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 doesn’t 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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9 § Differential Form of Continuity Equation

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14 § Cylindrical Polar Coordinates

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16 § The Stream Function

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18 Example 6.3 Stream Function The velocity component in a steady, incompressible, two dimensional flow field areThe 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. 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 For simplicity, we set C=0 Ψ=0 Ψ≠0Ψ≠0Ψ≠0Ψ≠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 § Equation of Motion

25 § 6.4 Inviscid Flow

26 § Euler’s Equations of Motion

27 § The Bernoulli Equation

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30 § Irrotational Flow

31 § The Velocity Potential

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37 § 6.5 Some Basic, Plane Potential Flows

38 § 6.8 Viscous Flow

39 § Stress - Deformation Relationships

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41 § The Navier–Stokes Equations