= Free viscous decay between parallel plates
{title2=$u(y,t)=\sum_{n\geq1}b_n\sin(n\pi y/H)e^{-\nu(n\pi/H)^2t}$}
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, with
$$
b_n=\frac2H\int_0^H g(y)\sin\frac{n\pi y}{H}\,dy.
$$
Each <normal mode> decays at a rate proportional to $n^2$. The lowest mode present dominates at late times; when $b_1\ne0$, 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.
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