For example, water surface depressed by putting floating needle on water, or feet of water strider insect, whose surface can be expressed by an equation
I don't know the formula describing the shape of the surface but I know a possible starting point. If the liquid surface is still, that means the forces on it are in balance. The pressure difference between the two sides of the surface is balanced by surface tension. The pressure difference due to surface tension is calculated with https://en.wikipedia.org/wiki/Young%E2%80%93Laplace_equation and (in the case of water strider) it equals the pressure difference due to hydrostatic pressure.
Dear Sumit Bhowmick,
Let me make a small clarification adjustment to your question.
For clarity of explanation and simplicity of the mathematical model, which will be given below, I will confined myself to considering only a non-stationary plane flow, and then the desired fluid profile h is as function depending on the time t and two spatial coordinates x (length) and y (width) of a Cartesian coordinate system XOY: h=h(x,y,t).
The liquid profile depends not only on the geometric shape of the external force (for example, a non-wetting solid on the liquid surface, etc.), but this also depends on other parameters listed in the attached PDF-file titled "Sh.E.Guseynov. Answer – 2D mathematical model".
The attached other PDF-file titled "Sh.E. Guseynov. Evaluation of influence of terrain profile on the physical characteristics of air flow" contains the proof of a fact that the profile irregularities of the underlying surface have a significant effect to many physical characteristics, such as pressure, density, speed, temperature.
Best regards,
Sharif Guseynov
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