SAE 30 oil at 20°C (density p-918 kg/m3, viscosity -0.440 Pa- s) flows between two horizontal, infinite, parallel plates separated by 40 mm. The bottom plate is fixed, and the upper plate moves parallel to the bottom plate with a velocity 5 cm/s. The pressure gradient in the direction of flow is 500 Pa/m. (a) (5 points) Determine the velocity distribution between the plates. (b) (5 points) Determine the flowrate passing between the plates (per a unit width).
The correct answer and explanation is :
To solve this problem, we use the Navier-Stokes equation for laminar flow between two infinite parallel plates, known as Couette-Poiseuille flow.
Given Data:
- Density, ( \rho = 918 ) kg/m³
- Dynamic viscosity, ( \mu = 0.440 ) Pa·s
- Plate separation, ( h = 40 ) mm = 0.04 m
- Upper plate velocity, ( U = 0.05 ) m/s
- Pressure gradient, ( \frac{dP}{dx} = 500 ) Pa/m
(a) Velocity Distribution:
For flow between two parallel plates, with one plate moving and a pressure gradient present, the velocity profile is given by:
[
u(y) = -\frac{1}{2\mu} \frac{dP}{dx} y^2 + \frac{U}{h} y + C_1
]
Since the bottom plate is fixed (( u(0) = 0 )), we determine ( C_1 = 0 ). The velocity profile simplifies to:
[
u(y) = -\frac{1}{2(0.440)} (500) y^2 + \frac{0.05}{0.04} y
]
[
u(y) = -\frac{500}{0.88} y^2 + 1.25y
]
[
u(y) = -568.18 y^2 + 1.25 y
]
(b) Flow Rate Per Unit Width:
The volumetric flow rate per unit width is:
[
Q = \int_0^h u(y) dy
]
[
Q = \int_0^{0.04} (-568.18 y^2 + 1.25 y) dy
]
Evaluating the integral:
[
Q = \left[ -\frac{568.18}{3} y^3 + \frac{1.25}{2} y^2 \right]_{0}^{0.04}
]
[
Q = \left[ -189.39 (0.04)^3 + 0.625 (0.04)^2 \right]
]
[
Q = \left[ -189.39 (0.000064) + 0.625 (0.0016) \right]
]
[
Q = (-0.01213 + 0.001) = -0.01113 \text{ m}^2\text{/s}
]
Thus, the flow rate per unit width is ( 0.01113 ) m²/s.
Explanation:
This problem involves both Couette flow (due to the moving upper plate) and Poiseuille flow (due to the pressure gradient). The velocity profile is parabolic due to pressure-driven flow but is skewed due to the moving plate. The negative sign in the flow rate indicates that the pressure-driven flow opposes the upper plate movement.