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

4th Edition

ISBN: 9781259696534

Author: Yunus A. Cengel Dr., John M. Cimbala

Publisher: McGraw-Hill Education

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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…

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Chapter 4, Problem 71P

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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).

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Chapter 4, Problem 70P

Chapter 4, Problem 72P

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Fluid Mechanics: Fundamentals and Applications

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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 ) .

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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).

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Chapter 4 Solutions