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 4, Problem 37P
Consider the following steady, incompressible, two-dimensional velocity field:
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4. A steady, incompressible, and two-dimensional velocity field is given by the following
components in the xy-plane:
Vxu = 2.65 + 3.12x + 5.46y
=
Vy=
=v=0.8+ 5.89x² + 1.48y
=
Calculate the acceleration field (find expressions for acceleration components ax and ay and
calculate the acceleration at the point (x,y) = (-1,3).
2. Consider the two-dimensional time-dependent velocity field u(x, t) = (sint, cost, 0),
in the basis of Cartesian coordinates.
a) Determine the streamlines passing through the point x = 0 at the times t =
0, π/2, π and 3π/2.
b) Determine the paths of fluid particles passing through the point x = 0 at the
same times, to = 0, π/2, 7 and 37/2. Hence, describe their motion.
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c) Find the streakline produced by tracer particles continuously released at the point
xo = 0 and find its position at t = 0, π/2, π and 37/2. Hence describe its motion.
1. For a velocity field described by V = 2x2i − zyk, is the flow two- or threedimensional? Incompressible? 2. For an Eulerian flow field described by u = 2xyt, v = y3x/3, w = 0, find the slope of the streamline passing through the point [2, 4] at t = 2. 3. Find the angle the streamline makes with the x-axis at the point [-1, 0.5] for the velocity field described by V = −xyi + 2y2j
Chapter 4 Solutions
Fluid Mechanics: Fundamentals and Applications
Ch. 4 - What does the word kinematics mean? Explain what...Ch. 4 - Briefly discuss the difference between derivative...Ch. 4 - Consider the following steady, two-dimensional...Ch. 4 - Consider the following steady, two-dimensional...Ch. 4 - -5 A steady, two-dimensional velocity field is...Ch. 4 - Consider steady flow of water through an...Ch. 4 - What is the Eulerian description of fluid motion?...Ch. 4 - Is the Lagrangian method of fluid flow analysis...Ch. 4 - A stationary probe is placed in a fluid flow and...Ch. 4 - A tiny neutrally buoyant electronic pressure probe...
Ch. 4 - Define a steady flow field in the Eulerian...Ch. 4 - Is the Eulerian method of fluid flow analysis more...Ch. 4 - A weather balloon is hunched into the atmosphere...Ch. 4 - A Pilot-stalk probe can often be seen protruding...Ch. 4 - List at least three oiler names for the material...Ch. 4 - Consider steady, incompressible, two-dimensional...Ch. 4 - Converging duct flow is modeled by the steady,...Ch. 4 - A steady, incompressible, two-dimensional velocity...Ch. 4 - A steady, incompressible, two-dimensional velocity...Ch. 4 - For the velocity field of Prob. 4-6, calculate the...Ch. 4 - Consider steady flow of air through the diffuser...Ch. 4 - For the velocity field of Prob. 4-21, calculate...Ch. 4 - A steady, incompressible, two-dimensional (in the...Ch. 4 - The velocity field for a flow is given by...Ch. 4 - Prob. 25CPCh. 4 - What is the definition of a timeline? How can...Ch. 4 - What is the definition of a streamline? What do...Ch. 4 - Prob. 28CPCh. 4 - Consider the visualization of flow over a 15°...Ch. 4 - Consider the visualization of ground vortex flow...Ch. 4 - Consider the visualization of flow over a sphere...Ch. 4 - Prob. 32CPCh. 4 - Consider a cross-sectional slice through an array...Ch. 4 - A bird is flying in a room with a velocity field...Ch. 4 - Conversing duct flow is modeled by the steady,...Ch. 4 - The velocity field of a flow is described by...Ch. 4 - Consider the following steady, incompressible,...Ch. 4 - Consider the steady, incompressible,...Ch. 4 - A steady, incompressible, two-dimensional velocity...Ch. 4 - Prob. 41PCh. 4 - Prob. 42PCh. 4 - The velocity field for a line some in the r plane...Ch. 4 - A very small circular cylinder of radius Rtis...Ch. 4 - Consider the same two concentric cylinders of...Ch. 4 - The velocity held for a line vartex in the r...Ch. 4 - Prob. 47PCh. 4 - Name and briefly describe the four fundamental...Ch. 4 - Prob. 49CPCh. 4 - Prob. 50PCh. 4 - Prob. 51PCh. 4 - Prob. 52PCh. 4 - Prob. 53PCh. 4 - Converging duct flow is modeled by the steady,...Ch. 4 - Converging duct flow is modeled by the steady,...Ch. 4 - Using the results of Prob. 4—57 and the...Ch. 4 - Converging duct flow (Fig. P4—16) is modeled by...Ch. 4 - Prob. 60PCh. 4 - For the velocity field of Prob. 4—60, what...Ch. 4 - For the velocity field of Prob. 4—60, calculate...Ch. 4 - For the velocity field of Prob. 4—60, calculate...Ch. 4 - Prob. 64PCh. 4 - Prob. 65PCh. 4 - Consider steady, incompressible, two-dimensional...Ch. 4 - Prob. 67PCh. 4 - Consider the steady, incompressible,...Ch. 4 - Prob. 69PCh. 4 - Prob. 70PCh. 4 - Prob. 71PCh. 4 - Prob. 72PCh. 4 - Prob. 73PCh. 4 - A cylindrical lank of water rotates in solid-body...Ch. 4 - Prob. 75PCh. 4 - A cylindrical tank of radius rrim= 0.354 m rotates...Ch. 4 - Prob. 77PCh. 4 - Prob. 78PCh. 4 - Prob. 79PCh. 4 - For the Couette flow of Fig. P4—79, calculate the...Ch. 4 - Combine your results from Prob. 4—80 to form the...Ch. 4 - Consider a steady, two-dimensional, incompressible...Ch. 4 - A steady, three-dimensional velocity field is...Ch. 4 - Consider the following steady, three-dimensional...Ch. 4 - Prob. 85PCh. 4 - A steady, three-dimensional velocity field is...Ch. 4 - Briefly explain the purpose of the Reynolds...Ch. 4 - Prob. 88CPCh. 4 - True or false: For each statement, choose whether...Ch. 4 - Consider the integral ddtt2tx2. Solve it two ways:...Ch. 4 - Prob. 91PCh. 4 - Consider the general form of the Reynolds...Ch. 4 - Consider the general form of the Reynolds...Ch. 4 - Prob. 94PCh. 4 - Prob. 95PCh. 4 - Prob. 96PCh. 4 - Prob. 97PCh. 4 - The velocity field for an incompressible flow is...Ch. 4 - Consider fully developed two-dimensional...Ch. 4 - For the two-dimensional Poiseuille flow of Prob....Ch. 4 - Combine your results from Prob. 4—100 to form the...Ch. 4 - Prob. 103PCh. 4 - Prob. 107PCh. 4 - Prob. 108PCh. 4 - Prob. 109PCh. 4 - Prob. 110PCh. 4 - Prob. 112PCh. 4 - Prob. 113PCh. 4 - Prob. 114PCh. 4 - Prob. 116PCh. 4 - Based on your results of Prob. 4—116, discuss the...Ch. 4 - Prob. 118PCh. 4 - In a steady, two-dimensional flow field in the...Ch. 4 - A steady, two-dimensional velocity field in the...Ch. 4 - A velocity field is given by u=5y2,v=3x,w=0 . (Do...Ch. 4 - The actual path traveled by an individual fluid...Ch. 4 - Prob. 123PCh. 4 - Prob. 124PCh. 4 - Prob. 125PCh. 4 - Water is flowing in a 3-cm-diameter garden hose at...Ch. 4 - Prob. 127PCh. 4 - Prob. 128PCh. 4 - Prob. 129PCh. 4 - Prob. 130PCh. 4 - Prob. 131PCh. 4 - An array of arrows indicating the magnitude and...Ch. 4 - Prob. 133PCh. 4 - Prob. 134PCh. 4 - Prob. 135PCh. 4 - A steady, two-dimensional velocity field is given...Ch. 4 - Prob. 137PCh. 4 - Prob. 138PCh. 4 - Prob. 139PCh. 4 - Prob. 140PCh. 4 - Prob. 141P
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- 4. Consider the steady, two-dimensional velocity field given by: u = 2xy-y²; v=x-y². Show that it is a possible 2d incompressible flow. Find the component of acceleration in x direction of a fluid particle at point (x, y) = (1,2)arrow_forwarda. Given the velocity field u=(u,v,w) in Cartesian coordinates with u=2x+y, v=2zt, w=0. i. Find the equations of the corresponding streamlines (Eulerian concept) ii. Find the equations of the corresponding particle paths, i.e., the pathlines (Lagrangian concept). b. Show that the Vu=0 everywhere implies that volumes are conserved, i.e., the volume of red particles at t-0 is the same as at t=t. Hint: Write out what you must prove and use the theorems to get there.arrow_forwardAn unsteady flow has velocity field v = t2 (x2y, xy2) in cartesian plane. 1. is the flow compresible or incompresible? 2. Find a streamfunction of the flow. 3. Find the pattern of instantaneous streamline for the flow.arrow_forward
- The velocity components of a flow field are given by: = 2x² – xy + z², v = x² – 4xy + y², w = 2xy – yz + y² (i) Prove that it is a case of possible steady incompressible fluid flow (ii) Calculate the velocity and acceleration at the point (2,1,3)arrow_forwardUse Eq. dx/u =dy/v=dz/w=dr/V to find and sketch the streamlines of the followingfl ow field:u = Kx; v = -Ky; w = 0, where K is a constant.arrow_forward1. Let V = Vx ((x+ yz)i) be the velocity field for a fluid flow. (a) Verify that this fluid flow is two-dimensional and incompressible. (b) Describe the streamlines for this flow.arrow_forward
- Need correctly.arrow_forwardConsider the incompressible, irrotational, 2D flow. where the stream function is given by: 4 = 424 1) determine the Velocity field and prove the flow is Physically Possible and irrotutional ₂) Calculate and graph (do not sketch) the streamline pattern. 3) find the velocity potential for this flow and graph the lines of constant potential on the same graph as the streamlines.arrow_forwardA steady, incompressible, two-dimensional velocity field is given by the following components in the xy-plane: u = 1.85 + 2.05x + 0.656y ? = 0.754 − 2.18x − 2.05yCalculate the acceleration field (find expressions for acceleration components ax and ay), and calculate the acceleration at the point (x, y) = (−1, 3).arrow_forward
- 1. Find the stream function for a parallel flow of uniform velocity V0 making an angle α with the x-axis. 2. A certain flow field is described by the stream function ψ = xy. (a) Sketch the flow field. (b) Find the x and y velocity components at [0, 0], [1, 1], [∞, 0], and [4, 1]. (c) Find the volume flow rate per unit width flowing between the streamlines passing through points [0, 0] and [1, 1], and points [1, 2] and [5, 3].arrow_forwardPlease answer botharrow_forwardA steady, incompressible, two-dimensional velocity field is given by the following components in the xy-plane: u = 0.205 + 0.97x + 0.851y ? = −0.509 + 0.953x − 0.97y Calculate the acceleration field (find expressions for acceleration components ax and ay) and calculate the acceleration at the point (x, y) = (2, 1.5).arrow_forward
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