Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2007 / Paper 2 / Section B / 10 / Solution

Substitute $u(y,t)=Y(y)T(t)$ into the diffusion equation and divide on a region where both factors are nonzero. This gives $Y^{\prime \prime }/Y=T^{\prime }/(\nu T)$. The two sides depend on different variables, so they equal a separation constant $a$:These identities then extend through the zeros of the separated factors. The zero Dirichlet boundary conditions require $Y(0)=Y(1)=0$. For a nontrivial solution, $a$ must be negative: $a=0$ gives only a vanishing linear function and $a>0$ gives only a vanishing hyperbolic-sine solution. Writing $a=-k^2$ gives $Y=A\sin (ky)$, and $k=n\pi$, $n=1,2,\dots$. Consequently the separated modes of free viscous decay between parallel plates are $\sin (n\pi y)e^{-\nu n^2{\pi }^{2}t}$.

The first initial condition selects the lowest mode:The superposition principle for a homogeneous linear differential equation states that any linear combination of solutions is again a solution, with the same combination of homogeneous boundary conditions and initial data. It therefore givesThe higher spatial mode decays nine times as fast, so the velocity profile approaches a multiple of the first sine mode before vanishing altogether.