= Solution
{c}
For an isotropic <Newtonian fluid> in <incompressible flow>, the <constitutive equation> is
$$
\boxed{\sigma_{ij}=-p\delta_{ij}+2\mu E_{ij},
\qquad E_{ij}=\frac12(\partial_j u_i+\partial_i u_j).}
$$
Here $p$ is <pressure>, $\delta_{ij}$ is the <Kronecker delta>, $\mu$ is the constant <dynamic viscosity>, and $E$ is the <strain-rate tensor>. The more general compressible <constitutive equation> can include a term proportional to $(\nabla\cdot\mathbf u)\delta_{ij}$; that term vanishes here.
Differentiate the <Cauchy stress tensor>, retaining the constant <dynamic viscosity> outside the <derivatives>:
$$
\partial_j\sigma_{ij}
=-\partial_i p+\mu\partial_j\partial_j u_i
+\mu\partial_i(\partial_j u_j)
=-\partial_i p+\mu\nabla^2u_i.
$$
The last equality uses <incompressibility>. Substitution into the <Cauchy momentum equation> gives the <Navier-Stokes equation>
$$
\boxed{\rho\left(\frac{\partial\mathbf u}{\partial t}
+(\mathbf u\cdot\nabla)\mathbf u\right)
=-\nabla p+\mu\nabla^2\mathbf u.}
$$
The <pressure gradient> and the <Laplacian> of the <velocity field> therefore give the two contact <force> densities in this model.
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