Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 9, Problem 88P
Consider steady, incompressible, parallel, laminar flow falling between two infinite vertical walls (Fig. 9-88). The distance between the walls is h, and gravity acts in the negative z-direction (downward in the figure). There is no applied (forced) pressure driving the flow-the fluid falls by gravity alone. The pressure is constant everywhere in the flow field. Calculate the velocity field and sketch the velocity profile using appropriate nondimensionalized variable.
FIGURE P9-88
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Consider steady, incompressible, parallel, laminar flow of a viscous fluid falling between two infinite vertical walls. The distance between the walls is h, and gravity acts in the negative z-direction (downward in the figure). There is no applied (forced) pressure driving the flow—the fluid falls by gravity alone. The pressure is constant everywhere in the flow field. Calculate the velocity field and sketch the velocity profile using appropriate nondimensionalized variables.
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EXAMPLE 6-1
Momentum-Flux Correction Factor
for Laminar Pipe Flow
CV
Vavg
Consider laminar flow through a very long straight section of round pipe. It
is shown in Chap. 8 that the velocity profile through a cross-sectional area of
the pipe is parabolic (Fig. 6-15), with the axial velocity component given by
r4
V
R
V = 2V
1
avg
R2
(1)
where R is the radius of the inner wall of the pipe and Vavg is the average
velocity. Calculate the momentum-flux correction factor through a cross sec-
tion of the pipe for the case in which the pipe flow represents an outlet of
the control volume, as sketched in Fig. 6-15.
Assumptions 1 The flow is incompressible and steady. 2 The control volume
slices through the pipe normal to the pipe axis, as sketched in Fig. 6-15.
Analysis We substitute the given velocity profile for V in Eq. 6-24 and inte-
grate, noting that dA, = 2ar dr,
FIGURE 6–15
%3D
Velocity…
A viscous incompressible Newtonian fluid is contained between two fixed parallel plates
inclined at an angle 0, and the flow is driven by both constant pressure gradient
E = constant) and gravity. The distance between the two plates is 2H and the chosen
system of coordinates is shown in the figure. Assuming steady, 2D, and parallel flow
(v = w = 0) and using differential analysis: (a) Show that the flow is fully developed
using continuity equation; and (b) Find the velocity profile u(y) in terms of ,H,P,g,0,H
de
using Navier-Stokes equations with appropriate boundary conditions.
211
Chapter 9 Solutions
Fluid Mechanics: Fundamentals and Applications
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