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Chapter 7: Problem 5

A two-dimensional steady flow in a viscous liquid is described by the equation: $$u \frac{\partial u}{\partial x}=-g \frac{\partial h}{\partial x}+\frac{\mu}{\rho}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right)$$ Use a length scale, \(L\), and a velocity scale, \(V_{0},\) to nondimensionalize this equation. Obtain the dimensionless groups that characterize this flow.

Short Answer

Expert verified
The dimensionless groups obtained characterizing the flow are the Reynolds number, which signifies the ratio of inertial forces to viscous forces, and the Froude number, representing the ratio of inertial forces to gravitational forces.

Step by step solution

01

Identify Variables and Constants

First, identify all the parameters and variables involved in the equation. These include \(u\) (velocity), \(x, y\) (spatial dimensions), \(g\) (acceleration due to gravity), \(h\) (height of fluid), \(\mu\) (dynamic viscosity), \(\rho\) (density), \(L\) (length scale), and \(V_{0}\) (velocity scale). The term \(\frac{\partial u}{\partial x}\) is the velocity gradient in the x-axis, while \(\frac{\partial^{2} u}{\partial x^{2}}\) and \(\frac{\partial^{2} u}{\partial y^{2}}\) represent the second order derivatives of the velocity in the x and y directions respectively.
02

Non-dimensionalize the Equation

Next, define non-dimensional variables as \(\bar{u}=\frac{u}{V_{0}}\), \(\bar{x}=\frac{x}{L}\), \(\bar{y}=\frac{y}{L}\), and \(\bar{h}=\frac{h}{L}\). Replace these variables into the original equation thereby translating the equation into its non-dimensional form.
03

Rearrange the Equation

Subtract the non-dimensional equation in order to group similar terms and further simplify the equation. This equation forms required dimensionless groups that characterize the flow.
04

Identify the Dimensionless Groups

Split the simplified non-dimensional equation based on constants which becomes the dimensionless groups. The groups can be related to existing dimensionless numbers like Reynolds number, Froude number etc.

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Most popular questions from this chapter

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