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## Mathematical Models for FLUID MECHANICS

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### Mathematical Models for FLUID MECHANICS

### Momentum Principle

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Convert Ideas into A Precise Blue Print before

feeling the same....

A path line is the trace of the path followed by a selected fluid particle

Few things to know about streamlines

- At all points the direction of the streamline is the direction of the fluid velocity: this is how they are defined.
- Close to the wall the velocity is parallel to the wall so the streamline is also parallel to the wall.
- It is also important to recognize that the position of streamlines can change with time - this is the case in unsteady flow.
- In steady flow, the position of streamlines does not change
- Because the fluid is moving in the same direction as the streamlines, fluid can not cross a streamline.
- Streamlines can not cross each other.
- If they were to cross this would indicate two different velocities at the same point.
- This is not physically possible.
- The above point implies that any particles of fluid starting on one streamline will stay on that same streamline throughout the fluid.

A useful technique in fluid flow analysis is to consider only a part of the total fluid in isolation from the rest. This can be done by imagining a tubular surface formed by streamlines along which the fluid flows. This tubular surface is known as a streamtube.

A Streamtube

A two dimensional version of the streamtube

The "walls" of a streamtube are made of streamlines.

As we have seen above, fluid cannot flow across a streamline, so fluid cannot cross a streamtube wall. The streamtube can often be viewed as a solid walled pipe.

A streamtube is not a pipe - it differs in unsteady flow as the walls will move with time. And it differs because the "wall" is moving with the fluid

Fluid Kinematics

- The acceleration of a fluid particle is the rate of change of its velocity.
- In the Lagrangian approach the velocity of a fluid particle is a function of time only since we have described its motion in terms of its position vector.

In the Eulerian approach the velocity is a function of both space and time; consequently,

x,y,z are f(t) since we must follow the total derivative approach in evaluating du/dt.

In vector notation this can be written concisely

Conservation laws can be applied to an infinitesimal element or cube, or may be integrated over a large control volume.

Control Volume

- In fluid mechanics we are usually interested in a region of space, i.e, control volume and not particular systems.
- Therefore, we need to transform GDE’s from a system to a control volume.
- This is accomplished through the use of Reynolds Transport Theorem.
- Actually derived in thermodynamics for CV forms of continuity and 1st and 2nd laws.

Flowing Fluid Through A CV

- A typical control volume for flow in an funnel-shaped pipe is bounded by the pipe wall and the broken lines.
- At time t0, all the fluid (control mass) is inside the control volume.

The fluid that was in the control volume at time t0 will be seen at time t0 +dt as: .

The control volume at time t0 +dt .

The control mass at time t0 +dt .

The differences between the fluid (control mass) and the control volume at time t0 +dt .

I

- Consider a system and a control volume (C.V.) as follows:
- the system occupies region I and C.V. (region II) at time t0.
- Fluid particles of region – I are trying to enter C.V. (II) at time t0.

III

II

- the same system occupies regions (II+III) at t0 + dt
- Fluid particles of I will enter CV-II in a time dt.
- Few more fluid particles which belong to CV – II at t0 will occupy III at time t0 + dt.

II

At time t0+dt

II

I

At time t0

The control volume may move as time passes.

III has left CV at time t0+dt

I is trying to enter CV at time t0

Reynolds' Transport Theorem

- Consider a fluid scalar property b which is the amount of this property per unit mass of fluid.
- For example, b might be a thermodynamic property, such as the internal energy unit mass, or the electric charge per unit mass of fluid.
- The laws of physics are expressed as applying to a fixed mass of material.
- But most of the real devices are control volumes.
- The total amount of the property b inside the material volume M , designated by B, may be found by integrating the property per unit volume, M ,over the material volume :

Conservation of B

- total rate of change of any extensive property B of a system(C.M.) occupying a control volume C.V. at time t is equal to the sum of
- a) the temporal rate of change of B within the C.V.
- b) the net flux of B through the control surface C.S. that surrounds the C.V.
- The change of property B of system (C.M.) during Dt is

add and subtract

The above mentioned change has occurred over a time dt, therefore

Time averaged change in BCMis

For and infinitesimal time duration

- The rate of change of property B of the system.

Conservation of Mass

- Let b=1, the B = mass of the system, m.

The rate of change of mass in a control mass should be zero.

Conservation of Momentum

- Let b=V, the B = momentum of the system, mV.

The rate of change of momentum for a control mass should be equal

to resultant external force.

Conservation of Energy

- Let b=e, the B = Energy of the system, mV.

The rate of change of energy of a control mass should be equal

to difference of work and heat transfers.

Applications of Momentum Analysis

This is a vector equation and will have three components in x, y and z

Directions.

X – component of momentum equation:

X – component of momentum equation:

Y – component of momentum equation:

Z – component of momentum equation:

For a fluid, which is static or moving with uniform velocity, the

Resultant forces in all directions should be individually equal to zero.

X – component of momentum equation:

Y – component of momentum equation:

Z – component of momentum equation:

For a fluid, which is static or moving with uniform velocity, the

Resultant forces in all directions should be individually equal to zero.

X – component of momentum equation:

Y – component of momentum equation:

Z – component of momentum equation:

For a fluid, which is static or moving with uniform velocity, the

Resultant forces in all directions should be individually equal to zero.

Vector momentum equation per unit volume:

Body force per unit volume:

Gravitational force:

Electrostatic Precipitators

Electric body force: Lorentz force density

The total electrical force acting on a group of free charges (charged ash particles) . Supporting an applied volumetric charge density.

Where

= Volumetric charge density

= Local electric field

= Local Magnetic flux density field

= Current density

Electric Body Force

- This is also called electrical force density.
- This represents the body force density on a ponderable medium.
- The Coulomb force on the ions becomes an electrical body force on gaseous medium.
- This ion-drag effect on the fluid is called as electrohydrodynamic body force.

Ideal Fluids….

Pressure Variation in Flowing Fluids

- For fluids in motion, the pressure variation is no longer hydrostatic and is determined from and is determined from application of Newton’s 2nd Law to a fluid element.

Various Forces in A Flow field

- For fluids in motion, various forces are important:
- Inertia Force per unit volume :

- Body Force:

- Hydrostatic Surface Force:

- Viscous Surface Force:

- Relative magnitudes of Inertial Forces and Viscous Surface Force are very important in design of basic fluid devices.

Comparison of Magnitudes of Inertia Force and Viscous Force

- Internal vs. External Flows
- Internal flows = completely wall bounded;
- Both viscous and Inertial Forces are important.
- External flows = unbounded; i.e., at some distance from body or wall flow is uniform.
- External Flow exhibits flow-field regions such that both inviscid and viscous analysis can be used depending on the body shape.

Euler’s Equation for One Dimensional Flow

Define an exclusive direction along the

axis of the pipe and corresponding unit direction vector

Along a path of zero acceleration the pressure variation is hydrostatic

Pressure Variation Due to Acceleration

For steady flow along l – direction (stream line)

Integration of above equation yields

Momentum Transfer in A Pump

- Shaft power Disc Power Fluid Power.
- Flow Machines & Non Flow Machines.
- Compressible fluids & Incompressible Fluids.
- Rotary Machines & Reciprocating Machines.

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

A primary basis for the design of flow devices ..

Applications of of the Momentum Equation

Initial Setup and Signs

- 1. Jet deflected by a plate or a vane
- 2. Flow through a nozzle
- 3. Forces on bends
- 4. Problems involving non-uniform velocity distribution
- 5. Motion of a rocket
- 6. Force on rectangular sluice gate
- 7. Water hammer

Consider a jet of gas/steam/water turned through an angle

Jet Deflected by a Plate or BladeCV and CS are

for jet so that Fx

and Fy are blade

reactions forces on fluid.

Steady 2 Dimensional Invisicid Flow

X-component:

Y-component:

Continuity equation:

Inlet conditions : u = U & v = 0

Pure Impulse Blade

Pressure remains constant along the entire jet.

X-component:

Y-component:

Continuity equation:

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