Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version
8th Edition
ISBN: 9781119080701
Author: Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein
Publisher: WILEY

Videos

Question
Chapter 5.1, Problem 1P
To determine

Use Equation 4.19 and obtain the continuity equation for steady or unsteady incompressible flow through a fixed control volume.

The continuity equation for steady or unsteady incompressible flow through a fixed control volume is csVn^dA=0

Explanation of Solution

Consider the Reynolds transport theorem.

DBsysDt=tcvρbdV+csρbVn^dA (I)

Here, control surface is cs, control surface is cs, velocity is V, product of the component of velocity is Vn^dA and density is ρ.

Substitute V for B and 1ρ for b in Equation (I).

DVsysDt=tcvρ1ρdV+csρ1ρVn^dA=tcvdV+csVn^dA=Vcvt+csVn^dA (II)

For incompressible flow, the volume of the system Vsys is constant with time.

DVsysDt=0 (III)

Also for a fixed control volume, Vcv does not change and therefore

Vcvt=0 (IV)

Combine Equations (II), (III) and (IV) to get,

csVn^dA=0

Hence, the continuity equation for steady or unsteady incompressible flow through a fixed control volume is csVn^dA=0.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
02:54