MECH 221 FLUID MECHANICS (Fall 06/07) Tutorial 5

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MECH 221 FLUID MECHANICS(Fall 06/07)Tutorial 5
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Outline

1. Equations of motion for inviscid flow
2. Conservation of mass
3. Conservation of momentum
4. Bernoulli Equation
5. Bernoulli equation for steady flow
6. Static, dynamic, stagnation and total pressure
7. Example

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1. Equations of Motion for Inviscid Flow

• Conservation of Mass
• Conservation of Momentum

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1.1. Conservation of Mass

• Mass in fluid flows must conserve. The total mass
  in V(t) is given by
• Therefore, the conservation of mass requires that
• dm/dt 0.
• where the Leibniz rule was invoked.

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1.1. Conservation of Mass

• Hence
• This is the Integral Form of mass conservation
  equation.

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1.1. Conservation of Mass

• As V(t)?0, the integrand is independent of V(t)
  and therefore,
• This is the Differential Form of mass
  conservation and also called as continuity
  equation.

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1.2. Conservation of Momentum

• The Newtons second law,
• is Lagrangian in a description of momentum
  conservation. For motion of fluid particles that
  have no rotation, the flow is termed
  irrotational. An irrotational flow does not
  subject to shear force, i.e., pressure force
  only. Because the shear force is only caused by
  fluid viscosity, the irrotational flow is also
  called as inviscid flow

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1.2. Conservation of Momentum

• For fluid subjecting to earth gravitational
  acceleration, the net force on fluids in the
  control volume V enclosed by a control surface S
  is
• where s is out-normal to S from V and the
  divergence theorem is applied for the second
  equality.
• This force applied on the fluid body will leads
  to the acceleration which is described as the
  rate of change in momentum.

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1.2. Conservation of Momentum

• where the Leibniz rule was invoked.

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1.2. Conservation of Momentum

• Hence
• This is the Integral Form of momentum
  conservation equation.

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1.2. Conservation of Momentum

• As V?0, the integrands are independent of V.
  Therefore,
• This is the Differential Form of momentum
  conservation equation for inviscid flows.

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1.2. Conservation of Momentum

• By invoking the continuity equation,
• The momentum equation can take the following
  alternative form
• which is commonly referred to as Eulers
  equation of motion.

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2.1. Bernoulli Equation for Steady Flows

• From differential form of the momentum
  conservation equation
• g-gVz
• By vector identity,
• Therefore, we get,

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2.1. Bernoulli Equation for Steady Flows

• Assumption,
• Steady flow
• v and t are independent
• Irrotational flow
• Vxv0

0 (Steady flow)
0 (irrotational flow)
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2.1. Bernoulli Equation for Steady Flows

• Finally, we can get,
• Or
• where vmagnitude of velocity vector v,
• i.e. vv(u2v2w2)

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2.1. Bernoulli Equation for Steady Flows

• Since, for dr in any direction,
  we have
• For anywhere of irrotational fluids
• For anywhere of incompressible fluids

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2.1. Bernoulli Equation for Steady Flows

• Bernoulli Equation in different form
• Energy density
• Total head (H)

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2.2. Static, Dynamic, Stagnation and Total
Pressure

• Consider the Bernoulli equation,
• The static pressure ps is defined as the pressure
  associated with the gravitational force when the
  fluid is not in motion. If the atmospheric
  pressure is used as the reference for a gage
  pressure at z0.

(for incompressible fluid)

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2.2. Static, Dynamic, Stagnation and Total
Pressure

• Then we have as also from chapter
  2.
• The dynamic pressure pd is then the pressure
  deviates from the static pressure, i.e., p
  pdps.
• The substitution of p pdps. into the
  Bernoulli equation gives

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2.2. Static, Dynamic, Stagnation and Total
Pressure

• The maximum dynamic pressure occurs at the
  stagnation point where v0 and this maximum
  pressure is called as the stagnation pressure p0.
  Hence,
• The total pressure pT is then the sum of the
  stagnation pressure and the static pressure,
  i.e., pT p0 - ?gz. For z -h, the static
  pressure is ?gh and the total pressure is p0
  ?gh.

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2.3.1. Example (1)

• Determine the flowrate through the pipe.

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2.3.1. Example (1)

• Procedure
• Choose the reference point
• From the Bernoulli equation
• P, V, Z all are unknowns
• For same horizontal level, Z1Z2
• V V(P1, P2)
• From the balance of static pressure
• P ?gh
• ?h is given, ?m, ?water are known
• V V(?h, ?m, ?water)
• Q AV pD2V/4

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2.3.1. Example (1)

• From the Bernoulli equation,

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2.3.1. Example (1)

• From the balance of static pressure,

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2.3.1. Example (1)

• Volume flow rate (Q),

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2.3.2. Example (2)

• A conical plug is used to regulate the air flow
  from the pipe. The air leaves the edge of the
  cone with a uniform thickness of 0.02m. If
  viscous effects are negligible and the flowrate
  is 0.05m3/s, determine the pressure within the
  pipe.

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2.3.2. Example (2)

• Procedure
• Choose the reference point
• From the Bernoulli equation
• P, V, Z all are unknowns
• For same horizontal level, Z1Z2
• Flowrate conservation
• QAV

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2.3.2. Example (2)

• From the Bernoulli equation,

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2.3.2. Example (2)

• From flowrate conservation,

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2.3.2. Example (2)

• Sub. into the Bernoulli equation,

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The End