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How to derive the Cauchy stress tensor term in the momentum equations?

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Old   September 9, 2022, 01:49
Default How to derive the Cauchy stress tensor term in the momentum equations?
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William
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I am learning about the source term in the momentum equation, and I got to the point where the momentum equation looks like this:

\rho\frac{D\mathbf{u}}{Dt} = -\nabla p + \nabla\cdot\bar{\bar{\tau}} + \mathbf{f}.

However, from this point onwards every source I came about provides the expression for \bar{\bar{\tau}} = \mu\left(\nabla\mathbf{u}+(\nabla\mathbf{u})^T\right)+\lambda(\nabla\cdot\mathbf{u})\mathbf{I} without providing a derivation. I understand that constitutive laws are involved here that states the shear stress is proportional to the spatial velocity gradient, but there's a lot more to it. Could someone please point me to a source where a detailed derivation, or at least an explanation of where each term comes from is given? Thanks a lot in advance!
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