Viscosity — Newton's Law, Stokes' Law, Reynolds Number, Streamline vs Turbulent

Question

What is viscosity, how do Newton’s law and Stokes’ law describe it, and how does the Reynolds number determine flow type?

Solution — Step by Step

Viscosity ( $\eta$ ) is a fluid’s resistance to flow — its internal friction. Honey has high viscosity; water has low viscosity.

Newton’s law of viscosity:

F = -\eta A \frac{dv}{dx}

where $F$ is the viscous force, $A$ is the area of the fluid layer, and $\frac{dv}{dx}$ is the velocity gradient perpendicular to the flow.

SI unit of $\eta$ : Pa $\cdot$ s (Pascal-second) or equivalently N $\cdot$ s/m $^2$ . CGS unit: poise (1 Pa $\cdot$ s = 10 poise).

When a sphere of radius $r$ moves through a viscous fluid at velocity $v$ :

F = 6\pi\eta rv

At terminal velocity ( $v_t$ ), the drag force balances the net downward force:

6\pi\eta r v_t = \frac{4}{3}\pi r^3(\rho_s - \rho_f)g

v_t = \frac{2r^2(\rho_s - \rho_f)g}{9\eta}

Terminal velocity increases with $r^2$ — larger drops fall faster. This explains why raindrops of different sizes hit the ground at different speeds.

The Reynolds number (Re) is a dimensionless quantity that predicts whether flow is streamline or turbulent:

Re = \frac{\rho v d}{\eta}

where $\rho$ = fluid density, $v$ = flow speed, $d$ = pipe diameter, $\eta$ = viscosity.

graph TD
    A["Calculate Re = rho v d / eta"] --> B{Value of Re?}
    B -->|"Re < 1000"| C[Streamline/Laminar Flow]
    C --> C1[Fluid layers slide smoothly]
    C --> C2[Velocity profile is parabolic]
    B -->|"1000 < Re < 2000"| D[Transitional Flow]
    D --> D1[Unstable, can switch between types]
    B -->|"Re > 2000"| E[Turbulent Flow]
    E --> E1[Chaotic eddies and mixing]
    E --> E2[Higher energy loss]

    F[To reduce turbulence] --> G[Decrease velocity]
    F --> H[Decrease pipe diameter]
    F --> I[Increase viscosity]

Why This Works

Viscosity arises from intermolecular forces between fluid layers. When one layer moves faster than an adjacent one, the molecules in the slower layer pull back on the faster layer (and vice versa). This internal friction is what makes thick fluids flow slowly.

Stokes’ law follows from solving the Navier-Stokes equations for a sphere in slow, viscous flow. The terminal velocity formula is used extensively in physics — from measuring viscosity (falling ball viscometer) to understanding sedimentation and cloud formation.

NEET frequently asks for the terminal velocity formula and its dependence on radius ( $v_t \propto r^2$ ). JEE Main may ask you to use Stokes’ law to find the viscosity of a liquid given experimental data. Both require knowing $v_t = \frac{2r^2(\rho_s - \rho_f)g}{9\eta}$ .

Alternative Method

To find the viscosity of an unknown liquid, drop a steel ball of known radius and density into the liquid. Measure the terminal velocity (constant speed zone). Plug values into the terminal velocity formula and solve for $\eta$ . This is the principle of the falling ball viscometer.

Common Mistake

Students forget the density difference $(\rho_s - \rho_f)$ in the terminal velocity formula, using only the sphere’s density. The buoyant force reduces the net downward force, so we must subtract the fluid density. If the sphere is less dense than the fluid (like an air bubble in water), $v_t$ is negative — meaning the sphere rises, which makes physical sense.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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