Discussion
The velocity gradient in the Transverse direction for a fluid flow equals :
Answer: C
The velocity gradient equals the rate of shear strain.
Consider a fluid element between two layers a distance dy apart, moving with a velocity difference du. In time dt the element's angle of shear changes by \(du\,dt/dy\), so the rate of angular deformation is
\[ \frac{d\phi}{dt} = \frac{du}{dy} \]
which is the transverse velocity gradient. Newton's law of viscosity, \(\tau = \mu\,\frac{du}{dy}\), relates it to the shear stress: the stress is proportional to the gradient, not equal to it, so the shear stress option is wrong.
The gradient measures a rate of strain, not the strain itself, because a fluid deforms continuously. The pressure gradient is a different quantity, the change of pressure along the flow.
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