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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!

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    $\begingroup$ My impression is that MO is not the suitable site. Have you looked at other sites of StackExchange ? $\endgroup$
    – Denis Serre
    Commented Feb 18, 2022 at 20:27
  • $\begingroup$ it's a symmetric tensor, which term contributes depends on the geometry. $\endgroup$
    – Carlo Beenakker
    Commented Feb 18, 2022 at 21:37
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    $\begingroup$ The geometry is in the strain, not stress. Newtonian fluid assumption that stress is simply proportional to strain gives this relation for stress. The derivation of strain is in many textbooks. Basically you need to compute shear strain ( angle) as a function of velocity gradienta $\endgroup$
    – Piyush Grover
    Commented Feb 19, 2022 at 19:24
  • $\begingroup$ This video exactly solves my question. $\endgroup$
    – Plum
    Commented Mar 4, 2022 at 1:54

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