Is there a consistent formula for the Cauchy stress tensor in fluid mechanics?

Question

Recently, my undergraduate thesis advisor gave me a formula for the Cauchy stress tensor in fluid mechanics: $$\boldsymbol{\sigma}=-P\mathbf{I}+\mu\big(\nabla\boldsymbol u+(\nabla\boldsymbol u)^{\intercal}\big)$$ However, I recall that the Cauchy stress tensor is supposed to be second order covariant, having shape $(0,2)$. But, the objects on the right hand side all have shape $(1,1)$. The distinction is irrelevant when working in rectangular coordinates, but in general, one needs to be careful. I am wondering if the correct formula involves raising or lowering an index on one side or the other? Like, would we compute $$\sigma^i_j=-P\delta^i_j+\mu\big((\nabla u)^i_j+(\nabla u)^j_i\big)$$ And then index lower the result, $$\sigma_{ij}=g_{ik}\sigma^k_j$$ Or perhaps instead of this, the correct formula is actually $$\sigma_{ij}=-P g_{ik}\delta^k_j+\mu\big(g_{ik}(\nabla u)^k_j+g_{kj}(\nabla u)^k_i\big)\\ -P g_{ij}+\mu\big(g_{ik}(\nabla u)^k_j+g_{kj}(\nabla u)^k_i\big)$$ I hope somebody can clear this up for me.

Answer 1

You are on the right track, but it would be easier if you combined the second term into a single tensor (the rate-of-strain tensor) rather than treating it as multiple terms.

A good reference on this subject is Rutherford Aris's book titled "Vectors, Tensors, and the Basic Equations of Fluid Mechanics". It is available in paperback and is my go-to reference for things like this. Page 180 gives the general equation for the contravariant components of the stress tensor of a Newtonian fluid as

$$ \sigma^{ij} = ( - p + \lambda e^m{}_m ) g^{ij} + 2 \mu e^{ij} \,, $$

where the covariant components of the rate-of-strain tensor are defined as

$$ e_{ij} \equiv \tfrac{1}{2} ( u_{j,i} + u_{i,j} ) \,. $$

Here the commas just refer to the covariant derivative. It seems that in your case, the flow is incompressible, so $e^m{}_m = 0$, and the 2nd term drops out leaving

$$ \sigma^{ij} = - p g^{ij} + 2 \mu e^{ij} \,, $$

This basically is your answer but written in terms of the contravariant components and using the covariant derivative.

Unfortunately most books either use vector notation or use Cartesian index notation (like G. K. Batchelor's "Introduction to Fluid Mechanics" or Landau/Lifshitz's book). Aris's book is one of the few that I know that uses general tensor notation, so it is a good reference for this topic.