### Derivations - Fluid Mechanics Questions

- Derive a general equation for the variation of pressure due to gravity from point to point in a static fluid.
- Derive an expression for the pressure difference across two limbs of a differential manometer containing two gauge fluids, mutually immissible. What factors influence the sensitivity?
- Derive the equation of continuity for an incompressible flow. State the importance of the equation.
- By applying momentum balance to the steady flow of a fluid inside a pipeline, obtain the Bernoulli's equation. Indicate the corrections necessary to the equation for application to real situations.
- Derive Euler's equation of motion along a streamline in differential form and obtain Bernoulli's equation in the integral form for the flow of an incompressible fluid.
- Derive the Hagen-Poiseuille equation for laminar flow through a circular pipe. State the limitations of the equation.
- Show that the average velocity of the fluid flowing through a circular pipe under laminar conditions is half that of the maximum velocity.
- Consider the flow of a Newtonian fluid in a pipe having roughness 'k'. By dimensional analysis, develop an equation relating the frictional pressure drop in terms of the following:
r - density of fluid

m - viscosity of fluid

v - velocity of fluid

D - diameter of pipe

L - length of pipe

k - roughness