Fox and McDonald's Introduction to Fluid Mechanics
9th Edition
ISBN: 9781118912652
Author: Philip J. Pritchard, John W. Mitchell
Publisher: WILEY
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Chapter 5, Problem 68P
Consider a steady, laminar, fully developed incompressible flow between two infinite parallel plates as shown. The flow is due to a pressure gradient applied in the x direction. Given that
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3. Consider a viscous flow between two flat plates separated by a distance H. The bottom plate is
located at y = 0 and moves to the right with speed Uplate. The top plate is located at y = H and
moves to the left with speed 2Uplate. The pressure in the flow is constant. The fluid between the
plates has dynamic viscosity μ. Assume that the flow is steady, incompressible, fully-developed, two
dimensional, and has negligible body forces. Starting from the differential form of the conservation
of mass and momentum equations:
a. Derive an equation for the velocity between the plates in terms of Uplate, H, and y.
b. Determine the average velocity between the plates in terms of Uplate and H.
C. Determine the shear stress acting on the bottom plate in terms of Uplate, H, and µ.
2Uplate
y
fluid
H
Uplate
b) In a flow field, there are two sources having equal strengths located along the x-axis at
x = 0 and x = 3 m, and a sink located on the y -axis at y = 3 m as illustrated in Figure
Q2(b). If the strength for each source is 0.5 m²/s and for the sink is 1.0 m²/s, determine
the magnitude and direction of the fluid velocity at point P (x, y) = (6 m, 0 m).
Y
B
P
3 m
A
3 m
6m
Figure Q2(b)
X
1. For a flow in the xy-plane, the y-component of velocity is given by v = y2 −2x+ 2y. Find a possible x-component for steady, incompressible flow. Is it also valid for unsteady, incompressible flow? Why? 2. The x-component of velocity in a steady, incompressible flow field in the xy-plane is u = A/x. Find the simplest y-component of velocity for this flow field.
Chapter 5 Solutions
Fox and McDonald's Introduction to Fluid Mechanics
Ch. 5 - Which of the following sets of equations represent...Ch. 5 - Which of the following sets of equations represent...Ch. 5 - In an incompressible three-dimensional flow field,...Ch. 5 - In a two-dimensional incompressible flow field,...Ch. 5 - The three components of velocity in a velocity...Ch. 5 - The x component of velocity in a steady,...Ch. 5 - The y component of velocity in a steady...Ch. 5 - The velocity components for an incompressible...Ch. 5 - The radial component of velocity in an...Ch. 5 - A crude approximation for the x component of...
Ch. 5 - A useful approximation for the x component of...Ch. 5 - A useful approximation for the x component of...Ch. 5 - For a flow in the xy plane, the x component of...Ch. 5 - Consider a water stream from a jet of an...Ch. 5 - Which of the following sets of equations represent...Ch. 5 - For an incompressible flow in the r plane, the r...Ch. 5 - A viscous liquid is sheared between two parallel...Ch. 5 - A velocity field in cylindrical coordinates is...Ch. 5 - Determine the family of stream functions that...Ch. 5 - The stream function for a certain incompressible...Ch. 5 - Determine the stream functions for the following...Ch. 5 - Determine the stream function for the steady...Ch. 5 - Prob. 23PCh. 5 - A parabolic velocity profile was used to model...Ch. 5 - A flow field is characterized by the stream...Ch. 5 - A flow field is characterized by the stream...Ch. 5 - Prob. 27PCh. 5 - A flow field is characterized by the stream...Ch. 5 - In a parallel one-dimensional flow in the positive...Ch. 5 - Consider the flow field given by V=xy2i13y3j+xyk....Ch. 5 - Prob. 31PCh. 5 - The velocity field within a laminar boundary layer...Ch. 5 - A velocity field is given by V=10ti10t3j. Show...Ch. 5 - The y component of velocity in a two-dimensional,...Ch. 5 - A 4 m diameter tank is filled with water and then...Ch. 5 - An incompressible liquid with negligible viscosity...Ch. 5 - Sketch the following flow fields and derive...Ch. 5 - Consider the low-speed flow of air between...Ch. 5 - As part of a pollution study, a model...Ch. 5 - As an aircraft flies through a cold front, an...Ch. 5 - Wave flow of an incompressible fluid into a solid...Ch. 5 - A steady, two-dimensional velocity field is given...Ch. 5 - A velocity field is represented by the expression...Ch. 5 - A parabolic approximate velocity profile was used...Ch. 5 - A cubic approximate velocity profile was used in...Ch. 5 - The velocity field for steady inviscid flow from...Ch. 5 - Consider the incompressible flow of a fluid...Ch. 5 - Consider the one-dimensional, incompressible flow...Ch. 5 - Expand (V)V in cylindrical coordinates by direct...Ch. 5 - Determine the velocity potential for (a) a flow...Ch. 5 - Determine whether the following flow fields are...Ch. 5 - The velocity profile for steady flow between...Ch. 5 - Consider the velocity field for flow in a...Ch. 5 - Consider the two-dimensional flow field in which u...Ch. 5 - Consider a flow field represented by the stream...Ch. 5 - Fluid passes through the set of thin, closely...Ch. 5 - A two-dimensional flow field is characterized as u...Ch. 5 - A flow field is represented by the stream function...Ch. 5 - Consider the flow field represented by the stream...Ch. 5 - Consider the flow field represented by the stream...Ch. 5 - Consider the velocity field given by V=Ax2i+Bxyj,...Ch. 5 - Consider again the viscometric flow of Example...Ch. 5 - The velocity field near the core of a tornado can...Ch. 5 - A velocity field is given by V=2i4xjm/s. Determine...Ch. 5 - Consider the pressure-driven flow between...Ch. 5 - Consider a steady, laminar, fully developed,...Ch. 5 - Assume the liquid film in Example 5.9 is not...Ch. 5 - Consider a steady, laminar, fully developed...Ch. 5 - Consider a steady, laminar, fully developed...Ch. 5 - A linear velocity profile was used to model flow...Ch. 5 - A cylinder of radius ri rotates at a speed ...Ch. 5 - The velocity profile for fully developed laminar...Ch. 5 - Assume the liquid film in Example 5.9 is...Ch. 5 - The common thermal polymerase chain reaction (PCR)...Ch. 5 - A tank contains water (20C) at an initial depth y0...Ch. 5 - For a small spherical particle of styrofoam...Ch. 5 - Use Excel to generate the progression to an...
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- A two-dimensional flow field is given by the equations: u = ax + by V = Cx + dy Give one possible combination of constants a, b, c and d (none of which are zero) for which this describes the flow of an incompressible fluid. a = b = C = d=arrow_forward4. Consider a viscous flow inside an annular pipe. The inner surface of the pipe is located at r = R₁ and is stationary. The outer surface is located at r = Ro and moves to the right with speed Upipe. The pressure inside the pipe is constant. The fluid inside the pipe has dynamic viscosity μ. Assume that the flow is steady, incompressible, fully-developed, axisymmetric, non-rotating, and has negligible body forces. Starting from the differential form of the conservation of mass and momentum equations: a. Derive an equation for the velocity inside the pipe in terms of Upipe, R₁, Ro, and r. b. Determine the average velocity inside the pipe in terms of Upipe, R₁, and Ro. C. Determine the shear stress acting on the inner annulus in terms of Upipe, Ri, Ro, and u. Upipe er R₁ fluidarrow_forwardA two-dimensional flow field has an x-component of velocity given in Cartesian coordinates by u = 2x − 3y. (a) Find v, the y-component of velocity, if the flow is incompressible and v = 0 when x = 0. (b) If the flow follows the Bernoulli equation, find an expression for the pressure distribution as a function of x and y, given that the pressure is p0 at the stagnation point.arrow_forward
- 4. Show that v= 3xi+ (t- 6xy)j is a possible fluid velocity for two-dimensional unsteady flow of a homogeneous incompressible fluid. Hence, find the pressure p(x, y,t) at any point of the fluid using Euler's equation of motion given that the body force per unit mass is F = 6 yi+18x yj.arrow_forwardb) An incompressible, viscous fluid is placed between two infinite, parallel plates as shown in Figure Q1(b). The two plates move in opposite directions with constant velocities V, and V2. There is no applied pressure in the x-direction, and gravity acts in the negative y-direction. Assuming laminar flow, derive an expression for the velocity distribution between the plates. h V2 Figure Q1(b)arrow_forward2. In a two dimensional incompressible fluid flow, the velocity components are given by: u=(x-4y) and v=-(y+4x). If possible show that the potential exists and determine its form. Also determine whether the flow is irrotational?arrow_forward
- 4. The viscosity of incompressible viscous fluid is u = 0.96m Pa s. The velocity field is given by: vi = k(x² – x2); v2 = -2kx1x2; V3 = 0| where k = s: At the point with coordinates (1,2, 1)m, for the plane with normal in the direction of ê1: (a) define the excess of the normal compressive force on top of the pressure p. (b) determine the magnitude of the shear stresses.arrow_forwardA viscous incompressible liquid of density p and of dynamic viscosity n is carried upwards against gravity with the aid of moving side walls. This laminar flow is steady and fully developed in z-direction and there is no applied pressure gradient. The coordinate is fixed in the midway between the walls as shown below. 2d g=-gk liquid P, n x=-d x=d I. The velocity profile in z-direction is pg w(x) = (x2 – d) + U. 2n II. To be able to carry a net amount of liquid upwards, the wall velocities need to be greater than pgd²/(2n). III. If one of the walls stops, increasing the speed of the other wall to 3/2U would carry the same amount of liquid upwards. Which of the above statements are true?arrow_forwardConsider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D or radius R = D/2. We ignore the effects of gravity. A constant pressure gradient ∂P/∂x is applied in the x-direction, where x1 and x2 are two arbitrary locations along the x-axis, and P1 and P2 are the pressures at those two locations. Note that we adopt a modified cylindrical coordinate system here with x instead of z for the axial component, namely, (r, ?, x) and (ur, u?, u). Derive an expression for the velocity field inside the pipe and estimate the viscous shear force per unit surface area acting on the pipe wall.arrow_forward
- 2- Consider the flow in rectangular coordinates given by v=i(X3Y)+j(2yx2z). Based on the continuity equation, verify that the fluid is compressible.arrow_forwardIn a certain steady, two-dimensional flow field the fluid den- sity varies linearly with respect to the coordinate x; that is, p = Ax where A is a constant. If the x component of velocity u is given by the equation u = y, determine an expression for v.arrow_forwardFluid Mechanics There is an incompressible Newtoniel fluid between the parallel plates whose bottom plate is fixed and the top plate is moved at V speed.The velocity field is u = V (y / h) i, where V is the velocity of the moving plate and h is the distance between plates. Find the stresses in such a flow. What can the momentum equation tell us about this flow?arrow_forward
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