Stokes' first and second problems

Example

Let a viscous fluid of kinematic viscosity $\nu > 0$ fill the half-space $y > 0$ above a flat plate $y = 0$. Look for flows $u = (w(y, t), 0, 0)$ with constant pressure. Such a field is divergence-free, and $(u \cdot \nabla) u = w \, \partial_x u = 0$. So the Navier–Stokes equations reduce to the heat equation $\partial_t w = \nu \, \partial_y^2 w$.

1. First problem (impulsively started plate): the fluid is at rest for $t < 0$, and for $t > 0$ the plate moves with constant speed $U$. The solution is $$w(y, t) = U \operatorname{erfc}\Big( \frac{y}{2 \sqrt{\nu t}} \Big),$$ where $\operatorname{erfc}(s) = \frac{2}{\sqrt{\pi}} \int_s^\infty e^{-\sigma^2} d\sigma$.
2. Second problem (oscillating plate): the plate moves with velocity $U \cos(\omega t)$, with $\omega > 0$. The time-periodic solution that stays bounded as $y \to \infty$ is $$w(y, t) = U e^{-k y} \cos(\omega t - k y), \qquad k = \sqrt{\frac{\omega}{2 \nu}}.$$

Check of 2: $w$ is the real part of $W = U e^{i \omega t - (1 + i) k y}$. Then $\partial_t W = i \omega W$ and $\nu \, \partial_y^2 W = \nu (1 + i)^2 k^2 W = 2 i \nu k^2 W = i \omega W$. At $y = 0$, $w = U \cos(\omega t)$, so the no-slip condition holds. Check of 1: $\operatorname{erfc}(y / (2 \sqrt{\nu t}))$ is the standard similarity solution of the heat equation. It equals $1$ at $y = 0$ and tends to $0$ as $y \to \infty$ or $t \to 0^+$.