The Mechanism of Fluid Friction
When a fluid flows over a solid surface, the layer of fluid in immediate contact with the surface is at rest—this is known as the no-slip condition. As we move away from the surface, the velocity of the fluid layers increases, creating a velocity gradient. This relative motion between layers gives rise to a tangential force that opposes the flow, which we call viscous force.
According to Newton’s Law of Viscosity, the viscous force F between two parallel layers of area A is directly proportional to the velocity gradient (dv/dx). The mathematical expression is given by:
F = η A (dv/dx)
Here, η (eta) is the coefficient of viscosity. Its SI unit is the poiseuille (Pl) or N·s/m². In the CGS system, the unit is the poise, where 1 Pl = 10 poise. Remember that viscosity is a property of the fluid; it generally decreases with temperature for liquids but increases with temperature for gases due to the increase in molecular collisions.
Poiseuille’s Equation and Laminar Flow
When a fluid flows through a cylindrical pipe, the velocity profile is parabolic, with the maximum velocity at the center and zero at the walls. Poiseuille’s Law describes the rate of flow (volume per unit time) of an incompressible, Newtonian fluid through a long cylindrical pipe of radius r and length l under a pressure difference ΔP:
- Rate of flow (Q) = (π ΔP r⁴) / (8 η l)
This equation is critical for competitive exams. Notice the r⁴ dependency: even a small reduction in the radius of a pipe (like a blood vessel) causes a massive decrease in the flow rate. This principle explains why blood pressure regulation is so sensitive to the constriction or dilation of arteries.
Terminal Velocity of Falling Spheres
When an object falls through a viscous fluid, it experiences three forces: its weight (downward), the buoyant force (upward), and the viscous drag force (upward). As the object accelerates, the viscous drag—governed by Stokes’ Law—increases until the net force becomes zero. At this point, the object reaches a constant speed known as terminal velocity (vₜ).
Stokes’ Law states that the drag force F on a sphere of radius r moving with velocity v is F = 6πηrv. By equating the forces at equilibrium, we derive the formula for terminal velocity:
vₜ = [2r²(ρ – σ)g] / (9η)
In this formula, ρ is the density of the object, σ is the density of the fluid, and g is the acceleration due to gravity. This formula is a favorite for numerical problems in IIT JEE, especially those involving the calculation of viscosity coefficients from experimental data.
Important Facts / Formulas
| Concept | Formula / Value |
|---|---|
| Newton’s Law | F = η A (dv/dx) |
| Poiseuille’s Flow | Q = (π ΔP r⁴) / (8 η l) |
| Stokes’ Law (Drag) | F = 6πηrv |
| Terminal Velocity | vₜ ∝ r² |
| Unit Conversion | 1 Pl = 10 Poise |
Previous Year Question Hints
- Question Type 1: You may be asked to compare the terminal velocities of two spheres of different radii falling through the same liquid. Remember that vₜ is proportional to r², so the ratio of velocities will be the square of the ratio of their radii.
- Question Type 2: Questions involving the “Poiseuille’s Law” often ask for the effect of doubling the radius of a pipe on the flow rate. Since Q ∝ r⁴, doubling the radius increases the flow rate by a factor of 16.
Quick Revision Summary
- Viscosity is fluid friction arising from relative motion between layers.
- Newton’s Law defines the viscous force based on the velocity gradient.
- Temperature effect: Liquids become less viscous when heated; gases become more viscous.
- Poiseuille’s Law shows that flow rate is extremely sensitive to pipe radius (r⁴).
- Stokes’ Law provides the drag force for spherical objects in a fluid.
- Terminal velocity occurs when the drag force and buoyancy balance the weight.
- Dimensional formula for η is [ML⁻¹T⁻¹].
- Always check units; ensure SI units (meters, kilograms, seconds) are used unless otherwise specified.
