# Chapter 6 Flow Analysis Using Differential Methods (Differential Analysis of Fluid Flow) - PowerPoint PPT Presentation

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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.

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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
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Figure 6.9 (p. 287) Components of force acting
on an arbitrary differential area.
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Figure 6.10 (p. 287) Double subscript notation
for stresses.
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Figure 6.11 (p. 288) Surface forces in the x
direction acting on a fluid element.
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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
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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