Viscosity — Definition, Formula, and Example

Viscosity measures how strongly a fluid resists flowing or having one layer slide past another. For a Newtonian fluid in simple shear flow, the standard relation is

$$\tau = \mu \frac{du}{dy}$$

where $\tau$ is shear stress, $\mu$ is dynamic viscosity, and $\frac{du}{dy}$ is the velocity gradient perpendicular to the flow.

The quick intuition is that water pours easily because its viscosity is low, while honey pours slowly because its viscosity is high. The formula needs conditions, but the core idea stays the same: larger viscosity means more resistance to deformation.

What viscosity means

Viscosity describes internal friction in a fluid. If one layer of fluid tries to move past another, viscosity is the property that resists that sliding motion.

That is why viscosity matters in both liquids and gases. It affects how easily a fluid flows, how much drag develops near a surface, and how much energy is lost in real fluid motion.

When the viscosity formula applies

For a Newtonian fluid in simple shear flow, shear stress is proportional to the velocity gradient:

$$\tau = \mu \frac{du}{dy}$$

This is often called Newton's law of viscosity. It does not say every fluid behaves this way in every situation. It says that if the fluid is Newtonian, then the ratio between shear stress and shear rate stays constant and that constant is $\mu$.

In this equation:

• $\tau$ is shear stress
• $\mu$ is dynamic viscosity
• $\frac{du}{dy}$ is the rate at which velocity changes from one layer to the next

The SI unit of dynamic viscosity is $\mathrm{Pa \cdot s}$, which is equivalent to $\mathrm{N \cdot s/m^2}$ or $\mathrm{kg/(m \cdot s)}$.

Dynamic viscosity vs. kinematic viscosity

Dynamic viscosity $\mu$ tells you how much shear stress is needed for a given rate of shear. Kinematic viscosity $\nu$ also accounts for density:

$$\nu = \frac{\mu}{\rho}$$

where $\rho$ is the fluid density.

This matters because two fluids can have the same dynamic viscosity but different densities, so their kinematic viscosities are not necessarily the same. Kinematic viscosity is especially common in fluid-flow tables and Reynolds-number calculations.

Worked example: shear stress between two plates

Suppose a Newtonian fluid fills the gap between two large parallel plates. The bottom plate is fixed, the top plate moves at $0.30\ \mathrm{m/s}$, and the gap between the plates is $0.005\ \mathrm{m}$. Let the fluid's dynamic viscosity be $\mu = 0.80\ \mathrm{Pa \cdot s}$.

If we assume a nearly linear velocity profile across the gap, then

$$\frac{du}{dy} = \frac{0.30}{0.005} = 60\ \mathrm{s^{-1}}$$

Now use the viscosity relation:

$$\tau = \mu \frac{du}{dy}$$ $$\tau = (0.80)(60)$$ $$\tau = 48\ \mathrm{Pa}$$

So the fluid needs a shear stress of $48\ \mathrm{Pa}$ to maintain that motion. If the same setup used a larger viscosity, the required shear stress would increase in the same proportion.

This example shows the role of viscosity clearly: it links how fast neighboring layers slide to how much shear stress is needed to keep them moving.

Common mistakes in viscosity problems

Treating viscosity as one universal formula

The equation $\tau = \mu \frac{du}{dy}$ is a model for Newtonian fluids. Many real fluids, such as blood, paint, or toothpaste, can behave non-Newtonian, so the relation between stress and shear rate is not always this simple.

Mixing up dynamic and kinematic viscosity

$\mu$ and $\nu$ are different quantities with different units. If density matters in the problem, make sure you are using the right one.

Forgetting the condition behind the worked formula

The standard shear formula is easiest to apply in simple shear flow, such as fluid between nearby moving layers or plates. In more complicated flows, the same basic idea remains, but the mathematics can be more involved.

Assuming high viscosity always means slow speed

Higher viscosity often makes flow harder, but speed also depends on pressure difference, geometry, gravity, and boundary conditions. Viscosity is one part of the picture, not the whole picture.

Where viscosity is used in physics and engineering

Viscosity matters in pipe flow, lubrication, blood flow, aerodynamics, manufacturing, and geophysics. Engineers use it when estimating drag, pressure loss, flow regime, and how a fluid will behave near surfaces.

It also helps explain everyday observations, such as why motor oil behaves differently at different temperatures and why syrup spreads much more slowly than water.

Try a similar problem

Keep the same plate example, but double the gap while keeping the top-plate speed and fluid the same. Predict what happens to $\frac{du}{dy}$ and the shear stress before calculating it, then check whether both quantities are cut in half.