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#### Concept explainers

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

Use a length scale, *L*, and a velocity scale, *V*_{0}, to nondimensionalize this equation. Obtain the dimensionless groups that characterize this flow.

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:

**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

The velocity is

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:

Substitute

Here,

The dimensionless group is

Thus, the dimensionless groups that characterizes the given flow is

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Fox and McDonald's Introduction to Fluid Mechanics

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