If the velocity profile of a fluid over a plate is parabolic with the vertex 20 cm from the plate, where the velocity is 120 cm/sec. Calculate the velocity gradients and shear stresses at distances of 0, 10 and 20 cm from the plate, if the viscosity of the fluid is 8.5 poise.

Fluid Velocity Profile Analysis

Problem Statement

Given Data

Distance of vertex from plate = 20 cm
Velocity at vertex, u = 120 cm/sec
Viscosity, μ = 8.5 poise = 0.85 N·s/m²

Solution

1. Establish Velocity Profile Equation

Parabolic velocity profile:
\( u = ay^2 + by + c \)
Boundary conditions:
(a) At y = 0, u = 0 ⇒ c = 0
(b) At y = 20 cm, u = 120 cm/sec ⇒ 120 = a(20)² + b(20)
(c) At y = 20 cm, du/dy = 0 ⇒ 0 = 2a(20) + b

2. Solve for Constants

From condition (c): b = -40a
Substituting into condition (b):
120 = 400a + 20(-40a) ⇒ a = -0.3
Therefore b = 12
Final velocity equation:
\( u = -0.3y^2 + 12y \)

3. Calculate Velocity Gradients

\( \frac{du}{dy} = -0.6y + 12 \)
At y = 0 cm: \( \frac{du}{dy} = 12 \, \text{s}^{-1} \)
At y = 10 cm: \( \frac{du}{dy} = 6 \, \text{s}^{-1} \)
At y = 20 cm: \( \frac{du}{dy} = 0 \, \text{s}^{-1} \)

4. Calculate Shear Stresses

\( \tau = \mu \frac{du}{dy} \) (μ = 0.85 N·s/m²)
At y = 0 cm: τ = 10.2 N/m²
At y = 10 cm: τ = 5.1 N/m²
At y = 20 cm: τ = 0 N/m²

Final Results:

Velocity gradients: 12/s (0 cm), 6/s (10 cm), 0/s (20 cm)
Shear stresses: 10.2 N/m² (0 cm), 5.1 N/m² (10 cm), 0 N/m² (20 cm)

Explanation

1. Velocity Profile:
The parabolic velocity profile was determined using three boundary conditions: no-slip at the plate (u=0 at y=0), known velocity at the vertex (u=120 cm/s at y=20 cm), and zero velocity gradient at the vertex (du/dy=0 at y=20 cm).

2. Velocity Gradients:
The velocity gradient decreases linearly from the plate (y=0) to the vertex (y=20 cm), reflecting how the fluid's deformation rate changes with distance from the plate.

3. Shear Stress Calculation:
Shear stress was calculated using Newton's law of viscosity, showing maximum stress at the plate and zero stress at the vertex where there's no velocity gradient.

Physical Meaning

1. Velocity Gradient Interpretation:
The decreasing velocity gradient indicates the fluid's internal friction is strongest near the plate and diminishes toward the vertex, where fluid layers move with uniform velocity.

2. Shear Stress Distribution:
The maximum shear stress at the plate surface (10.2 N/m²) is crucial for designing plate materials and understanding drag forces in applications like lubrication systems.

3. Zero Stress at Vertex:
The zero shear stress at the vertex confirms this point as the location of maximum velocity in the parabolic profile, important for flow rate calculations in channels.