Exact plane Poiseuille and Couette flows of the incompressible Newtonian model, with flux and ansatz uniqueness

The flux integrals below are ordinary Riemann integrals of continuous profiles on $[0,h]$.

Let $I\subseteq \mathbb{R}$ be a nonempty open time interval, let $\rho >0$, $\eta >0$ and $h>0$ be constants, and let $O:=\mathbb{R}\times (0,h)\times \mathbb{R}$ with spatial coordinates $(x,y,z)$. Let $v\in C([0,h])\cap C^2((0,h))$, with endpoint values given by its continuous extension, and $p_{\mathrm{s}\mathrm{p}}\in C^1(O)$. Define stationary spacetime fields $u:I\times O\to {\mathbb{R}}^3$ and $p:I\times O\to \mathbb{R}$ by $u(t,x,y,z):=(v(y),0,0)$ and $p(t,x,y,z):=p_{\mathrm{s}\mathrm{p}}(x,y,z)$. Here $D_{x}u$ denotes the spatial total derivative at fixed $t$, $(u\cdot \mathrm{\nabla })u:=D_{x}uu$, and $\mathrm{\Delta }u:=({\sum }_{j=1}^3{\mathrm{\partial }}_j^2{u}_i{)}_{i=1}^3$ acts componentwise (Euclidean spatial calculus: gradient, divergence and material derivative). The incompressible Newtonian equations of Incompressible Newtonian and barotropic inviscid fluid constitutive models with no body force and steady data read

$$\begin{array}{cccc}& \mathrm{div}u=0,\rho (u\cdot \mathrm{\nabla })u=-\mathrm{\nabla }p+\eta \mathrm{\Delta }u,& & \text{(NS)}\end{array}$$

where the second equation is the adopted stationary momentum equation with $\sigma =-p{I}_3+\eta (D_{x}u+(D_{x}u{)}^{\mathsf{T}})$.

For this item, “irrotational” means that the spatial velocity gradient $D_{x}u$ is symmetric.

(a) Plane Poiseuille flow. Let $G\in \mathbb{R}$, $p_0\in \mathbb{R}$ and

$v(y):=\frac{Gy(h-y)}{2\eta },p_{\mathrm{s}\mathrm{p}}(x,y,z):=p_0-Gx\mathrm{.}$

Then $u$ is $C^{\mathrm{\infty }}$ on $I\times O$, satisfies (NS) there, and $v(0)=v(h)=0$. Its flux per unit width is

${\int }_0^{h}v(y)dy=\frac{G{h}^3}{12\eta }\mathrm{.}$

Moreover, within the ansatz $u=(v(y),0,0)$, $p=p_0-Gx$ with the same constant $G$ and the no-slip conditions $v(0)=v(h)=0$, this is the only solution: any two such profiles differ by a linear function vanishing at both endpoints, hence coincide.

(b) Plane Couette flow. Let $V\in \mathbb{R}$, let $v(y):=Vy\mathrm{/}h$, and let $p_{\mathrm{s}\mathrm{p}}$ be constant. Then the stationary $u,p$ defined above satisfy (NS) on $I\times O$ and $v(0)=0$, $v(h)=V$. If $V\mathrm{\ne }0$, the velocity gradient $D_{x}u$ is not symmetric, so the flow is not irrotational by the definition above.

• The no-slip boundary conditions are adopted model conditions, not consequences of the equations; the uniqueness statement is only within the stated steady parallel ansatz with the same driving constant $G$. General nonlinear initial-value problems, finite-Reynolds stability, entrance effects and three-dimensional perturbations are not treated.

• The Couette computation exhibits explicitly that a laminar shear flow has nonzero antisymmetric velocity gradient; this is why the irrotational Bernoulli branch of the barotropic item does not apply to it. The two flows here are exact solutions of the adopted constant-coefficient model and carry no empirical claim about any real fluid.