Solution
= Solution
<Momentum> balance gives $\rho u_t=-p_x+\mu u_{yy}$. Steadiness forces $p_x=-G$ and no slip gives $u=Gy(H-y)/(2\mu)$, hence $Q=GH^3/(12\mu)$. After removal, separation gives
$$
u(y,t)=\sum_{\substack{n\ge1\\n\text{ odd}}}\frac{4GH^2}{\mu n^3\pi^3}\sin(n\pi y/H)\exp[-(\mu/\rho)(n\pi/H)^2t].
$$