Periodic shear fields: an exact steady Euler solution, failure for the viscous branch, and the classification of steady periodic shears

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the cited periodic Sobolev and local-evolution suppliers.

Use the dimensionless variables of Adopted fluid constitutive branches: incompressible Newtonian, barotropic inviscid, and the periodic-cell idealization clause (c): $t,x,u$ in this example denote the scaled time, unit-torus position and velocity, $\nu =\eta \mathrm{/}(\rho U_{\ast }{L}_{\ast })$, and $p=P_{\mathrm{m}}\mathrm{/}(\rho U_{\ast }^2)$. The physical fields are restored by $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=U_{\ast }u(t\mathrm{/}{T}_{\ast },x\mathrm{/}{L}_{\ast })$, $P_{\mathrm{m}}=\rho U_{\ast }^{2}p(t\mathrm{/}{T}_{\ast },x\mathrm{/}{L}_{\ast })$. Work on the periodic cell ${\mathbb{T}}^3=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^3$ of Adopted fluid constitutive branches: incompressible Newtonian, barotropic inviscid, and the periodic-cell idealization clause (c), with constant density $\rho >0$, no body force and the stress of the branch. The variables and field values here are dimensionless. Let $A>0$ be a dimensionless velocity amplitude and $\phi \in \mathbb{R}$ a dimensionless phase, and

$u(t,x):=(A\sin (2\pi x_2+\phi ),0,0)\mathrm{.}$

Then:

1. $\mathrm{div}u=0$ and $(u\cdot \mathrm{\nabla })u=0$, so with $p\equiv 0$ the field is an exact steady solution of the Euler branch ($\nu =0$) of the model on the periodic cell: ${\mathrm{\partial }}_{t}u+P((u\cdot \mathrm{\nabla })u)=\nu \mathrm{\Delta }u$ holds as $0=0$, and the primitive equation ${\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u=-\mathrm{\nabla }p+\nu \mathrm{\Delta }u$ holds with $p\equiv 0$;
2. the field satisfies the hypotheses of Local existence, uniqueness and pressure reconstruction for the incompressible fluid model on the periodic cell: it is smooth, divergence-free, periodic and has finite $H^s$ norm ($s\ge 4$), and the displayed global solution is consistent with the local existence-uniqueness statement;
3. for the viscous branch ($\nu >0$) the field is not a solution: $\mathrm{\Delta }u=-(2\pi {)}^{2}u\mathrm{\ne }0$ and $P\mathrm{\Delta }u=\mathrm{\Delta }u$ because $u$ is divergence-free, so the projected equation would require $-\nu \cdot (2\pi {)}^{2}u=0$, which fails for $A>0$;
4. among classical periodic shears $u=(v(x_2),0,0)$ with $v\in C^3(\mathbb{T})$ and periodic pressure $p\in C^2({\mathbb{T}}^3)$, every steady solution of the viscous branch with no body force and constant density is constant: the dimensionless equations force $\mathrm{\nabla }p=\nu (v^{\mathrm{\prime }\mathrm{\prime }}(x_2),0,0)$, and periodicity of $p$ in $x_1$ together with periodicity of $v$ in $x_2$ forces $v^{\mathrm{\prime }\mathrm{\prime }}\equiv 0$ and then $v\equiv \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$.

• The example separates the two branches sharply on the same data: the shear is an exact Euler solution and not even a solution of the viscous projected equation, because constant positive viscosity diffuses the shear on the periodic cell. The classified steady viscous shears are exactly the constants, i.e. the trivial shears.

• The pressure in clause 4 is forced to be linear in $x_1$ before periodicity kills the slope; this is the periodic-cell analogue of the constant pressure gradient of the channel example, and it shows that on a periodic cell a steady viscous shear cannot be driven. No measurement and no stability claim is made.