Couette flow with infinite depth Consider a fluid below $x$ axis or $xy$ plane. Its top layer starts to move with velocity $v$ at and after time $t_0$.
If the flow is not fully developed and evolving from standing water, I receive $v_x=0$ from continuity equation due to zero velocity in $y$ direction (and $z$ in 3D case). Which corresponds to fully developed flow since the acceleration is zero.
$$\nabla \cdot v = 0
\\v_t + (v \cdot \nabla) v = -(1/\rho) \nabla p + \nu \Delta v$$
Task itself was to calculate velocity at certain level $h$ below the surface after given time $t_1$. (Pressure at the surface is atmospheric).
I ended up reducing system to 1D heat equation whose solution is implicit.
Could one advise on what I’m doing wrong?
 A: Let the axes be oriented so that the fluid on the $xy$-plane moves impulsively at $t = 0+$ with velocity $U$ in the positive $x$-direction and the positive $y$-axis points into the body of fluid.
An analytical solution can be obtained assuming unsteady, unidirectional flow with one non-vanishing velocity component $u$ in the $x$-direction. The continuity equation requires that $\frac{\partial u}{\partial x} = 0$. Assuming the fluid is bounded by an infinite planar surface, we have that $u(y,t)$ is a function only of $y$ and $t$, and the Navier-Stokes equations and boundary conditions reduce to
$$\tag{*}\frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial y^2}, \\  u(0,t) = U \,\,\text{ for }\,\, t \geqslant 0,\\ u(y,t) \to 0 \text{ as  }y \to \infty,$$
where  $\nu = \mu/\rho$ is the kinematic viscosity.
We can solve with a similarity transformation where
$$u(y,t) = f(\eta), \quad \eta = \frac{y}{\sqrt{\nu t}}$$
Then we have
$$\frac{\partial u }{\partial t}= - f'(\eta)\frac{y}{2 \sqrt{\nu}t^{3/2}}=  - \frac{1}{2t}\eta f'(\eta),\\\frac{\partial u}{\partial y} = \frac{1}{\sqrt{\nu t}}f'(\eta),\\\frac{\partial^2 u}{\partial y^2}=\frac{1}{\sqrt{\nu t}}f''(\eta)\frac{1}{\sqrt{\nu t}} = \frac{1}{\nu t}f''(\eta)$$
Substituting for $u$ and its partial derivatives in (*) and canceling the $1/t$ factors , we get the second-order linear ordinary differential equation and conditions
$$f''(\eta) + \frac{1}{2}\eta f'(\eta) = 0,\\f(0) = U ,\\ f(\eta) \to 0 \,\, \text{ as } \, \, \eta \to \infty$$
It is not difficult to solve this by separating variables and integrating twice, yielding the solution
$$u = U \text{ erfc} \left(\frac{\eta}{2}\right) = U \text{ erfc} \left(\frac{y}{\sqrt{4\nu t}}\right),$$
where the complementary error function is given by
$$\text{erfc}(z) = \frac{2}{\sqrt{\pi}}\int_z^\infty e^{-t^2} \, dt$$
