Plane Poiseuille flow

For a wall shear $u\simeq sz$, heating over streamwise length $L$ gives an outer temperature rise proportional to $L/(sz)$. Its transverse thermal conduction grows as $z^{-3}$ relative to the local source. If $\chi$ is thermal diffusivity, comparing conduction with heating gives $\delta \sim (\chi L/s{)}^{1/3}$, up to a numerical constant depending on the precise comparison. The same scale follows by comparing transverse diffusion time $z^2/\chi$ with residence time $L/(sz)$.

For a unidirectional incompressible flow between stationary plates a distance $H>0$ apart, removing the driving pressure gradient leaves $u_t=\nu u_{yy}$ when the kinematic viscosity $\nu >0$ is constant. The no-slip boundary condition gives $u(0,t)=u(H,t)=0$. For an initial profile $g$ in L2 space, separation of variables and the Fourier sine series give the displayed solution, withEach normal mode decays at a rate proportional to $n^2$. The lowest mode present dominates at late times; when $b_1\ne 0$, its time constant is $H^2/(\nu {\pi }^2)$, consistent with the viscous diffusion time. The solution approaches the initial data in L2 space; compatible smooth profiles also give the familiar pointwise initial condition. This is a heat equation for momentum diffusion, rather than temperature.

At large Reynolds number $\mathcal{R}$, a long indentation of relative height $ϵ$ has nonlinear near-wall interaction when the streamwise scale ratio satisfies $\lambda /\mathcal{R}={ϵ}^3$. The regime ${\mathcal{R}}^{1/7}\ll \lambda \ll \mathcal{R}$ gives a thin viscous layer and an effectively common pressure across the core. The core displacement is $u_1=A(x)U_0^{\prime }(y)$ and $v_1=-A^{\prime }(x)U_0(y)$. A lower wall displaced to $ϵF(x)$ produces layer displacement $F+A$, while the upper wall produces $-A$. Uniqueness for their common pressure forcing and upstream data makes the two layer displacements equal, so $A=-F/2$.

For indentation $F=b{x}^{1/3}$, the common layer displacement is $H=b{x}^{1/3}/2$. The boundary layer stream function $\psi =x^{2/3}G(z{x}^{-1/3})$ gives velocity $U=x^{1/3}{G}^{\prime }$ and pressure $P=\bar{p}{x}^{2/3}$. Substitution into the Prandtl boundary-layer equation yields the displayed equation, with $G(0)=G^{\prime }(0)=0$ and $G^{\prime }\sim 6\eta +3b$. This is a conditional similarity construction: the constant $\bar{p}$ must permit a solution of the boundary value problem.

Two viscous layers occupy $-a<y<0$ and $0<y<a$, with viscosities ${\mu }_{-}$ and ${\mu }_{+}$. A uniform driving pressure gradient $G=-p_x>0$, a stationary flat interface and no-slip outer walls give the displayed plane Poiseuille flow, where $U_0=G{a}^2/({\mu }_{-}+{\mu }_{+})$ and $C=Ga({\mu }_{-}-{\mu }_{+})/[2({\mu }_{-}+{\mu }_{+})]$. These constants follow from continuity of velocity and shear stress at the interface and zero wall velocities. The upper-to-lower volume-flux ratio is $(1+7r)/[r(7+r)]$, with $r={\mu }_{+}/{\mu }_{-}$. As $r\to \infty$, the interface becomes effectively stationary, the upper flux vanishes, and the lower layer approaches flow between two stationary plates a distance $a$ apart. The upper wall stress remains finite despite its vanishing velocity gradient.