Fluid Mechanics - 2

A stagnation point is a point in a flow field where the local velocity of the fluid is zero. The Bernoulli equation shows that the static pressure is highest when the velocity is zero and hence static pressure is at its maximum value at stagnation points. This static pressure is called the stagnation pressure.

Total no of π terms = m - n

Here, m = total parameter = 5

n = fluid property, flow property and geometric property (ν, v, L) = 3

∴ no of π – terms = 5 – 3 = 2

In fluid dynamics, turbulent flow is characterized by the irregular movement of particles of the fluid. In contrast to laminar flow the fluid does not flow in parallel layers, the lateral mixing is very high, and there is a disruption between the layers. In turbulent flow the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction.

For three dimensional steady incompressible flow the continuity equation is

Which is nothing but Δ²ϕ  = 0. The “Laplace equation“. Thus any function that satisfied Laplace equation is a possible case of fluid flow.

The flow in a circular pipe is

Laminar Flow: Re ≤ 2300

Transitional flow: 2300 ≤ Re ≤ 4000

Turbulent flow: Re ≥ 4000

The dynamic viscosity of mercury (1.52 cp) is greater than water (0.894 cp). The kinematic viscosity is the dynamic viscosity divided by the density. Mercury is a lot denser than water, so its kinematic viscosity is lower than the kinematic viscosity of water.

Dynamic viscosity of water at 20°C

= 1.005 centipoise

Dynamic viscosity of mercury at 20°C

= 1.6 centipoise

Hence shear stress is proportional to velocity gradient.

Maximum Reynolds’s number for laminar flow = 2000

We know that, RE=

∴Maximumvelocity

The point ' M ' at which the line of action of the new buoyant force intersects the original vertical through the CG of the body, is called the metacentre. It is a point about which a floating body starts oscillating, when given a small angular displacement.

GM > 0 (M is above G)                                     Stable equilibrium

GM = 0 (M coinciding with G)                           Neutral equilibrium

GM < 0 (M is below G)                                     Unstable equilibrium

Density (ρ)=

Manometric head is the head against which head is required to be produced by the pump to deliver water to the destination. Manometric head is higher than the sum of suction and delivery heads because it accounts for head losses due to friction.

Flow separation occurs when the pressure gradient is positive and velocity gradient is negative.

Displacement thickness,

Momentum thickness,

Energy thickness,

δ > δ_E > δ_θ 

Bernoulli's equation states that the summation of pressure head, kinetic head and datum/potential head is constant for steady, incompressible, irrotational and non-viscous flow. In other words an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy i.e. the total energy of a flowing system remain constant until external force is applied. So Bernoulli’s equation refers to conservation of energy.

All flow measuring devices like Venturimeter, Orifice meter, Pitot tube meter works on the Bernoulli’s theorem.

Dynamic viscosity (μ)

μ = Pa – S

Surface tension is responsible for spherical shape of droplets. Surface tension for mercury is 0.485 N/m and that for water is 0.072 N/m.

Stream function Ψ = x² – y²

Φ = -2xy + C

An accumulator is a device to store sufficient energy in case of machines which work intermittently to supplement the discharge from the normal source.

Efficiency of power transmission is given by

For maximum efficiency

We get

η_max = 66.66%