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In Wikipedia https://en.wikipedia.org/wiki/Newtonian_fluid, it is said that for the incompressible isotropic case of Newtonian fluid, the viscous stress equation is

$$ \tau_{i j}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}\right) $$ where $x_{j}$ is the $j$ th spatial coordinate

$v_{i}$ is the fluid's velocity in the direction of axis $i$

$\tau_{i j}$ is the $j$ th component of the stress acting on the faces of the fluid element perpendicular to axis $i$.

But I don't understand why there is a $\frac{\partial v_{j}}{\partial x_{i}}$ term here. It seems that $v_{j}$ penetrates the face, which means it can not provide sheer stress. Can anyone show me by graphs? Thank you!

In this Wikipedia entry, it is said that for the incompressible isotropic case of Newtonian fluid, the viscous stress equation is

$$ \tau_{i j}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}\right) $$ where

• $x_{j}$ is the $j$ th spatial coordinate
• $v_{i}$ is the fluid's velocity in the direction of axis $I$
• $\tau_{i j}$ is the $j$ th component of the stress acting on the faces of the fluid element perpendicular to axis $i$.

But I don't understand why there is a $\frac{\partial v_{j}}{\partial x_{i}}$ term here. It seems that $v_{j}$ penetrates the face, which means it can not provide sheer stress. Can anyone show me by graphs? Thank you!

In Wikipedia https://en.wikipedia.org/wiki/Newtonian_fluid, it is said that for the incompressible isotropic case of Newtonian fluid, the viscous stress equation is

$$ \tau_{i j}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}\right) $$ where $x_{j}$ is the $j$ th spatial coordinate

$v_{i}$ is the fluid's velocity in the direction of axis $i$

$\tau_{i j}$ is the $j$ th component of the stress acting on the faces of the fluid element perpendicular to axis $i$.

But I don't understand why there is a $\frac{\partial v_{j}}{\partial x_{i}}$ term here. It seems that $v_{j}$ penetrates the face, which can not provide sheer stress. Can anyone show me by graphs? Thank you!

In Wikipedia https://en.wikipedia.org/wiki/Newtonian_fluid, it is said that for the incompressible isotropic case of Newtonian fluid, the viscous stress equation is

$$ \tau_{i j}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}\right) $$ where $x_{j}$ is the $j$ th spatial coordinate

$v_{i}$ is the fluid's velocity in the direction of axis $i$

$\tau_{i j}$ is the $j$ th component of the stress acting on the faces of the fluid element perpendicular to axis $i$.

But I don't understand why there is a $\frac{\partial v_{j}}{\partial x_{i}}$ term here. It seems that $v_{j}$ penetrates the face, which means it can not provide sheer stress. Can anyone show me by graphs? Thank you!