 Free viscous decay between parallel plates

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.