Consider a column of fluid of length $L$, with initial density $\rho_0$ and initial velocity ($u_0 =0$) everywhere. Now at time $t=0$ gravity is switched on. No-slip boundary conditions are assumed at both end of the fluid column.
We know that after a while column will attain a steady state with fluid everywhere at rest and density as exponential function of distance from either end.
Continuity equation is \begin{eqnarray} \frac{\partial\rho}{\partial t} + \frac{\partial(\rho u)}{\partial x} = 0 \end{eqnarray}
Navier-stokes equation for fluid in one dimension is
\begin{eqnarray} \rho\left[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} \right] &=& -\frac{\partial P}{\partial x} + f_{external} \nonumber \\ &=& -\frac{\partial\rho}{\partial x}c^2_s - \rho g \end{eqnarray} Here I assume shear forces are zero since the system is one dimensional.
In the steady state $u=0$ and $\frac{d\rho}{dt} = 0$ so we get, \begin{eqnarray} \frac{d\rho}{dx}c^2_s &=& - \rho g \\ \frac{d\rho}{\rho} &=& - dx \frac{g}{c^2_s} \\ \rho &=& \rho'\exp\left(-\frac{g}{c^2_s}x \right) \end{eqnarray}
where $\rho'$ is evaluated by mass conservation equation. \begin{eqnarray} \rho_{0} L = \int^{L}_0\rho'\exp\left(-\frac{g}{c^2_s}x \right)dx \end{eqnarray}
Where I assume hydrostatic pressure($P$) is proportional to density ($\rho$). Is it possible to solve these equations(assuming they are correct) as a function of time? To start with, I tried to get velocity ($u$) profile for the time very close to initial time. When the time is really small $t<<1$, For the Navier-Stokes equation we assume spatial variation in density($\rho$) and velocity($u$) is yet to develop, so that we get \begin{eqnarray} \frac{du}{dt} &=& -g \\ u &=& -gt \hspace{0.5cm} t <<1 \end{eqnarray}
I am not sure if it is allowed to assume initial spatial variation small compared to time variations in the system. Even if allowed, I am not able to go any further.
Also I feel the solution for density and velocity depend upon viscosity of the fluid but viscosity appears nowhere in the formulation. Do I need to include shear forces?