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Rivlin–Ericksen Tensor
A Rivlin–Ericksen temporal evolution of the strain rate tensor such that the derivative translates and rotates with the flow field. The first-order Rivlin–Ericksen is given by :\mathbf_= \frac+\frac where :v_i is the fluid's velocity and :A_ is n-th order Rivlin–Ericksen tensor. Higher-order tensor may be found iteratively by the expression : A_=\fracA_. The derivative chosen for this expression depends on convention. The upper-convected time derivative In continuum mechanics, including fluid dynamics, an upper-convected time derivative or Oldroyd derivative, named after James G. Oldroyd, is the rate of change of some tensor property of a small parcel of fluid that is written in the coordinate ..., lower-convected time derivative, and Jaumann derivative are often used. References * {{DEFAULTSORT:Rivlin-Ericksen tensor Multivariable calculus Fluid dynamics Non-Newtonian fluids ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strain-rate Tensor
In continuum mechanics, the strain-rate tensor or rate-of-strain 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 Jacobian matrix (derivative with respect to position) of the flow velocity. In fluid mechanics it also can be described as the velocity gradient, a measure of how the velocity of a fluid changes between different points within the fluid. Though the term can refer to the differences in velocity between layers of flow in a pipe, it is often used to mean the gradient of a flow's velocity with respect to its coordinates. The concept has implications in a variety of areas of physics and engineering, including magnetohydrodynamics, mining and water treatment. The strain rate tensor is a purely kinematic concept that describes the macroscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper-convected Time Derivative
In continuum mechanics, including fluid dynamics, an upper-convected time derivative or Oldroyd derivative, named after James G. Oldroyd, is the rate of change of some tensor property of a small parcel of fluid that is written in the coordinate system rotating and stretching with the fluid. The operator is specified by the following formula: : \stackrel = \frac \mathbf - (\nabla \mathbf)^T \cdot \mathbf - \mathbf \cdot (\nabla \mathbf) where: * is the upper-convected time derivative of a tensor field \mathbf *\frac is the substantive derivative *\nabla \mathbf=\frac is the tensor of velocity derivatives for the fluid. The formula can be rewritten as: : _ = \frac + v_k \frac - \frac A_ - \frac A_ By definition, the upper-convected time derivative of the Finger tensor is always zero. It can be shown that the upper-convected time derivative of a spacelike vector field is just its Lie derivative by the velocity field of the continuum. The upper-convected deriva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jaumann Derivative
Gustav Andreas Johannes Jaumann (1863–1924) was an Austrian physicist. An assistant to the physicist Ernst Mach, he had a talent for mathematics, but disbelieved the existence of small particles like electrons and atoms. Between 1901 and 1924 he taught physics at the German Technical University in Brno. He won the Haitinger Prize of the Austrian Academy of Sciences in 1911. Remembered for * "Corotational derivative" expresses the stress tensor in a rotating body. Han-Chin Wu (2005) "''Continuum Mechanics and Plasticity''" in: David Gao and Ray W. Ogden (Eds.); ''CRC Series: Modern Mechanics and Mathematics''; Chapman & Hall / CRC, Boca Raton, U.S.A.; 2005; 676 pp. — pages 170ff., 172ff. * Jaumann was offered a professorship at Prague University in 1911, but refused the position. The candidate who was the faculty's first choice, Albert Einstein, would accept the offer after it was turned down by Jaumann, who is alleged to have said in an unsubstantiated quotation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariable Calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one. Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus. Typical operations Limits and continuity A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the function. :f(x,y) = \frac approaches zero whenever the point (0,0) is approached along lines through the origin (y=kx). However, when the origin is appr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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