# Fluid Mechanics

**Stream line flow: -** Flow of a liquid fluid is said to be streamlined if the velocity of a molecule, at any point, coincides with that of the preceding one.

**Laminar flow: -** It is a special case of streamline flow in which velocities of all the molecules on one streamline is same throughout its motion.

**Turbulent flow: -** Whenever the velocity of a fluid is very high or it rushes past an obstacle so that there is a sudden change in its direction of motion, the motion of fluid becomes irregular, forming eddies or whirlpools. This type of motion of fluid is called turbulent flow.

**Rate of flow (Equation of continuity): -**

AV = Constant, A – Area, V – velocity

A1V1=A2V2

Equation of continuity can be considered to be a statement of conservation of mass.

So, V ∝ 1/A

Velocity of flow of liquid varies inversely as the area of cross-section of the opening from where the liquid comes out.

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**(a) Kinetic Energy: -** It is the energy possessed by a liquid by virtue of its velocity.

**(b) Potential Energy: -** It is the energy possessed by a liquid by virtue of its position.

**(c) Pressure Energy: -** It is the energy possessed by a liquid by virtue of its pressure.

**Total energy of a liquid is the sum total of kinetic energy, potential energy and pressure energy.**

**Bernoulli’s Equation: -**

It states that the total energy of a small amount of an incompressible non-viscous liquid flowing without friction from one point to another, in a streamlined flow, remains constant throughout the displacement.

Therefore, Bernoulli’s theorem states that in case of an incompressible, non-viscous fluid, flowing from one point to another in a streamlined flow, the sum total of velocity head, gravitational head and the pressure head is a constant quantity.

**Torricelli’s theorem (velocity of efflux)**

It states that the velocity of efflux of a liquid (V), from an orifice, is equal to the velocity acquired by a body, falling freely (v), from the surface of liquid to the orifice.

**Viscosity: -**Viscosity is the property of fluids by virtue of which they tend to destroy any relative motion between their layers.

**Velocity Gradient: -**Velocity gradient is defined as the rate of change of velocity with respect to distance.

Velocity gradient = dv/dr

**Newton’s law of Viscosity**

In accordance to Newton’s law of viscosity, the viscous drag force depends upon the nature of fluid along with following factors: -

**Unit of η: -**

**Fluidity: -** Reciprocal of coefficient of viscosity of a fluid is called its fluidity.

**Kinematic viscosity: -** Kinematic viscosity of a fluid is defined as the ration between its coefficient of viscosity to the density of fluid.

Kinetic viscosity of a fluid having its dynamic viscosity one poise and density one g cm-3 is said to be 1 stoke.

**Critical velocity (Reynold’s Number)**

Critical velocity (vc) is the maximum velocity of the flow of liquid flowing in a streamlined flow.

𝑣𝑐`= 𝑁𝑅𝜂/𝜌𝐷

Here η is the coefficient of viscosity of liquid, ρ is the density of liquid and D is the diameter of the tube.

Reynold’s Number, 𝑁𝑅 = 𝜌𝑣𝑐𝐷/𝜂

**Stokes Law: -** In accordance to Stokes’ law, force of viscosity F depend upon, force of viscosity,

𝐹=6𝜋𝜂𝑟𝑣

**(a) Effect of temperature on η**:

**(b) Effect of pressure on η**: Co-efficient of viscosity of liquids increases due to an increase in pressure but there is no relation, so far, to explain the effect.

**Change in viscosity of gases: -**

**(a) Effect of temperature: -** Co-efficient of viscosity of a gas at a given temperature is given by,

Here T is the absolute temperature of gas.

**(b) Effect of pressure:** - At low pressure, co-efficient of viscosity of a gas varies directly with pressure.

1. Rate of flow of liquid through a liquid through a capillary tube of radius r and length l,

Pressure exerted by a column of liquid of height h: - 𝑃 = h𝜌𝑔

Here, ρ is the density of liquid.

2. Pressure at a point within the liquid: - 𝑃 = 𝑃0 + h𝜌𝑔

Here, P0 is the atmospheric pressure and the depth of point with respect to free surface of liquid.