Mathematical Models for FLUID MECHANICS

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Mathematical Models for FLUID MECHANICS

• P M V Subbarao
• Associate Professor
• Mechanical Engineering Department
• IIT Delhi

Convert Ideas into A Precise Blue Print before
feeling the same....
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A path line is the trace of the path followed by
a selected fluid particle
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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.

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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
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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.

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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.
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Similarly for ay az,
In vector notation this can be written concisely
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x
Conservation laws can be applied to an
infinitesimal element or cube, or may be
integrated over a large control volume.
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Basic Control-Volume Approach
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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 GDEs 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.

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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.

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

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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        .
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• 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 (IIIII) 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.

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The control volume may move as time passes.
III has left CV at time t0dt
I is trying to enter CV at time t0
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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

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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
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The above mentioned change has occurred over a
time dt, therefore Time averaged change in BCM is
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For and infinitesimal time duration

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

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Conservation of Mass

• Let b1, the B mass of the system, m.

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

• Let bV, the B momentum of the system, mV.

The rate of change of momentum for a control mass
should be equal to resultant external force.
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Conservation of Energy

• Let be, 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.
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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
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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.
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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.
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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.
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Vector equation for momentum
Vector momentum equation per unit volume
Body force per unit volume
Gravitational force
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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
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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.

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0
Ideal Fluids.
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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 Newtons 2nd
  Law to a fluid element.

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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.

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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.

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Ideal or Inviscid Flows
Eulers Momentum Equation
X Momentum Equation
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Eulers 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
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Pressure Variation Due to Acceleration
For steady flow along l direction (stream line)
Integration of above equation yields
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Momentum Transfer in A Pump

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

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Pump

• Rotate a cylinder containing fluid at constant
  speed.
• Supply continuously fluid from bottom.
• See What happens?

• Any More Ideas?

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Momentum Principle

• P M V Subbarao
• Associate Professor
• Mechanical Engineering Department
• IIT Delhi

A primary basis for the design of flow devices ..
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Momentum Equation
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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

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Navier-Stokes EquationsDifferential form of
momentum equation
X-component
Y-component
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z-component
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Applications of Momentum Equation
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Generation of Motive Power Through Newtons
Second Law
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Jet Deflected by a Plate or Blade
Consider a jet of gas/steam/water turned through
an angle
CV and CS are for jet so that Fx and Fy are blade
reactions forces on fluid.
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Steady 2 Dimensional Flow
X-component
Y-component
Continuity equation
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Steady 2 Dimensional Invisicid Flow
X-component
Y-component
Continuity equation
Inlet conditions u U v 0
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Pure Impulse Blade
Pressure remains constant along the entire jet.
X-component
Y-component
Continuity equation
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