Problem Statement
In a stream of glycerine in motion, at a certain point the velocity gradient is 0.25 metre per sec per metre. The mass density of the fluid is 1268.4 kg per cubic metre and kinematic viscosity is 6.30 x 10⁻⁴ square metre per second. Calculate the shear stress at the point.
Given Data
- Velocity Gradient, \(\frac{du}{dy} = 0.25 \, \text{s}^{-1}\)
- Mass Density, \(\rho = 1268.4 \, \text{kg/m}^3\)
- Kinematic Viscosity, \(\nu = 6.30 \times 10^{-4} \, \text{m}^2/\text{s}\)
Solution
1. Calculate Dynamic Viscosity (μ)
To find the shear stress, we first need the dynamic viscosity (\(\mu\)). It can be calculated from the kinematic viscosity (\(\nu\)) and density (\(\rho\)) using the formula:
Substituting the given values:
2. Calculate the Shear Stress (τ)
Now we can use Newton's law of viscosity to calculate the shear stress (\(\tau\)) using the dynamic viscosity (\(\mu\)) and the velocity gradient (\(\frac{du}{dy}\)).
Substitute the calculated and given values:
The shear stress at the point is \( \tau \approx 0.1998 \, \text{N/m}^2 \).
Explanation of Viscosity Types
It's important to distinguish between the two types of viscosity used in this problem:
- Dynamic Viscosity (\(\mu\)): Also known as absolute viscosity, this is the true measure of a fluid's internal resistance to shear forces. It answers the question, "How much force is required to shear the fluid?" Its units are N·s/m² or Pa·s.
- Kinematic Viscosity (\(\nu\)): This is the ratio of dynamic viscosity to the fluid's density (\(\nu = \mu / \rho\)). It describes how easily a fluid flows under the influence of gravity. It answers the question, "How fast will the fluid flow?" Its units are m²/s.
The problem provides the kinematic viscosity, which we must first convert to dynamic viscosity before calculating the shear stress.
Physical Meaning
The calculated shear stress of approximately 0.2 N/m² is the internal force per unit area that exists within the flowing glycerine at that specific point. This internal friction arises because adjacent layers of the glycerine are moving at slightly different velocities (as defined by the velocity gradient).
This value is significant for engineers designing systems that transport viscous fluids like glycerine. It helps determine:
- Pumping Power: The total shear force over the surface area of a pipe contributes to the overall pressure drop, which dictates the power required to pump the fluid.
- Fluid Forces: It allows for the calculation of the drag force that the moving fluid exerts on any submerged objects or the pipe walls.