Explanation Given information: The two-dimensional strain tensor in the x-y plane is ( ε x x ε x y ε y x ε y y ) . Write the expression for the two-dimensional velocity field in the vector form. V → = ( U + a 1 x + b 1 y ) i → + ( V + a 2 x + b 2 y ) j → ...... (I) Here, the constants are, U , V , a 1 , b 1 , a 2 and b 2 , the distance in x direction is x , and the distance in y direction is y . Write the expression for the velocity along x direction. u = U + a 1 x + b 1 y ...... (II) Write the expression for the velocity along y direction. v = V + a 2 x + b 2 y ...... (III) Write the expression for the shear strain rate along x-y plane. ε x y = 1 2 ( ∂ u ∂ y + ∂ v ∂ x ) ...... (IV) Here, the shear strain rate along x-y plane is ε x y . Write the expression for the normal strain rate along x axis. ε x x = ∂ u ∂ x ...... (V) Here, the normal strain rate along x direction is ∂ u ∂ x . Write the expression for the strain rate along y axis. ε y y = ∂ u ∂ y ...... (VI) Here, the normal strain rate along y direction is ∂ u ∂ y . Write the expression for the strain rate tensor. ε i j = ( ε x x ε x y ε y x ε y y ) ...... (VII) Write the expression for the strain rate along y-x plane. ε y x = ε x y ...... (VIII) Calculation: Substitute U + a 1 x + b 1 y for u and V + a 2 x + b 2 y for v in Equation (IV). ε x y = 1 2 ( ∂ ∂ y ( U + a 1 x + b 1 y ) + ∂ ∂ x ( V + a 2 x + b 2 y ) ) = 1 2 ( ( 0 + 0 + b 1 ) + ( 0 + a 2 + 0 ) ) = 1 2 ( b 1 + a 2 ) Substitute U + a 1 x + b 1 y for u in Equation (V)

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 64P

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The conditions for x and y axes to be principal axes.

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Chapter 4, Problem 63P

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Fluid Mechanics: Fundamentals and Applications

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The conditions for x and y axes to be principal axes.

Solution Summary: The author describes the conditions for x and y to be principal axes, and the expression for the two-dimensional velocity field in the vector form.

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Chapter 4 Solutions