# Equations for a fluid in a rectangular domain, acted on by gravity

Suppose I have a rectangular domain, bounded by the $x$-axis, and the lines $x=a,x=b$ for some numbers $a<b$. Next, I pour in some amount $A$, of a fluid into the domain, and arrange it so that it takes the shape $$\{(x,y):a \le x \le b,0 \le y \le f(x) \}$$ for some continuous function $f(x)$, with $\int_a^b f(x) \mathrm{d} x=A$. I then let gravity act on the fluid, uniformly in the $- \mathbf{j}$ direction. I know that the fluid will approach a uniform distribution given by $$g(x)= \frac{A}{b-a},$$ but I'd like to have a differential equation for the height of the fluid, $u(x,t)$, with $u(x,0)=f(x),\lim_{t \to +\infty}u(x,t)=g(x)$. Could anyone help me formulate such precise equation? Thanks.

EDIT: Some desired properties of the PDE are

• If $u(x,t)$ is a solution, then so is $u(x,t)+h$ for any positive $h$ (with different initial conditions).
• $\int_a^b u(x,t) \mathrm{d} x \equiv A$ for all $t$ (conservation of mass).
• What do you know about the viscosity of the fluid?
– Paul
Oct 3, 2017 at 1:05
• @Paul Ideally, I'd like a model that can handle any constant viscosity. Oct 3, 2017 at 1:11

Idea: Given a function $f(x,t_0)$ write the differential equations for a small particle moving on this curve under a constant downwards force and a friction term proportional to the velocity. This particle has momentum and energy and we assume principle of extremal action. Update $f(x,t_0+\Delta t)$ by the displacement of the particle in the increment $\Delta t$.
Everything under $min(f(x,t_0))$ with $\frac{df(x,t_0)}{dt}=0$ is irrelevant in this model. Was this how you imagined it?
• Thanks for the idea. I indeed imagined that one could think of the area below $\min_x f$ as part of the "floor". Oct 3, 2017 at 13:39