Is the velocity gradient of a homogeneous viscous fluid constant?

Question

The formula for the viscosity of a fluid (or even a gas) is equal to \begin{align*} \frac{F}{A} =\mu\frac{du}{dz} \end{align*} where $F$ is the force applied to the top layer of the fluid, $A$ is the surface area between the top and bottom layers of the fluid, $\mu$ is the viscosity of the fluid, and $du/dz$ is the velocity gradient of the fluid. In general, is the velocity gradient of a homogeneous viscous fluid a constant, or are there cases where higher order terms are needed?

Answer 1

This formula is applicable only in the case of a stationary fluid sheared between two plates (Poiseuille flow), in the limit that the shear strain is small. In that case the shear strain is automatically spatially constant. If the shear strain is large (or of the fluid is non-Newtonian) then $\mu$ will depend on the shear strain.

The general formula is that the stress tensor is given by $$ \Pi_{ij} = P\delta_{ij} +\rho u_iu_j - \mu\left( \nabla_iu_j+\nabla_j u_i -\frac{2}{3}\delta_{ij} (\nabla\cdot u) \right) - \zeta \delta_{ij} (\nabla\cdot u) + O(\nabla^2) $$ where $O(\nabla^2)$ denotes terms that become important at higher strain rate. The force per area on a surface element is $$ dF_i = \Pi_{ij}dA_j $$ We can quickly check that this reproduces Newton's formula. Take a flow $u_x(z)$ bounded by plates in the $xy$ plane. The tangential force on the plates is $$ F_x/A=-(\hat{e}_z)_i \Pi_{xj} = -\Pi_{xz}=\eta\nabla_z u_x $$