Exact single-phase incompressible perturbation of a shear flow

For a perturbation ${\mathbf{u}}^{\prime }=\tilde{\mathbf{u}}f(\mathbf{k}\cdot \mathbf{x})e^{st}$ with constant $\tilde{\mathbf{u}},\mathbf{k}$ and nonconstant differentiable $f$, incompressibility forces $\mathbf{k}\cdot \tilde{\mathbf{u}}=0$. Consequently ${\mathbf{u}}^{\prime }\cdot \nabla {\mathbf{u}}^{\prime }=f{f}^{\prime }{e}^{2st}(\mathbf{k}\cdot \tilde{\mathbf{u}})\tilde{\mathbf{u}}=0$. About a shear depending only on $x,z$ and directed along $y$, an axisymmetric such perturbation therefore obeys its linear amplitude equations exactly, including at finite amplitude in this homogeneous model. For pressure $P^{\prime }=\tilde{P}g(\xi )e^{st}$, nonzero pressure requires $g^{\prime }=Cf$, and $C$ can be absorbed into $\tilde{P}$; choose $g^{\prime }=f$. This relation does not imply $g=f$ for an arbitrary profile.