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 ) )

4th Edition

ISBN: 2810022150991

Author: CENGEL

Publisher: MCG

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 9, Problem 115P

To determine

The conditions for flow to be incompressible.

The primary dimensions of constants.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9, Problem 114P

Chapter 9, Problem 116P

Students have asked these similar questions

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. ㅠ 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.

Consider the velocity field represented by V = K (yĩ + xk) Rotation about z-axis is

4. Consider a velocity field V = K(yi + ak) where K is a constant. The vorticity, z , is (A) -K (B) K (C) -K/2 (D) K/2

Chapter 9 Solutions

FLUID MECHANICS FUNDAMENTALS+APPS

Ch. 9 - Explain the fundamental differences between a flow...Ch. 9 - What does it mean when we say that two more...Ch. 9 - The divergence theorem is v.cdv=A c . n dA Ch. 9 - Prob. 4CP Ch. 9 - Prob. 5CP Ch. 9 - Prob. 6CP Ch. 9 - Prob. 7P Ch. 9 - Prob. 8P Ch. 9 - Let vector G=2xzi12x2jz2kk . Calculate the...Ch. 9 - Prob. 10P

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