Fluid Dynamics and Flow Measurements Study Notes for SSC JE & AE JE Exams

Fluid Dynamics

Dynamics is that branch of mechanics which treats with the motion of bodies and the action of forces in producing or changing their motion.

Flow rate

• Mass flow rate
  
  

• Volume flow rate - Discharge
  More commonly we use volume flow rate also known as discharge. The discharge is generally denoted by the symbol Q.

Continuity Equation

This principle of conservation of mass says matter neither be created nor destroyed. This is applied in fluids of fixed volumes, known as control volumes (or surfaces).

• For any control volume, from principle of conservation of mass

      Mass entering per unit time = Mass leaving per unit time + mass increase     (in control vol per unit time).                

• For steady flow, there is no increase of mass within the control volume,

        Rate of mass entering = Rate of mass leaving

Applying to a stream-tube:
Mass comes in and goes out only through the two ends (it cannot cross the stream tube wall).

For steady flow:

ρ₁A₁u₁ =  ρ₂A₂u₂= Constant= Mass flow rate

This is the continuity equation.

For the incompressible flows: ρ = constant

A₁u₁ =  A₂u₂

1. Some example applications of Continuity:

A liquid is flowing from left to right. By the continuity: 

ρ₁A₁u₁ =  ρ₂A₂u₂

As we are considering a liquid:ρ₁=ρ₂

Q₁ = Q₂ 

A₁u₁=A₂u₂

2. Velocities of fluid exiting a junction: 

Apply continuity equation:

Mass entering the junction = Mass exit out 

ρ₁Q₁= ρ₂Q₂ + ρ₃Q₃

When incompressible:

Q₁ = Q₂ + Q₃

A₁u₁=A₂u₂ + A₃u₃

Energy Equations:

• This is the equation of motion in which the forces due to pressure and gravity are taken into consideration . Following are the common fluid mechanics equations used in fluid dynamics.
• Let, Gravity force F_g, Pressure force F_p, Viscous force F_v, Compressibility force F_c , and Turbulent force F_t.
  F_net =  F_p +F_g + F_v + F_c + F_t

• When fluid flow is incompressible, then: F_c = 0
  ∴ F_net = F_p + F_g + F_v + F_t
  This is called as Reynolds equation of motion.

• When fluid flow is incompressible and turbulence is negligible, then: F_c = 0, F_t = 0
  ∴ F_net = F_p+ F_g + F_v
  This equation is called as Navier-Stokes equation.

• If fluid flow is assumed ideal then, a viscous effect will also be negligible. Then
  F_net = F_g + F_p
  This equation is known as Euler’s equation.

• Euler’s equation can be written as:
• 

Bernoulli’s Equation:

This is based on law of conservation of energy. The assumptions for validity of this law are 

• Flow is steady and irrotational.
• Fluid is ideal (non-viscous).
• Fluid is incompressible.

Statement: It states that in an ideal, steady flow of an incompressible fluid, the total energy at any point of the fluid is constant. The total energy comprising kinetic energy, pressure energy and potential energy or datum energy. The Bernoulli's equation per unit weight is 

Limitation of Bernoulli’s equation:

• Flow is steady.
• Density is constant (incompressible).
• Friction losses are negligible.
• It says the states of two points along a single streamline only, (no conditions for two different streamlines).

The Bernoulli equation is applied along a single streamline joining points 1 and 2:

Total energy head at point 1 = Total energy head at point 2

Note Point: The Bernoulli equation is generally combined with the continuity equation to find velocities and pressures at points in the flow connected by a streamline.

Vortex flow:

This is the flow of rotating mass of fluid or flow of fluid along curved path.

Free vortex flow:

• No external torque or energy required. The fluid rotating by certain energy previously given to them.
• In a free vortex mechanics, total energy flow remains constant. There is no energy interaction between an external source and the flow or no dissipation of external energy.
• Fluid mass rotates to conserve moment of momentum.
• Velocity is inversely proportional to the radius.
• For a free vortex flow:
  vr= constant 
  v= c/r
• At the center (r = 0) of rotation, velocity tends to infinite, that point is called singular point.
  
• The free vortex flow is irrotational, and therefore, also known as the irrotational vortex.
• Bernoulli’s equation is applicable in free vortex flow.
  Examples: a whirlpool in a river, water flow in a sink or a bathtub, flow in centrifugal pump at casing outlet and flow around the circular bend in a pipe.

Forced vortex flow

• To maintain a forced vortex flow, continuous supply of energy or external torque is required.
• All fluid particles rotates at the constant angular velocity ω as a solid body. Hence, a flow of forced vortex is called as a solid body rotation.
• Tangential velocity in forced vortex flow is directly proportional to the radius.
  • v = r ω           
  • ω = Angular velocity. 
  • r = radial distance of fluid particle from the axis of rotation.
• The surface profile is parabolic in forced vortex flow.

Forced vortex flow equation is

• In forced vortex total energy per unit weight increases with an increase in radius.
• Forced vortex is not irrotational, rather it is a rotational flow with constant vorticity 2ω.
  Examples: rotating a vessel containing a liquid with constant angular velocity, flow inside the centrifugal pump.

Note:

In forced vortex flow, the rise of liquid level at the ends is equal to the fall of liquid level at the axis of rotation.

i.e. Rise of liquid at the ends from O−O = Fall of liquid at the centre from O-O.

Thus, X = Y.

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