Explanation Given information: The flow is steady, three-dimensional and incompressible. Write the expression for the velocity field. V → = ( a x z 2 − b y ) i → + c x y z j → + ( d z 3 + e x z 2 ) k → ....... (I) Here, unit vector along x direction is i → , unit vector along y direction is j → and unit vector along y direction is k → , constants are a , b , c , d and e and position coordinates are z , y and z . Write the expression for the continuity equation. ∂ u ∂ x + ∂ v ∂ y + ∂ w ∂ z = 0 ...... (II) Here, velocity components are u , v and w in x , y and z -direction. Calculation: Substitute ( a x z 2 − b y ) for u , c x y z for v and ( d z 3 + e x z 2 ) for w in Equation (II). ∂ ( a x z 2 − b y ) ∂ x + ∂ ( c x y z ) ∂ y + ∂ ( d z 3 + e x z 2 ) ∂ z = 0 ∂ ∂ x ( a x z 2 − b y ) + ∂ ∂ y ( c x y z ) + ∂ ∂ z ( d z 3 + e x z 2 ) = 0 a z 2 − 0 + c x z + 3 d z 2 + 2 e x z = 0 a z 2 + c x z + 3 d z 2 + 2 e x z = 0 ...... (III) Equate similar terms in Equation (III). a = − 3 d c = − 2 e ...... (IV) The each component of velocity field in Equation (I) must be dimensionally homogenous and should have dimensions of velocity. Write dimensions of velocity u -component, u = ( a x z 2 − b y ) ( L t ) = ( a ( L 3 ) − b ( L ) )

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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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 9, Problem 115P

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The conditions for flow to be incompressible.

The primary dimensions of constants.

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Chapter 9, Problem 114P

Chapter 9, Problem 116P

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Fluid Mechanics: Fundamentals and Applications

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The conditions for flow to be incompressible. The primary dimensions of constants.