fluid_mechanics

6DD ifferential
Analysis of
Fluid Flow
CHAPTER OPENING PHOTO: Flow past an inclined plate: The streamlines of a viscous fluid flowing slowly
past a two-dimensional object placed between two closely spaced plates 1a Hele-Shaw cell2 approximate
inviscid, irrotational 1potential2 flow. 1Dye in water between glass plates spaced 1 mm apart.2 1Photography
courtesy of D. H. Peregrine.2
Learning Objectives
After completing this chapter, you should be able to:
■ determine various kinematic elements of the flow given the velocity field.
■ explain the conditions necessary for a velocity field to satisfy the continuity
equation.
■ apply the concepts of stream function and velocity potential.
■ characterize simple potential flow fields.
■ analyze certain types of flows using the Navier–Stokes equations.
In the previous chapter attention is focused on the use of finite control volumes for the solution
of a variety of fluid mechanics problems. This approach is very practical and useful, since it does
not generally require a detailed knowledge of the pressure and velocity variations within the control
volume. Typically, we found that only conditions on the surface of the control volume were needed,
and thus problems could be solved without a detailed knowledge of the flow field. Unfortunately,
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. 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. In these circumstances we need to develop relationships
that apply at a point, or at least in a very small infinitesimal region within a given flow field. This
approach, which involves an infinitesimal control volume, as distinguished from a finite control
volume, is commonly referred to as differential analysis, since 1as we will soon discover2 the
governing equations are differential equations.
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