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Newtonian Fluid
In continuum mechanics, a Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly[1] proportional to the local strain rate—the rate of change of its deformation over time.[2][3][4] That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions. More precisely, a fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow
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Spatial Derivative
A spatial gradient is a gradient whose components are spatial derivatives, i.e., rate of change of a given scalar physical quantity with respect to the position coordinates. Homogeneous regions have spatial gradient vector norm equal to zero
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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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Atmosphere Of Earth
The atmosphere of Earth
Earth
is the layer of gases, commonly known as air, that surrounds the planet Earth
Earth
and is retained by Earth's gravity. The atmosphere of Earth
Earth
protects life on Earth
Earth
by creating pressure allowing for liquid water to exist on the Earth's surface, absorbing ultraviolet solar radiation, warming the surface through heat retention (greenhouse effect), and reducing temperature extremes between day and night (the diurnal temperature variation). By volume, dry air contains 78.09% nitrogen, 20.95% oxygen,[2] 0.93% argon, 0.04% carbon dioxide, and small amounts of other gases. Air also contains a variable amount of water vapor, on average around 1% at sea level, and 0.4% over the entire atmosphere
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Differential Equation
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers
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Rotor (mathematics)
A rotor is an object in geometric algebra (or more generally Clifford algebra) that rotates any blade or general multivector about the origin.[1] They are normally motivated by considering an even number of reflections, which generate rotations (see also the Cartan–Dieudonné theorem). The term originated with William Kingdon Clifford,[2] in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension" (Ausdehnungslehre).[3] Hestenes[4] defined a rotor to be any element R displaystyle R of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies R ~ R = 1 displaystyle tilde R R=1 , where R ~ displaystyle tilde R is the "reverse" of R displaystyle R —that is, the product of t
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Vector Product
In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space (R3) and is denoted by the symbol ×. Given two linearly independent vectors a and b, the cross product, a ×
×
b, is a vector that is perpendicular to both a and b and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with dot product (projection product). If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero
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Taylor Series
In mathematics, a Taylor series
Taylor series
is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The concept of a Taylor series
Taylor series
was formulated by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor
Brook Taylor
in 1715. If the Taylor series
Taylor series
is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series
Taylor series
in the 18th century. A function can be approximated by using a finite number of terms of its Taylor series. Taylor's theorem
Taylor's theorem
gives quantitative estimates on the error introduced by the use of such an approximation
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Strain Rate Tensor
In continuum mechanics, the strain rate tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the gradient (derivative with respect to position) of the flow velocity. The strain rate tensor is a purely kinematic concept that describes the macroscopic motion of the material. Therefore, it does not depend on the nature of the material, or on the forces and stresses that may be acting on it; and it applies to any continuous medium, whether solid, liquid or gas. On the other hand, for any fluid except superfluids, any gradual change in its deformation (i.e. a non-zero strain rate tensor) gives rise to viscous forces in its interior, due to friction between adjacent fluid elements, that tend to oppose that change
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Vector Field
In vector calculus, a vector field is an assignment of a vector to each point in a subset of space.[1] A vector field in the plane (for instance), can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. The elements of differential and integral calculus extend naturally to vector fields
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Differential (mathematics)
In mathematics, differential refers to infinitesimal differences or to the derivatives of functions.[1] The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.Contents1 Basic notions 2 Differential geometry 3 Algebraic geometry 4 Other meanings 5 References 6 External linksBasic notions[edit]In calculus, the differential represents a change in the linearization of a function.The total differential is its generalization for functions of multiple variables.In traditional approaches to calculus, the differentials (e.g. dx, dy, dt, etc.) are interpreted as infinitesimals
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array[1] of numbers, symbols, or expressions, arranged in rows and columns.[2][3] For example, the dimensions of the matrix below are 2 × 3 (read "two by three"), because there are two rows and three columns: [ 1 9 − 13 20 5 − 6 ] . displaystyle begin bmatrix 1&9&-13\20&5&-6end bmatrix
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Coordinate System
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.[1][2] The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring
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Matrix Equation
In mathematics, a matrix (plural: matrices) is a rectangular array[1] of numbers, symbols, or expressions, arranged in rows and columns.[2][3] For example, the dimensions of the matrix below are 2 × 3 (read "two by three"), because there are two rows and three columns: [ 1 9 − 13 20 5 − 6 ] . displaystyle begin bmatrix 1&9&-13\20&5&-6end bmatrix
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Incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An equivalent statement that implies incompressibility is that the divergence of the flow velocity is zero (see the derivation below, which illustrates why these conditions are equivalent). Incompressible flow does not imply that the fluid itself is incompressible
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Derivative
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value
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