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Cauchy Momentum Equation
The Cauchy
Cauchy
momentum equation is a vector partial differential equation put forth by Cauchy
Cauchy
that describes the non-relativistic momentum transport in any continuum.[1] In convective (or Lagrangian) form it is written: D u D t = 1 ρ ∇ ⋅ σ + 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)
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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. 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. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives. 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
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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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Conservative Field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function, known in this context as a scalar potential.[1] Conservative vector fields have the property that the line integral is path independent, i.e., the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl
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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.[1] [2] 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.[3] 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
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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.[1][2] The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738.[3] Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonard Euler who derived Bernoulli's equation in its usual form in 1752.[4][5] The principle is only applicable for isentropic flows: so when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g. heat radiation) are small and can be neglected. 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
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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. 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
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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[1]), as would be seen by an observer located at that point and traveling along with the flow. Conceptually, vorticity could be determined by marking parts of a 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
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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
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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.[1] For liquids, it corresponds to the informal concept of "thickness"; for example, honey has higher viscosity than water.[2] Viscosity
Viscosity
is a property of the fluid which opposes the relative motion between the two surfaces of the fluid that are moving at different velocities. In simple terms, viscosity means friction between the molecules of fluid. 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
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Vector Calculus Identity
The following identities are important in vector calculus:Contents1 Operator notations1.1 Gradient 1.2 Divergence 1.3 Curl 1.4 Laplacian 1.5 Special
Special
notations2 Properties2.1 Distributive properties 2.2 Product rule for the gradient 2.3 Product of a scalar and a vector 2.4 Quotient rule 2.5 Chain rule 2.6 Vector dot product 2.7 Vector cross product3 Second derivatives3.1 Curl of the gradient 3.2
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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
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Inviscid Flow
Inviscid flow
Inviscid flow
is the flow of an inviscid fluid, in which the viscosity of the fluid is equal to zero.[1] Though there are limited examples of inviscid fluids, known as superfluids, inviscid flow has many applications in fluid dynamics.[1][2] The Reynolds number
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Gravity
Gravity, or gravitation, is a natural phenomenon by which all things with mass are brought toward (or gravitate toward) one another, including objects ranging from atoms and photons, to planets and stars. Since energy and mass are equivalent, all forms of energy (including light) cause gravitation and are under the influence of it. On Earth, gravity gives weight to physical objects, and the Moon's gravity causes the ocean tides. The gravitational attraction of the original gaseous matter present in the Universe
Universe
caused it to begin coalescing, forming stars – and for the stars to group together into galaxies – so gravity is responsible for many of the large scale structures in the Universe
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