Fox and McDonald's Introduction to Fluid Mechanics
9th Edition
ISBN: 9781118912652
Author: Philip J. Pritchard, John W. Mitchell
Publisher: WILEY

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Textbook Question
Chapter 7, Problem 1P

The slope of the free surface of a steady wave in one-dimensional flow in a shallow liquid layer is described by the equation

h x = u g u x

Use a length scale, L, and a velocity scale, V0, to nondimensionalize this equation. Obtain the dimensionless groups that characterize this flow.

To determine

The dimensionless groups that characterizes the given flow.

### Explanation of Solution

Given:

The slope of the free surface of a steady wave in one dimensional flow in a shallow liquid is as follows:

hx=ugux        (I)

Calculation:

From Equation (I),

The length dimensional quantity which are having same unit of measurement are h and x.

The velocity dimensional quantity is u.

Consider the reference dimensional quantities as follows:

The length is L.

The velocity is V0.

Dividing the dimensional quantity by its reference dimensional quantity yields non-dimensional quantity. The non-dimensional quantities are denoted with asterisk.

Divide the given dimensional quantities with their reference as follows:

hL=h*h=h*L

xL=x*x=x*L

uV0=u*u=u*V0

Substitute h=h*L, x=x*L, and u=u*V0 in Equation (I).

(h*L)(x*L)=u*V0g(u*V0)(x*L)h*x*=V02gLu*(u*x*)

Here,

The dimensionless group is V02gL which is equal to the square of the Froude’s number.

Thus, the dimensionless groups that characterizes the given flow is V02gL_.

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