Question

Using Newton's law of viscosity for streamline flow, derive an expression for coefficient of viscosity.

Hint:

Think of viscosity (\(\eta\)) as the "stickiness" or internal friction of a fluid. The formula shows it's the force required to slide layers past each other at a certain speed. High \(\eta\) means high force is needed (like for honey).

Answer 1

Newton's Law of Viscosity states that for a streamline flow of a fluid, the viscous force (\(F\)) acting between two adjacent layers is directly proportional to the area of the layers (\(A\)) and the velocity gradient (\(dv/dx\)) between them. \[ F \propto A \frac{dv}{dx} \] Introducing a constant of proportionality, \(\eta\), we get: \[ F = -\eta A \frac{dv}{dx} \] The negative sign indicates that the viscous force opposes the relative motion between the layers. The constant \(\eta\) is called the coefficient of viscosity.
Derivation of the Expression for \(\eta\):
From the formula above, we can rearrange the terms to solve for the coefficient of viscosity, \(\eta\). Considering only the magnitude of the force: \[ \eta = \frac{F}{A \frac{dv}{dx}} \] This is the expression for the coefficient of viscosity.
\(F\) is the tangential viscous force.
\(A\) is the area of the layer.
\(\frac{dv}{dx}\) is the velocity gradient, which is the rate of change of velocity with distance perpendicular to the direction of flow.
The SI unit of the coefficient of viscosity is Pascal-second (Pa·s) or N·s/m².