Problem Statement
A flat plate 0.3m2 in area moves edgewise through oil between large fixed parallels 10cm apart. If the velocity of the plate is 0.6m/s and the oil has a kinematic viscosity of 0.45 stokes and specific gravity 0.8, calculate the drag force when:
- The plate is 2.5cm from one of the planes.
- The plate is equidistant from both planes.
Solution
Given:
- Area of plate (A) = 0.3m2
- Velocity of plate (u) = 0.6m/s
- dv = 0.6m/s
- Kinematic viscosity (υ) = 0.45 stokes = 0.45 × 10-4 m2/s
- Specific gravity of fluid (S) = 0.8
Calculations:
1. Dynamic viscosity (μ):
μ = υ × ρ
Density of fluid (ρ) = S × 1000 = 0.8 × 1000 = 800kg/m3
μ = 0.45 × 10-4 × 800 = 0.036 NS/m2
(i) Plate is 2.5cm from one plane:
- dy1 = 2.5cm = 0.025m
- dy2 = 10 – 2.5 = 7.5cm = 0.075m
F = (μ × du/dy1 + μ × du/dy2) × A
F = [0.036 × 0.6/0.025 + 0.036 × 0.6/0.075] × 0.3
F = 0.345 N
(ii) Plate is equidistant from both planes:
- dy1 = dy2 = dy = 10/2 = 5cm = 0.05m
F = (τ1 + τ2) × A
F = 2 × μ × (du/dy) × A
F = 2 × 0.036 × 0.6/0.05 × 0.3
F = 0.26 N
Results:
- Drag force when plate is 2.5cm from one plane: 0.345 N
- Drag force when plate is equidistant: 0.26 N
Explanation
This problem calculates the drag force on a flat plate moving edgewise through a viscous fluid confined between two large parallel planes. Here’s how it works:
- Dynamic viscosity: The kinematic viscosity and specific gravity are used to compute the dynamic viscosity of the fluid, which quantifies its resistance to shear.
- Force on each side: The force due to fluid drag on each side of the plate is calculated using the shear stress, which depends on the velocity gradient (du/dy).
- Unequal distances: When the plate is closer to one plane, the velocity gradients differ for each side, resulting in different shear forces.
- Equal distances: When the plate is equidistant, the velocity gradients and hence the forces are symmetric, simplifying the calculation.
This solution demonstrates the principles of fluid mechanics in calculating viscous drag in constrained flow conditions.
