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Cauchy Momentum Equation
The CAUCHY MOMENTUM EQUATION is a vector partial differential equation put forth by Cauchy
Cauchy
that describes the non-relativistic momentum transport in any continuum . In convective (or Lagrangian) form it is written: D u D t = 1 + g {displaystyle {frac {Dmathbf {u} }{Dt}}={frac {1}{rho }}nabla cdot {boldsymbol {sigma }}+mathbf {g} } where ρ is the density at the point considered in the continuum (for which the continuity equation holds), σ is the stress tensor , and G contains all of the body forces per unit mass (often simply gravitational acceleration ). U is the flow velocity vector field, which depends on time and space
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Vorticity
In continuum mechanics , the VORTICITY is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate ), as would be seen by an observer located at that point and traveling along with the flow. Conceptually, vorticity could be determined by marking the part of continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow. The vorticity vector would be twice the mean angular velocity vector of those particles relative to their center of mass , oriented according to the right-hand rule . This quantity must not be confused with the angular velocity of the particles relative to some other point. More precisely, the vorticity is a pseudovector field ω→, defined as the curl (rotational) of the flow velocity u→ vector
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Gradient
In mathematics , the GRADIENT is a multi-variable generalization of the derivative . While a derivative can be defined on functions of a single variable, for functions of several variables , the gradient takes its place. The gradient is a vector-valued function , as opposed to a derivative, which is scalar-valued . Like the derivative, the gradient represents the slope of the tangent of the graph of the function . More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. This characterizing property of the gradient allows it to be defined independently of a choice of coordinate system, as a vector field whose components in a coordinate system will transform when going from one coordinate system to another
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Identity Matrix
In linear algebra , the IDENTITY MATRIX, or sometimes ambiguously called a UNIT MATRIX, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context. (In some fields, such as quantum mechanics , the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I.) Less frequently, some mathematics books use U or E to represent the identity matrix, meaning "unit matrix" and the German word "Einheitsmatrix", respectively
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Constitutive Relation
In physics and engineering , a CONSTITUTIVE EQUATION or CONSTITUTIVE RELATION is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance , and approximates the response of that material to external stimuli, usually as applied fields or forces . They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis , the connection between applied stresses or forces to strains or deformations . Some constitutive equations are simply phenomenological ; others are derived from first principles . A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant
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Curl (mathematics)
In vector calculus , the CURL is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space
Euclidean space
. At every point in the field, the curl of that point is represented by a vector . The attributes of this vector (length and direction) characterize the rotation at that point. The direction of the curl is the axis of rotation, as determined by the right-hand rule , and the magnitude of the curl is the magnitude of rotation. If the vector field represents the flow velocity of a moving fluid , then the curl is the CIRCULATION DENSITY of the fluid. A vector field whose curl is zero is called irrotational . The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes\' theorem , which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve
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Total Head
In fluid dynamics , BERNOULLI\'S PRINCIPLE states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid 's potential energy . The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738. Bernoulli's principle
Bernoulli's principle
can be applied to various types of fluid flow, resulting in various forms of BERNOULLI\'S EQUATION; there are different forms of Bernoulli's equation for different types of flow. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number
Mach number
). More advanced forms may be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation ). Bernoulli's principle
Bernoulli's principle
can be derived from the principle of conservation of energy
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Horace Lamb
SIR HORACE LAMB FRS (/læm/ ; 27 November 1849 – 4 December 1934) was an English applied mathematician and author of several influential texts on classical physics , among them Hydrodynamics (1895) and Dynamical Theory of Sound (1910). Both of these books remain in print. CONTENTS* 1 Biography * 1.1 Early life and education * 1.2 University of Cambridge, 1872-75 * 1.3 University of Adelaide, 1876-85 * 1.4 University of Manchester, 1885-1920 * 1.5 Later years, 1920-34 * 2 Honours and awards * 3 See also * 4 References * 5 Publications * 6 External links BIOGRAPHYEARLY LIFE AND EDUCATIONLamb was born in Stockport
Stockport
, Cheshire
Cheshire
, the son of John Lamb and his wife Elizabeth, née Rangeley, the latter a foreman in a cotton mill, who had gained some distinction by an invention for the improvement of spinning machines. John Lamb died while his son was a child
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Partial Differential Equation
In mathematics , a PARTIAL DIFFERENTIAL EQUATION (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives . (A special case are ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives .) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model . 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
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Streamlines, Streaklines And Pathlines
STREAMLINES, STREAKLINES AND PATHLINES are field lines in a fluid flow . They differ only when the flow changes with time, that is, when the flow is not steady . Considering a velocity vector field in three-dimensional space in the framework of continuum mechanics , we have that: * STREAMLINES are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction in which a massless fluid element will travel at any point in time. * STREAKLINES are the loci of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye
Dye
steadily injected into the fluid at a fixed point extends along a streakline. * PATHLINES are the trajectories that individual fluid particles follow. These can be thought of as "recording" the path of a fluid element in the flow over a certain period
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Triple Scalar Product
In vector algebra , a branch of mathematics , the TRIPLE PRODUCT is a product of three 3-dimensional vectors, usually Euclidean vectors . The name "triple product" is used for two different products, the scalar-valued SCALAR TRIPLE PRODUCT and, less often, the vector-valued VECTOR TRIPLE PRODUCT. CONTENTS* 1 Scalar triple product * 1.1 Geometric interpretation * 1.2 Properties * 1.3 Scalar or pseudoscalar * 1.4 As an exterior product * 1.5 As a trilinear functional * 2 Vector triple product * 2.1 Proof * 2.2 Using geometric algebra * 3 Interpretations * 3.1 Tensor calculus * 4 Notes * 5 References * 6 External links SCALAR TRIPLE PRODUCT Three vectors defining a parallelepiped The SCALAR TRIPLE PRODUCT (also called the MIXED PRODUCT, BOX PRODUCT, or TRIPLE SCALAR PRODUCT) is defined as the dot product of one of the vectors with the cross product of the other two
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Viscosity
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 informal concept of "thickness"; for example, honey has a much higher viscosity than water . Viscosity
Viscosity
is a property of the fluid which opposes the relative motion between the two surfaces of the fluid in a fluid that are moving at different velocities . When the fluid is forced through a tube, the particles which compose the fluid generally move more quickly near the tube's axis and more slowly near its walls; therefore some stress (such as a pressure difference between the two ends of the tube) is needed to overcome the friction between particle layers to keep the fluid moving. For a given velocity pattern, the stress required is proportional to the fluid's viscosity. A fluid that has no resistance to shear stress is known as an ideal or inviscid fluid
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Shear Velocity
SHEAR VELOCITY, also called FRICTION VELOCITY, is a form by which a shear stress may be re-written in units of velocity . It is useful as a method in fluid mechanics to compare true velocities, such as the velocity of a flow in a stream, to a velocity that relates shear between layers of flow. Shear velocity is used to describe shear-related motion in moving fluids. It is used to describe: * Diffusion
Diffusion
and dispersion of particles, tracers, and contaminants in fluid flows * The velocity profile near the boundary of a flow (see Law of the wall ) * Transport of sediment in a channel Shear velocity also helps in thinking about the rate of shear and dispersion in a flow. Shear velocity scales well to rates of dispersion and bedload sediment transport. A general rule is that the shear velocity is between 5% to 10% of the mean flow velocity
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Conservation Law
In physics , a CONSERVATION LAW states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy , conservation of linear momentum , conservation of angular momentum , and conservation of electric charge . There are also many approximate conservation laws, which apply to such quantities as mass , parity , lepton number , baryon number , strangeness , hypercharge , etc. These quantities are conserved in certain classes of physics processes, but not in all. A local conservation law is usually expressed mathematically as a continuity equation , a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume
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Perturbation Theory
PERTURBATION THEORY comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbation" parts. Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem. Perturbation theory leads to an expression for the desired solution in terms of a formal power series in some "small" parameter – known as a PERTURBATION SERIES – that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem
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Degrees Of Freedom
In many scientific fields, the DEGREES OF FREEDOM of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation : its two coordinates ; a non-infinitesimal object on the plane might have additional degrees of freedoms related to its orientation . In mathematics , this notion is formalized as the dimension of a manifold or an algebraic variety . When degrees of freedom is used instead of dimension, this usually means that the manifold or variety that models the system is only implicitly defined
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