Class 11 > Unit # 07: Fluid Dynamics > Viscosity

Talha's Physics Academy

Fluid Dynamics - Viscous Forces & Coefficient of Viscosity

Q. Define Viscous forces in fluids. Define Viscosity and derive expression for coefficient of viscosity.

Viscous Force in a Fluid

"Viscous force is an opposition between relative motion of layers of a fluid. Viscous forces in a fluid are proportional to the rate of change of velocity of fluid’s layers, with the viscosity serving as the proportionality constant."

Consider a viscous fluid contained between two parallel plates, where the lower plate is stationary and the upper plate is movable. The fluid molecules directly in contact with each plate are held firmly to the respective surface due to adhesive forces.

• Consequently, the topmost layer of fluid moves with the same speed $v$ as the upper plate, whereas the bottom layer in contact with the stationary plate remains at rest ($v = 0$).
• Each stationary or slower layer retards the flow of the layer directly above it. Thus, the velocity varies continuously from $0$ to $v$ across the fluid thickness $l$.
• Velocity Gradient: The increase in velocity divided by the distance over which the change occurs ($\frac{v}{l}$) is defined as the velocity gradient—meaning the rate of change of velocity with distance normal to the direction of flow.

Viscosity

Viscosity is a fluid’s inherent resistance to flow. Fluids resist both the relative motion of objects immersed within them and the relative sliding motion of fluid layers possessing differing velocities.

Derivation of the Expression for Coefficient of Viscosity ($\eta$)

For a given fluid, the viscous drag force $F$ required to maintain steady relative motion between parallel plates depends on:

1. Directly proportional to the area $A$ of the fluid layers in contact:
   $F \propto A$
2. Directly proportional to the velocity gradient ($\frac{v}{l}$):
   $F \propto \frac{v}{l}$

Combining these proportionalities:

$F \propto A \frac{v}{l}$

Introducing the proportionality constant $\eta$ (known as the coefficient of viscosity):

$F = \eta A \frac{v}{l}$

Rearranging the equation to solve for the coefficient of viscosity ($\eta$):

$\eta = \frac{F \cdot l}{A \cdot v}$

Units of Viscosity

• SI Unit: $\frac{\text{N} \cdot \text{m}}{\text{m}^2 \cdot (\text{m/s})} = \text{N} \cdot \text{s/m}^2 = \text{Pa} \cdot \text{s}$ (Pascal-second).
• CGS Unit: $\text{dyne} \cdot \text{s/cm}^2$, which is commonly termed a Poise (P). ($1 \text{ Pa}\cdot\text{s} = 10 \text{ Poise}$).

© 2026 Talha's Physics Academy. All rights reserved.