Question

Driven by a pressure gradient of 100 kPa/m, a fluid of dynamic viscosity 0.1 Pa.s flows between two fixed infinitely large parallel plates under steady, incompressible, and fully developed laminar conditions. The average velocity of the flow is 2 m/s. The gap between the parallel plates in mm (rounded off to 2 decimal places) is ..........

Hint:

For fully developed laminar flow between parallel plates, the average velocity and pressure gradient are directly related to the gap between the plates. The gap can be found using the formula for laminar flow between parallel plates.

Answer 1

For fully developed laminar flow between two parallel plates, the flow velocity profile is parabolic, and the relationship between the pressure gradient and the gap between the plates is given by the following formula: \[ \frac{dp}{dx} = \frac{12\mu U}{h^2} \] Where: - \( \frac{dp}{dx} = 100 \, {kPa/m} = 100 \times 10^3 \, {Pa/m} \) is the pressure gradient, - \( \mu = 0.1 \, {Pa.s} \) is the dynamic viscosity, - \( U = 2 \, {m/s} \) is the average velocity, - \( h \) is the gap between the plates. Rearranging for \( h \): \[ h^2 = \frac{12 \mu U}{\frac{dp}{dx}} \] Substituting the known values: \[ h^2 = \frac{12 \times 0.1 \times 2}{100 \times 10^3} \] \[ h^2 = \frac{2.4}{100000} = 2.4 \times 10^{-5} \] \[ h = \sqrt{2.4 \times 10^{-5}} = 4.89 \, {mm} \] Thus, the gap between the plates is approximately between 4.80 and 5.00 mm.