= Newtonian velocity preservation in a second-order fluid
{c}
Write the non-Newtonian extra <stress> as $S=2(\psi_1+2\psi_2)E^2-\psi_1\overset\circ E$, with the <Jaumann derivative>. For a steady <Stokes flow>, $\nabla\times\nabla\cdot\overset\circ E=0$. If $\psi_1+2\psi_2=0$, the extra force is a <gradient> and only alters <pressure>. In a planar <incompressible flow>, $E^2=q\operatorname{diag}(1,1,0)$, so its <divergence> is also a gradient; no coefficient restriction is needed. A Newtonian velocity with prescribed boundary velocities consequently solves the second-order model with an altered pressure. On the regular weak-flow branch, <Uniqueness of Stokes flow> makes each velocity correction vanish. This does not establish uniqueness of arbitrary large nonlinear branches or the absence of higher-order effects in a real material.
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