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Chapter 7: Problem 87

The basic equation that describes the motion of the fluid above a large oscillating flat plate is $$\frac{\partial u}{\partial t}=v \frac{\partial^{2} u}{\partial y^{2}}$$ where \(u\) is the fluid velocity component parallel to the plate, \(t\) is time, \(y\) is the spatial coordinate perpendicular to the plate, and \(v\) is the fluid kinematic viscosity. The plate oscillating velocity is given by \(U=U_{0} \sin \omega t .\) Find appropriate dimensionless parameters and the dimensionless differential equation.

Short Answer

Expert verified
The dimensionless parameters are \(u' = u/U_0\), \(t' = ωt\) and \(y' = y(ω/v)^{1/2}\). The dimensionless differential equation is \(∂u' ∂t' = ∂²u'/∂y'^2\).

Step by step solution

01

Determine the Variables

Start by pointing out the variables involved in the problem. These variables are the fluid velocity component parallel to the plate (u), the time (t), the spatial coordinate perpendicular to the plate (y), and the fluid kinematic viscosity (v). The plate oscillating velocity is given by \(U=U_{0} \sin \omega t\).
02

Choose Characteristic Units

After all variables have been identified, characteristic units need to be chosen for each variable. In this case \(U_0\) could be used as the characteristic unit of velocity and 1/ω as the characteristic unit of time. So, we have \(u' = u/U_0\), \(t' = ωt\) and \(y' = y(ω/v)^{1/2}\).
03

Derive the Dimensionless Differential Equation

The next step is to substitute the dimensionless parameters into the original differential equation. Derive the equation using the chain rule of differentiation so that every variable in the original equation can be expressed in terms of the dimensionless variables. Following through the differentiation, and after simplification, we arrive at the dimensionless differential equation \(∂u' ∂t' = ∂²u'/∂y'^2\)

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