Question 10.25: Find the rate of shear deformation in the flow described by u...
Find the rate of shear deformation in the flow described by u = kx²y + α/y, v = −kxy² + β/x, w = ε, where k, α, β, and ε are constants.
The shear deformation rate is given by Eq. 10.41 as
D=\left[\begin{matrix} \frac{\partial u}{\partial x}- \frac{1}{3}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\right)&\frac{1}{2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)&\frac{1}{2}\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x} \right) \\\frac{1}{2}\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y} \right)& \frac{\partial v}{\partial x} -\frac{1}{3}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\right)&\frac{1}{2}\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y} \right) \\\frac{1}{2}\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z} \right)&\frac{1}{2}\left(\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z} \right) &\frac{\partial w}{\partial z} -\frac{1}{3}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\right) \end{matrix} \right]Because of the many derivatives involved, it is best to solve this problem with a symbolic mathematics code or calculator. The result is
D=\left[\begin{matrix} 2kxy&\frac{1}{2}\left[\left(\frac{kx^2y^2 −α }{y^2} \right)-\left(\frac{kx^2y^2+\beta }{x^2} \right)\right]&0\\\frac{1}{2}\left[\left(\frac{kx^2y^2 −α }{y^2} \right)-\left(\frac{kx^2y^2+\beta }{x^2} \right)\right]&-2kxy&0\\0&0&0 \end{matrix} \right]We see that in this flow the shear deformation varies significantly with position. This is normally the case for all but the simplest flow fields.