Question

Consider steady fully developed flow of a liquid through two large horizontal flat parallel plates separated by a distance of 2 mm. One of the plates is fixed and the other plate moves at a speed of 0.5 m/s. What is the magnitude of the pressure gradient (in Pa/m) in the direction of the flow required to ensure that the net flow through the plates is zero? Dynamic viscosity of the liquid is $5 \times 10^{-4}$ Ns/m$^2$ (Round off to the nearest integer)

Hint:

For Couette–Poiseuille flow, zero net flow condition gives $\frac{dp}{dx} = \frac{6\mu U}{h^2}$ directly.

Answer 1

Correct Answer: 374

For zero net flow between parallel plates (Couette–Poiseuille flow), the required pressure gradient is obtained by setting the average velocity to zero: \[ \frac{dp}{dx} = \frac{6 \mu U}{h^2} \] Given: $\mu = 5 \times 10^{-4}$ Ns/m$^2$, $U = 0.5$ m/s, $h = 2~\text{mm} = 0.002$ m. \[ \frac{dp}{dx} = \frac{6 (5 \times 10^{-4})(0.5)}{(0.002)^2} \] \[ = \frac{1.5 \times 10^{-3}}{4 \times 10^{-6}} = 375~\text{Pa/m} \] Rounded to nearest integer: \[ \boxed{375} \]