MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 5: EQUATIONS OF MOTION OF VISCOUS FLOWS

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MECH 221 FLUID MECHANICS(Fall 06/07)Chapter 5
EQUATIONS OF MOTION OF VISCOUS FLOWS

• Instructor Professor C. T. HSU

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5.1 Flow Fields and Gradients

• The continuum assumption allows the treatment of
  fluid properties as fields, scalar vector or
  tensor, which are function of space (r) and time
  (t)
• Scalar fields density
• pressure
• temperature
• Vector fields velocity
• vorticity
• Tensor fields total stress
• viscous stress
• strain rate

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5.1 Flow Fields and Gradients

• Total change of a scalar field, , due to change
  in space only
• For the position vector given by
  , the change in is only cause by the
  change in r described by

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5.1 Flow Fields and Gradients

• The total derivative is then given by
• where
• is the gradient of . The gradient is a
  vector along the direction where the magnitude of
  has a maximum.

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5.1 Flow Fields and Gradients

• Consider P Q to be 2 points on a surface where
  . These points are chosen so
  that Q is a small distance from P. Then dr is
  tangential to the surface.
• Now lets move from P to Q. The change in
  
• is then given by,
• since the 2 points are on the surface with the
  same C.

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5.1 Flow Fields and Gradients

• Therefore, is perpendicular to dr from P.
• Since dr may be in any direction from P, as long
  as it is tangential to the surface , we
  conclude that has to be in normal to the
  surface .

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5.1 Flow Fields and Gradients

• Example
• For unsteady flows where may change with
  time, recall that the total derivative is

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5.2 Conservation of Mass

• We recall in Chapter 3 that the conservation of
  mass can be understood more easily in the
  Lagrangian frame. It states that the total mass m
  in control volume V has to be conserved if the
  control volume deformed with the flow to confine
  the same fluid particles.

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5.2 Conservation of Mass

• We now extend further to include the cases when
  there is a mass source in the control volume. If
• represents the rate of mass source per unit
  volume, the mass balance then read

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5.2 Conservation of Mass

• Above equation is the integral form of the mass
  conservation in the Eulerian description. Note
  that the control volume V can be either fixed or
  varying.
• The first term on the left hand side is the
  contribution caused by the density change in V
  and the second term caused by the mass flux enter
  the surface that define the control volume. The
  term on the right hand side then represents the
  rate of mass created or annihilated in V

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5.2 Conservation of Mass

• To obtain the differential form, we now employ
  the divergence theorem to the second term on the
  left of conservation of mass equation to give

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5.2 Conservation of Mass

• Again, as V(t)?0 the integrand is independent of
  V and therefore,
• which is the differential form of equation for
  mass conservation.
• Note that in the above equation, all terms are in
  rate of mass change per unit volume.

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5.2 Conservation of Mass

• For flow fields without mass sources, the
  integral and differential forms of conservation
  of mass equation reduce to
• and
• respectively, which were given in Chapter 3

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5.2.1 Derivation of the Differential Equation
in Eulerian frame and Cartesian Coordinate

• Consider the fixed control volume as shown below

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5.2.1 Derivation of the Differential Equation
in Eulerian frame and Cartesian Coordinate

• Net mass leaving the control volume/time

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5.2.1 Derivation of the Differential Equation
in Eulerian frame and Cartesian Coordinate

• Net mass increase in the control volume/time
• Conservation of mass states that the net mass
  entering the control volume/unit time is equal to
  the rate of increase of mass in the differential
  control volume

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5.2.1 Derivation of the Differential Equation
in Eulerian frame and Cartesian Coordinate
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5.2.2 Special Cases

• Steady flow,
• Incompressible flow,
• Cartesian
• Polar
• Spherical
• 2-D flow,

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5.3 Conservation of Momentum (Navier-Stokes
Equations)

• In Chapter 3, for inviscid flows, only pressure
  forces act on the control volume V since the
  viscous forces (stress) were neglected and the
  resultant equations are the Eulers equations.
  The equations for conservation of momentum for
  inviscid flows were derived based on Newtons
  second law in the Lagrangian form.

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5.3 Conservation of Momentum (Navier-Stokes
Equations)

• Here we should include the viscous stresses to
  derive the momentum conservation equations.
• With the viscous stress, the total stress on the
  fluid is the sum of pressure stress(
  , here the negative sign implies that tension is
  positive) and viscous stress ( ), and is
  described by the stress tensor given by

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5.3 Conservation of Momentum (Navier-Stokes
Equations)

• Here, we generalize the body force (b) due to all
  types of far field forces. They may include those
  due to gravity , electromagnetic force,
  etc.
• As a result, the total force on the control
  volume in a Lagrangian frame is given by

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5.3 Conservation of Momentum (Navier-Stokes
Equations)

• The Newtons second law then is stated as
• Hence, we have

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5.3 Conservation of Momentum (Navier-Stokes
Equations)

• By the substitution of the total stress into the
  above equation, we have
• which is integral form of the momentum
  equation.

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5.3 Conservation of Momentum (Navier-Stokes
Equations)

• For the differential form, we now apply the
  divergence theorem to the surface integrals to
  reach
• Hence, V?0, the integrands are independent of V.
  Therefore,
• which are the momentum equations in differential
  form for viscous flows. These equations are also
  named as the Navier-Stokes equations.

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5.3 Conservation of Momentum (Navier-Stokes
Equations)

• For the incompressible fluids where
  constant.
• If the variation in viscosity is negligible
  (Newtonian fluids), the continuity equation
  becomes , then the shear stress tensor
  reduces to .

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5.3 Conservation of Momentum (Navier-Stokes
Equations)

• The substitution of the viscous stress into the
  momentum equations leads to
• where is the Laplacian operator
  which in a Cartesian coordinate system reads

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5.3 Conservation of Momentum (Navier-Stokes
Equations)

• For inviscid flow where , the above
  equation reduces to the Eulers equation given in
  Chapter 3 where the body force is also taken the
  form due to gravity.