= Solution
Use <conservation of mass> and Newton's <momentum balance for a material fluid element>[momentum balance], in addition to the supplied <constitutive equation>[constitutive law]. For a <material volume>,
$$
\frac d{dt}\int_{V(t)}\rho\mathbf u\,dV=\int_{\partial V(t)}\boldsymbol\sigma\mathbf n\,dS+\int_{V(t)}\rho\mathbf g\,dV.
$$
The <Reynolds transport theorem> and $\rho_t+\nabla\cdot(\rho\mathbf u)=0$ reduce the left side to $\int\rho D\mathbf u/Dt$. The <divergence theorem> then gives $\rho D u_i/Dt=\partial_j\sigma_{ij}+\rho g_i$. For constant $\mu$,
$$
\partial_j\sigma_{ij}=-\partial_ip+\mu\partial_j\partial_ju_i+\mu\partial_i(\partial_ju_j).
$$
<Incompressibility> removes the last term. Thus
$$
\boxed{\rho\frac{D\mathbf u}{Dt}=-\nabla p+\mu\nabla^2\mathbf u+\rho\mathbf g.}
$$
The stress formula and zero divergence alone are not a dynamical law: the <force> balance supplies the required physical input.
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