Pressure on a liquid semisphere

[rinale84]

Imagine to have a drop of water(semisphere volume) on a flat membrane covered by a polymer so that the liquid can't excape from this envelope. If I apply a pressure upon this sphere (imagine a plane at a constant pressure that step by step would squeeze down the central part of the semisphere and therefore the volume of liquid in the external part would increase). This keep going until a point in which the fluid is freezed in a shape almost parallelepiped and can't move anymore. How could I be able to describe at discrete point on the plane surface (the bottom of the semisphere) the variation in height? So imagine to have the x-y axis on the flat surface , for a chosen point (x,y) can I be able to get an equation or something to explain how his related height (z) varies depending on the pressure??

 

[LightHero]

The equation describing the height variation, z, as a function of pressure, p, and position (x,y) is called Laplace's equation. It is given by:$$\Delta z = \frac{\partial^2z}{\partial x^2} + \frac{\partial^2z}{\partial y^2} = -\frac{p}{\rho g},$$where $\rho$ is the density of the liquid and $g$ is the acceleration due to gravity. This equation describes how the height of the water surface changes in response to an applied pressure, and can be solved numerically using finite difference methods.