Discussion
In a two-dimensional flow field, the velocities in the x- and y- directions are u and v, respectively. The shear stress for a Newtonian fluid having dynamic viscosity \(\mu\) is given by
Answer: D
The shear stress is \(\mu\left(\dfrac{\partial v}{\partial x} + \dfrac{\partial u}{\partial y}\right)\).
For a Newtonian fluid the shear stress is the viscosity times twice the shear strain rate. In two dimensions the rate of angular deformation of a fluid element is the sum of the two cross derivatives, so \(\tau_{xy} = \mu(\partial v/\partial x + \partial u/\partial y)\). For a simple parallel flow this reduces to the familiar \(\mu\,du/dy\).
The difference of the derivatives is the vorticity, which measures rotation, not deformation. \(2\mu\,\partial u/\partial x\) and \(2\mu\,\partial v/\partial y\) are the viscous parts of the normal stresses, not shear stress.
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