Explanation Assume the fluid element at any point P ( x . y . z ) has velocity components u along the x direction and v along y direction. Assume any two line segments P A of length δ x parallel to x axis and P B of length δ y parallel to y axis. Write the expression for the velocity at A along y direction. v A = ( v + ∂ v ∂ x ∂ x ) ...... (I) Here, the velocity component along y direction is v . Write the expression for the velocity at B along x direction. u A = ( u + ∂ u ∂ y ∂ y ) ...... (II) Here, the velocity component along x direction is u . The velocities at P and A along the y direction are different so there will be an angular velocity developed for the linear element P A . Similarly, the velocities at P and B in the x direction are different. Thus, there will be an angular velocity developed for the linear element P B . During a time interval of d t , the element P A and P B will have moved relative to P to new positions P A ′ and P B ′ . Write the expression for the angular velocity of the element P A about z axis. ω P A = lim δ t → 0 α a δ t = lim δ t → 0 ( ( v + ∂ v ∂ x δ x ) − v ) δ t δ x δ t ω P A = ∂ v ∂ x rad / s Here, the angular velocity of P A element is ω P A

Fluid Mechanics Fundamentals And Applications

3rd Edition

ISBN: 9780073380322

Author: Yunus Cengel, John Cimbala

Publisher: MCGRAW-HILL HIGHER EDUCATION

See similar textbooks

Concept explainers

Fluid Kinematics

The branch of science that deals with objects which are either in motion or stationary are studied under the branch of classical mechanics. Fluid mechanics is a subcategory of classical mechanics that deals with the behavior of fluids at rest (fluid stat…

Videos

Question

Chapter 4, Problem 69P

To determine

The magnitude of the rate of rotation (angular velocity) about point P in the xy plane is given by ωz=12(∂v∂x−∂u∂y).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4, Problem 68P

Chapter 4, Problem 70P

Students have asked these similar questions

Given answers to be: i) 14.65 kN; 6.16 kN; 8.46 kN ii) 8.63 kN; 9.88 kN iii) Bearing 6315 for B1 & B2, or Bearing 6215 for B1

(b) A steel 'hot rolled structural hollow section' column of length 5.75 m, has the cross-section shown in Figure Q.5(b) and supports a load of 750 kN. During service, it is subjected to axial compression loading where one end of the column is effectively restrained in position and direction (fixed) and the other is effectively held in position but not in direction (pinned). i) Given that the steel has a design strength of 275 MN/m², determine the load factor for the structural member based upon the BS5950 design approach using Datasheet Q.5(b). [11] ii) Determine the axial load that can be supported by the column using the Rankine-Gordon formula, given that the yield strength of the material is 280 MN/m² and the constant *a* is 1/30000. [6] 300 600 2-300 mm wide x 5 mm thick plates. Figure Q.5(b) L=5.75m Pinned Fixed

Help ارجو مساعدتي في حل هذا السؤال

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 steady flow of water through an...Ch. 4 - Consider the following steady, two-dimensional...Ch. 4 - -5 A steady, two-dimensional velocity field is...Ch. 4 - Consider the following steady, two-dimensional...Ch. 4 - What is the Eulerian description of fluid motion?...Ch. 4 - Is the Lagrangian method of fluid flow analysis...Ch. 4 - Prob. 9CP 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 - List at least three oiler names for the material...Ch. 4 - A weather balloon is hunched into the atmosphere...Ch. 4 - A Pilot-stalk probe can often be seen protruding...Ch. 4 - Is the Eulerian method of fluid flow analysis more...Ch. 4 - Consider steady, incompressible, two-dimensional...Ch. 4 - Converging duct flow is modeled by the steady,...Ch. 4 - Prob. 19P Ch. 4 - A steady, incompressible, two-dimensional velocity...Ch. 4 - The velocity field for a flow is given by...Ch. 4 - Consider steady flow of air through the diffuser...Ch. 4 - For the velocity field of Prob. 422, calculate the...Ch. 4 - A steady, incompressible, two-dimensional (in the...Ch. 4 - For the velocity field of Prob. 4-6, calculate the...Ch. 4 - Prob. 26CP Ch. 4 - Prob. 27CP Ch. 4 - What is the definition of a streamline? What do...Ch. 4 - Prob. 29CP Ch. 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 - What is the definition of a timeline? How can...Ch. 4 - Consider a cross-sectional slice through an array...Ch. 4 - Prob. 35P 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. 41P Ch. 4 - Prob. 42P Ch. 4 - The velocity held for a line vartex in the r...Ch. 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 - Conversing duct flow is modeled by the steady,...Ch. 4 - Prob. 48CP Ch. 4 - Name and briefly describe the four fundamental...Ch. 4 - Converging duct flow (Fig. P4—16) is modeled by...Ch. 4 - Prob. 51P Ch. 4 - Prob. 52P Ch. 4 - Prob. 53P Ch. 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 - Prob. 57P Ch. 4 - Prob. 58P Ch. 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. 62P Ch. 4 - Prob. 63P Ch. 4 - Consider steady, incompressible, two-dimensional...Ch. 4 - Prob. 65P Ch. 4 - Consider the steady, incompressible,...Ch. 4 - Prob. 67P Ch. 4 - Prob. 68P Ch. 4 - Prob. 69P Ch. 4 - Prob. 70P Ch. 4 - Prob. 71P Ch. 4 - Prob. 72P Ch. 4 - A cylindrical lank of water rotates in solid-body...Ch. 4 - Prob. 74P Ch. 4 - A cylindrical tank of radius rrim= 0.354 m rotates...Ch. 4 - Prob. 76P Ch. 4 - Prob. 77P Ch. 4 - Consider the following steady, three-dimensional...Ch. 4 - Prob. 79P Ch. 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 - A steady, three-dimensional velocity field is...Ch. 4 - Prob. 83P Ch. 4 - Prob. 84P Ch. 4 - A steady, three-dimensional velocity field is...Ch. 4 - Prob. 88CP Ch. 4 - Briefly explain the purpose of the Reynolds...Ch. 4 - True or false: For each statement, choose whether...Ch. 4 - Consider the integral ddtt2tx2. Solve it two ways:...Ch. 4 - Prob. 92P Ch. 4 - Consider the general form of the Reynolds...Ch. 4 - Consider the general form of the Reynolds...Ch. 4 - Prob. 95P Ch. 4 - Prob. 96P Ch. 4 - Prob. 97P Ch. 4 - Prob. 98P 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. 103P Ch. 4 - Prob. 107P Ch. 4 - The velocity field for an incompressible flow is...Ch. 4 - Prob. 109P Ch. 4 - Prob. 110P Ch. 4 - Prob. 111P Ch. 4 - Prob. 112P Ch. 4 - Prob. 114P Ch. 4 - In a steady, two-dimensional flow field in the...Ch. 4 - Prob. 116P Ch. 4 - Prob. 117P Ch. 4 - Prob. 119P Ch. 4 - Based on your results of Prob. 4—116, discuss the...Ch. 4 - Prob. 121P Ch. 4 - Prob. 122P Ch. 4 - Water is flowing in a 3-cm-diameter garden hose at...Ch. 4 - Prob. 124P Ch. 4 - Prob. 125P Ch. 4 - Prob. 126P Ch. 4 - Prob. 127P Ch. 4 - Prob. 128P Ch. 4 - The actual path traveled by an individual fluid...Ch. 4 - Prob. 130P Ch. 4 - Prob. 131P Ch. 4 - An array of arrows indicating the magnitude and...Ch. 4 - Prob. 133P Ch. 4 - Prob. 134P Ch. 4 - Prob. 135P Ch. 4 - Prob. 136P Ch. 4 - A steady, two-dimensional velocity field is given...Ch. 4 - Prob. 138P Ch. 4 - Prob. 139P Ch. 4 - Prob. 140P Ch. 4 - Prob. 141P Ch. 4 - Prob. 142P

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The magnitude of the rate of rotation (angular velocity) about point P in the x y plane is given by ω z = 1 2 ( ∂ v ∂ x − ∂ u ∂ y ) .

Solution Summary: The author explains the magnitude of the rate of rotation (angular velocity) about point P in the xy plane.

Concept explainers

Videos

The magnitude of the rate of rotation (angular velocity) about point P in the xy plane is given by ωz=12(∂v∂x−∂u∂y).

Want to see the full answer?

Chapter 4 Solutions