Fluid Mechanics Formulas

1. Pressure inside the liquid

The pressure due to liquid act on the surface below depth h is given by
P = hρg
where p is the density of liquid and g acceleration due to gravity.

2. Pascal’s law

According to Pascal’s law the pressure at every point inside the liquid is same in the absence of gravity.

3. Absolute pressure & gauge pressure

• Absolute pressure at a point A is the total pressure at that point including the pressure of liquid and that of atmosphere.
• Gauge pressure is the relative pressure between two points of which one is at the surface and other is at some distance below inside the liquid. Thus

P_absolute = P_Gauge + P_atmospheric, P_gauge = ρgh
P_abs = ρgh + P₀

4. Archimede’s principle

It states that when a body is immersed wholly or partly in a liquid at rest, it losses some of its weight. The loss in weight of the body in the liquid is equal to the weight of the liquid displaced by the immersed part of the body. Let the true weight of the body be W_b then
W_b = M_bg = V_bρ_bg
weight of the liquid displaced
W_L = m_Lg = V_Lρ_Lg
Then observed weight of the body
W = W_b – W_L
= (V_bρ_b – V_Lρ_L)g

5. Laws of flotation

If ρ_b → density of the body & ρ_L → density of the liquid. Then
Case I
ρ_b > ρ_L the body will sink to the bottom of the liquid.

Case II
ρ_b < ρ_L the body will rise above the surface of liquid to such an extent that the weight of the liquid displaced by immersed part of the body becomes equal to the weight of the body.

Case III
ρ_b = ρ_L In this the resultant force acting on the body fully immersed in liquid is zero, The body is at rest anywhere within the liquid.

(B) Hydrodynamics

6. Critical velocity

The critical velocity is that velocity of liquid flow, upto which its flow is streamlined and above which its flow becomes turbulent.
v_c = \(\frac{N_{R} \eta}{\rho D}\)
N_R → Reynold Number.
η → Cofficient of viscosity,
ρ → density of liquid.
D → Diameter of the tube.

7. Physical significance of N_R

• If the value of Reynold number lies between 0 to 2000, the flow of liquid is stream line or laminar.
• For value of N_R above 3000, the flow of liquid is turbulent.
• For value of N_R between 2000 to 3000, the flow of liquid is unstable changing from stream line to turbulent.

8. Equation of continuity

A₁V₁ = A₂ v₂ = v/t
A v = constant

9. Bernoulli’s equation

It is a mathematical expression of the law of conservation of mechanical energy in fluid dynamics. Every point in an ideal fluid flow is associated with three kinds of energy.
(i) Kinetic energy per unit volume at a point.
\(\frac{\mathrm{K.E.}}{\mathrm{Volume}}=\frac{1}{2} \rho \mathrm{v}^{2}\)

(ii) Potential energy per unit volume at a point with respect to an assumed datum is
\(\frac{\text { P.E. }}{\text { Volume }}=\rho g h\)

(iii) Pressure energy per unit volume at a point.
\(\frac{\text { Pressure Energy }}{\text { Volume }}=\mathrm{P}\)

According to conservation of energy flow, the sum total of these energy remains constant along a stream line in a steady flow of an ideal liquid.
\(\frac{1}{2}\) ρv² + ρgh + P = constant ← Bemoullis equation.

10. Torricelli’s theorem

According to this theorem, velocity with which the liquid flows out of on orifice (i.e. a narrow hole) is equal to that which a freely falling body would acquire in falling through a vertical distance equal to the depth of orifice below the free surface of liquid.
v = \(\sqrt{2 \mathrm{gh}}\)

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