Viscous Polymer Pipeline Flow (Chemical & Petroleum Engineering)

A chemical engineer monitors the laminar flow of a highly viscous polymer solution through a rectangular processing channel in a refinery plant. Due to no-slip boundary conditions at the solid duct walls, fluid velocity drops to zero at the channel perimeter and peaks along the central core line.

The axial velocity $v(x, y)$ (in meters per second, $\text{m/s}$) across the rectangular cross-section $0 \le x \le 4\text{ cm}$ and $0 \le y \le 2\text{ cm}$ is modeled by the separable polynomial profile:

1. Explain how to compute $\frac{\partial v}{\partial x}$ and $\frac{\partial v}{\partial y}$ by treating terms involving the other coordinate as constant scaling factors.
2. Determine the analytical expressions for the transverse velocity gradients $\frac{\partial v}{\partial x}$ and $\frac{\partial v}{\partial y}$.
3. In Newtonian fluid mechanics, shear stress across fluid layers is proportional to the velocity gradient:
   $$\tau_{zx} = \mu \frac{\partial v}{\partial x}, \quad \tau_{zy} = \mu \frac{\partial v}{\partial y}$$
   Evaluate $\frac{\partial v}{\partial x}$ and $\frac{\partial v}{\partial y}$ at the cross-sectional evaluation point $(x, y) = (1, 0.5)\text{ cm}$. State their units and identify which coordinate axis experiences greater internal fluid shear stress.