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My question relates to the attached pictures, which contain text from the book noted in the question title. In particular, my question is related to the first assumption made in the text (See bottom of attached picture). For context, these are assumptions made about the stress tensor describing a fluid as the author builds up to deriving the Navier-Stokes Equations. The assumption is that the stress tensor depends linearly on the velocity gradients.

Question - Can someone express mathematically what is meant by the first assumption? I understand physically what this assumption means but I think my understanding would be richer if I could see this assumption stated precisely. Does this just mean that the stress tensor is the Jacobian of u, the fluid velocity field,enter image description here multiplied by some matrix A?

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No, not a matrix, but a rank 4 tensor. In index notation it means:

$\sigma_{ij}=A_{ijkl}(\nabla u)_{kl}$

You can think of it this way: A linear transformation taking one vector (1-index object) to another vector (1-index object) requires a 2-index object, namely a matrix. A linear transformation taking 2-index objects to 2-index objects requires a 4-index object (rank 4 tensor).

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  • $\begingroup$ I'm not so familiar with tensors and the book i'm using does not seem to be emphasizing them. Is there some way to express this transformation in notation involving just the matrix (i.e, notation where the matrix is treated as its own object rather than being dealt with via its components)? For example, the final form for the stress tensor that the author lands on is A = wD +qC where A is the stress tensor, D is the deformation tensor and C is the identity matrix. This algebraically resembles a "linear equation" like y=mx+b for a line in two dimensions. Can it be expressed in this way? $\endgroup$
    – JRR
    Commented Jul 19, 2022 at 19:19
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Assumption 1) in the book means that each component $\sigma_{ij}$ is a linear and homogeneous function of each $\frac{\partial u_k}{\partial x_l}$. To express that mathematically you need $3^4$ numbers say $A_{ijkl}$. You do not think of tensors if you don't want to $$\sigma_{ij}=\sum_{kl} A_{ijkl}\frac{\partial u_k}{\partial x_l}$$

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