The fluid mechanics is the subject that is all about the study of fluid flow. The fluid flow has lot more concepts like velocity, viscosity, pressure, density, etc. It has lots more formulas. Some of them are:

Bernoulli principle: It states that the total energy in a steadily flowing system is equal to the sum of potential energy, kinetic energy and pressure energy. It is given by,

$\frac{1}{2}$ $\rho$ V$^{2}$ + $\rho$ g h + p = Constant
Where, $\rho$ is density,
V is fluid flow velocity,
g is acceleration due to gravity,
h is height,
p is pressure.

If height in two energy of the system are equal (h1 = h2) then the Bernoulli principle is given by
p$_{2}$ + $\frac{1}{2}$ $\rho$ V$_{2}^{2}$ = p$_{1}$ + $\frac{1}{2}$ $\rho$ V$_{1}^{2}$
Where, V$_{1}$ and V$_{2}$ are the volumes in two systems,
p$_{1}$ and p$_{2}$ are the pressures,
$\rho$ is the density.

Continuity equation: It states that in a steady state process the rate at which mass is entering the system is equal to the mass is leaving the system. The continuity equation is given by,
p$_{1}$ - p$_{2}$ = $\frac{1}{2}$ $\rho$ (V$_{2}^{2}$ - V$_{1}^{2}$)

and
A$_{1}$V$_{1}$ = A$_{2}$V$_{2}$
Where, V$_{1}$ and V$_{2}$ are the velocity of the fluid,
A$_{1}$ and A$_{2}$ are the area of fluid,
P$_{1}$ and P$_{2}$ are the pressures,
$\rho$ is the density.

Poiseuille's Principle: The velocity of the fluid that flows through the capillary is directly proportional to the fluid's pressure and the fourth power of capillary radius and inversely proportional to the viscosity of fluid and capillary length.The Poiseuille's formula for resistance R is given by
R = $\frac{\pi r^4 \Delta p}{8 \eta L}$
Where, r is radius of the tube,
$\Delta$ p is the change in pressure from one end to other,
$\eta$ is the viscosity of the fluid,
L is the length of the tube.

Below are given some examples on fluid mechanics. You can go through it:

Question 1: A fluid flows in a tube of radius 0.5 cm and has a length of 2 m.If the pressure of 20 Pa is applied on fluid having viscosity of 0.6. Calculate the resistance.
Solution:
Given: radius r = 0.5 cm = 0.005 m, length l = 2m, pressure p = 20 Pa, viscosity $\eta$ = 0.6

The resistance to flow is given by,
R = $\frac{\pi r^4 \Delta p}{8 \eta L}$

R = $\frac{\pi \times 0.005^4 \times 20 Pa}{8 \times 0.6 \times 2m}$

R = 2.45 $\times$ 10-9 N.

Therefore, the resistance of the fluid flow in a tube is 2.45 $\times$ 10$^{-9}$ N.

Question 2: A fluid moves with velocity 2 m/s in an area of 10 cm2. If its velocity becomes 4 m/s then its area would be?
Solution:
Given: area A$_{1}$ = 10 cm2, velocity V$_{1}$ = 2m/s, velocity V$_{2}$ = 4m/s

The area A2 is given by,
A$_{1}$V$_{1}$ = A$_{2}$V$_{2}$
A$_{2}$ = $\frac{A_1 v_1}{v_2}$
    
A$_{2}$ = $\frac{0.1 m^2 \times 2 m/s}{4 m/s}$

A$_{2}$ = 0.05 m2.