilates electrons and holes are called generation and recombination. Light emission In certain semiconductors, excited electrons can relax by emitting light instead of producing heat; these semiconductors are used in the construction of light-emitting diodes and fluorescent quantum dots. High thermal conductivitySemiconductors with high thermal conductivity can be used for heat dissipation and improving thermal management of electronics. Thermal energy conversion Semiconductors have large thermoelectric power factors making them useful in thermoelectric generators, as well as high thermoelectric figures of merit making them useful in thermoelectric coolers.
A large number of elements and compounds have semiconducting properties, including: Certain pure elements are found in Group 14 of the p
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