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Chapter 6 Flow Analysis Using Differential Methods (Differential Analysis of Fluid Flow)

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Chapter 6 Flow Analysis Using Differential Methods (Differential Analysis of Fluid Flow) * * * * * 6.5 Some Basic, Plane Potential Flows * 6.8 Viscous Flow ... – PowerPoint PPT presentation

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Title: Chapter 6 Flow Analysis Using Differential Methods (Differential Analysis of Fluid Flow)


1
Chapter 6 Flow Analysis Using Differential
Methods (Differential Analysis of Fluid Flow)
2
  • In the previous chapter--
  • Focused on the use of finite control volume for
    the solution of a variety of fluid mechanics
    problems.
  • The approach is very practical and useful since
    it doesnt generally require a detailed knowledge
    of the pressure and velocity variations within
    the control volume.
  • Typically, only conditions on the surface of the
    control volume entered the problem.
  • There are many situations that arise in which the
    details of the flow are important and the finite
    control volume approach will not yield the
    desired information

3
  • For example --
  • We may need to know how the velocity varies over
    the cross section of a pipe, or how the pressure
    and shear stress vary along the surface of an
    airplane wing.
  • ? we need to develop relationship that apply at a
    point,
  • or at least in a very small region (
    infinitesimal volume)
  • within a given flow field.
  • ? involve infinitesimal control volume (instead
    of finite
  • control volume)
  • ? differential analysis (the governing equations
    are
  • differential equation)

4
  • In this chapter
  • (1) We will provide an introduction to the
    differential equation that describe (in detail)
    the motion of fluids.
  • (2) These equation are rather complicated,
    partial differential equations, that cannot be
    solved exactly except in a few cases.
  • (3) Although differential analysis has the
    potential for supplying very detailed information
    about flow fields, the information is not easily
    extracted.
  • (4) Nevertheless, this approach provides a
    fundamental basis for the study of fluid
    mechanics.
  • (5) We do not want to be too discouraging at this
    point, since there are some exact solutions for
    laminar flow that can be obtained, and these have
    proved to very useful.

5
  • (6) By making some simplifying assumptions, many
    other analytical solutions can be obtained.
  • for example , µ? small? 0
    neglected
  • ? inviscid
    flow.
  • (7) For certain types of flows, the flow field
    can be conceptually divided into two regions
  • (a) A very thin region near the boundaries
    of the system in which viscous effects are
    important.
  • (b) A region away from the boundaries in
    which the flow is essentially inviscid.
  • (8) By making certain assumptions about the
    behavior of the fluid in the thin layer near the
    boundaries, and
  • using the assumption of inviscid flow
    outside this layer, a large class of problems can
    be solved using differential analysis .
  • the boundary problem is discussed in chapter 9.
  • Computational fluid dynamics (CFD) ? to solve
    differential eq.

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6.2.1 Differential Form of Continuity Equation
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6.2.2 Cylindrical Polar Coordinates
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6.2.3 The Stream Function
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Example 6.3 Stream Function
  • The velocity component in a steady,
    incompressible, two dimensional flow field are
  • Determine the corresponding stream function
    and show on a sketch several streamlines.
    Indicate the direction of glow along the
    streamlines.

19
Example 6.3 Solution
From the definition of the stream function
?0
For simplicity, we set C0
??0
20
6.3 Conservation of Linear Momentum
21
Figure 6.9 (p. 287) Components of force acting
on an arbitrary differential area.
22
Figure 6.10 (p. 287) Double subscript notation
for stresses.
23
Figure 6.11 (p. 288) Surface forces in the x
direction acting on a fluid element.
24
6.3.2 Equation of Motion
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6.4 Inviscid Flow
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6.4.1 Eulers Equations of Motion
27
6.4.2 The Bernoulli Equation
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6.4.3 Irrotational Flow
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6.4.5 The Velocity Potential
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6.5 Some Basic, Plane Potential Flows
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6.8 Viscous Flow
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6.8.1 Stress - Deformation Relationships
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6.8.2 The NavierStokes Equations
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