Velocity gradient

In fluid mechanics and continuum mechanics, the velocity gradient describes how the velocity of a fluid changes between different points within the fluid.^[1] Though the term can refer to the differences in velocity between layers of flow in a pipe,^[2] it is often used to mean the gradient of a flow's velocity with respect to its coordinates.^[3] The concept has implications in a variety of areas of physics and engineering, including magnetohydrodynamics, mining and water treatment. ^[4] ^[5] ^[6]

A simple example

Consider the velocity field of a fluid flowing through a pipe. The layer of fluid in contact with the pipe tends to be at rest with respect to the pipe. This is called the no slip condition.^[7] If the velocity difference between fluid layers at the centre of the pipe and at the sides of the pipe is sufficiently small, then the fluid flow is observed in the form of continuous layers. This type of flow is called laminar flow.

The flow velocity difference between adjacent layers can be measured in terms of a velocity gradient, given by $\Delta u/\Delta y$ . Where $\Delta u$ is the difference in flow velocity between the two layers and $\Delta y$ is the distance between the layers.

Dimensional analysis

By performing dimensional analysis, the dimensions of velocity gradient can be determined. The dimensions of velocity are $M^{0}L^{1}T^{-1}$ , and the dimensions of distance are $M^{0}L^{1}T^{0}$ . Since the velocity gradient can be expressed as ${\frac {\Delta velocity}{\Delta distance}}$ . Therefore, the velocity gradient has the same dimensions as this ratio, i.e. $M^{0}L^{0}T^{-1}$ .

Velocity gradient in continuum mechanics

In 3 dimensions, the gradient, $\nabla {\bf {v}}$ , of the velocity $\textbf{v}$ is a second-order tensor which can be expressed as the matrix L:

{\bf {L}}={\nabla {\bf {v}}}=\left[{\begin{matrix}{\frac {\partial v_{x}}{\partial x}}&{\frac {\partial v_{x}}{\partial y}}&{\frac {\partial v_{x}}{\partial z}}\\\\{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{y}}{\partial y}}&{{\frac {\partial v_{y}}{\partial z}}\ }\\\\{\frac {\partial v_{z}}{\partial x}}&{\frac {\partial v_{z}}{\partial y}}&{\frac {\partial v_{z}}{\partial z}}\end{matrix}}\right]

${\textbf {L}}$ can be decomposed into the sum of a symmetric matrix ${\textbf {E}}$ and a skew-symmetric matrix ${\textbf {W}}$ as follows

{\textbf {E}}={\frac {1}{2}}\left({\textbf {L}}+{\textbf {L}}^{\textbf {T}}\right)

{\textbf {W}}={\frac {1}{2}}\left({\textbf {L}}-{\textbf {L}}^{\textbf {T}}\right)

${\textbf {E}}$ is called the strain rate tensor and describes the rate of stretching and shearing. ${\textbf {W}}$ is called the spin tensor and describes the rate of rotation.^[8]

Relationship between shear stress and the velocity field

Sir Isaac Newton proposed that shear stress is directly proportional to the velocity gradient: ^[9]

$\tau =\mu {\frac {\partial u}{\partial y}}$ .

The constant of proportionality, $\mu$ , is called the dynamic viscosity.

Applications

The study of velocity gradients is useful in analysing path dependent materials and in the subsequent study of stresses and strains, e.g. Plastic deformation of metals.^[3] The near-wall velocity gradient of the unburned reactants flowing from a tube is a key parameter for characterising flame stability.^[5]^:1-3 The velocity gradient of a plasma can define conditions for the solutions to fundamental equations in magnetohydrodynamics.^[4]

References

↑ Carl Schaschke (2014). A Dictionary of Chemical Engineering. Oxford University Press. ISBN 9780199651450.
↑ "Infoplease: Viscosity: The Velocity Gradient".
1 2 "Velocity gradient at continuummechanics.org".
1 2 Zhang, Zujin (June 2017), "Generalized MHD System with Velocity Gradient in Besov Spaces of Negative Order", Acta Applicandae Mathematicae, 149 (1): 139–144, doi:10.1007/s10440-016-0091-0, ISSN 1572-9036
1 2 Grumer, J.; Harris, M. E.; Rowe, V. R. (Jul 1956), Fundamental Flashback, Blowoff, and Yellow-Tip Limits of Fuel Gas-Air Mixtures (PDF), Bureau of Mines (RI 5225)
↑ Rojas, J.C.; Moreno, B.; Garralón, G.; Plaza, F.; Pérez, J.; Gómez, M.A. (2010), "Influence of velocity gradient in a hydraulic flocculator on NOM removal by aerated spiral-wound ultrafiltration membranes (ASWUF)", Journal of Hazardous Materials, 178 (1): 535–540, doi:10.1016/j.jhazmat.2010.01.116, ISSN 0304-3894
↑ Levicky, R. "Review of fluid mechanics terminology" (PDF).
↑ Gonzalez, O.; Stuart, A. M. (2008). A First Course in Continuum Mechanics. Cambridge Texts in Applied Mathematics. Cambridge University Press. pp. 134–135.
↑ Batchelor, G.K. (2000). An Introduction to Fluid Dynamics. Cambridge Mathematical Library. Cambridge University Press. p. 145. ISBN 9780521663960.

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[1] Carl Schaschke (2014). A Dictionary of Chemical Engineering. Oxford University Press. ISBN 9780199651450.

[2] "Infoplease: Viscosity: The Velocity Gradient".

[continuum-3] 1 2 "Velocity gradient at continuummechanics.org".

[MHD-4] 1 2 Zhang, Zujin (June 2017), "Generalized MHD System with Velocity Gradient in Besov Spaces of Negative Order", Acta Applicandae Mathematicae, 149 (1): 139–144, doi:10.1007/s10440-016-0091-0, ISSN 1572-9036

[Mines-5] 1 2 Grumer, J.; Harris, M. E.; Rowe, V. R. (Jul 1956), Fundamental Flashback, Blowoff, and Yellow-Tip Limits of Fuel Gas-Air Mixtures (PDF), Bureau of Mines (RI 5225)

[Watertreatment-6] Rojas, J.C.; Moreno, B.; Garralón, G.; Plaza, F.; Pérez, J.; Gómez, M.A. (2010), "Influence of velocity gradient in a hydraulic flocculator on NOM removal by aerated spiral-wound ultrafiltration membranes (ASWUF)", Journal of Hazardous Materials, 178 (1): 535–540, doi:10.1016/j.jhazmat.2010.01.116, ISSN 0304-3894

[7] Levicky, R. "Review of fluid mechanics terminology" (PDF).

[stuart-8] Gonzalez, O.; Stuart, A. M. (2008). A First Course in Continuum Mechanics. Cambridge Texts in Applied Mathematics. Cambridge University Press. pp. 134–135.

[9] Batchelor, G.K. (2000). An Introduction to Fluid Dynamics. Cambridge Mathematical Library. Cambridge University Press. p. 145. ISBN 9780521663960.