Viscosity and Bernoulli’s Principle

Viscosity measures resistance to layers of fluid sliding past one another. Bernoulli's principle relates pressure, speed and height in an ideal steady flow.

• Fluid: a liquid or gas that can flow.
• Viscosity: resistance to relative motion within a fluid.
• Steady flow: velocity at each fixed position remains unchanged over time.
• Streamline: a curve whose tangent follows the local fluid velocity.

Think of pouring water and honey at the same temperature. Honey usually resists flowing more strongly. That comparison concerns viscosity, rather than electrical conductivity or attraction to unrelated substances.

Neighbouring layers exert shearing forces when their velocities differ. Imagine fluid between two parallel plates. The lower plate stays still while the upper plate moves.

Keeping the upper plate moving requires an applied force. For a Newtonian fluid, viscous stress is proportional to the velocity gradient:

F/A = η × Δv/L

Here, F/A is shearing force per unit area. Δv is the speed difference across separation L. Dynamic viscosity, η, has the International System of Units (SI) unit pascal-second, Pa·s.

For the same geometry and speed difference, a larger viscosity requires a larger force. Viscosity is not density: density measures mass per volume.

A liquid's viscosity generally falls as temperature rises. Gas viscosity generally increases with temperature, so the liquid rule is not universal.

Colder lubricating oil can resist flow more strongly. Oil selection therefore considers operating temperature. That connection does not mean every winter machinery problem is caused by viscosity.

Yes. Steady flow fixes velocity at a position, rather than along a moving particle's entire journey.

Imagine water passing through a narrowing pipe. Successive particles have the same velocity when they reach the same position. A particle can still speed up as it travels toward the narrow section.

Laminar flow has orderly layers. Turbulent flow contains irregular fluctuations and mixing. These descriptions concern the flow pattern, while “steady” concerns change over time.

A steady incompressible flow carries the same volume per second through each section of an unbranched pipe. This gives the continuity equation:

Q = Av and A₁v₁ = A₂v₂

Q is volume flow rate, measured in m³/s. A is cross-sectional area, and v is average speed.

Suppose one section has area 4 cm² and speed 1 m/s. A second section has area 2 cm².

The second speed is (4/2) × 1 = 2 m/s. The area halves, so the speed doubles. Neither section loses water from the flow.

Bernoulli's equation expresses mechanical energy conservation along a streamline. Its basic form assumes steady, incompressible flow with negligible viscous losses.

P + ½ρv² + ρgh = constant

At equal height, increased speed corresponds to lower static pressure under these assumptions. If height changes, the gravitational term also matters.

The equation does not say that any faster fluid anywhere must have lower pressure. Compare points within the applicable flow, rather than unrelated streams.

Use the ideal equal-height model for water of density 1000 kg/m³. Let speed increase from 1 m/s to 2 m/s.

P₁ − P₂ = ½ρ(v₂² − v₁²)

P₁ − P₂ = ½ × 1000 × (4 − 1) = 1500 Pa

The faster section has 1500 Pa lower static pressure. This is a pressure difference, rather than either section's absolute pressure.

Real pipes also dissipate energy through viscosity. The calculation illustrates the stated ideal model.

A venturimeter uses a constriction's pressure difference to determine flow rate. Continuity links the narrower section with greater speed. Bernoulli's equation then connects that speed change with pressure.

In a simple air-jet atomiser, fast air passes across the liquid tube's opening. The pressure difference helps draw liquid into the jet. The liquid breaks into small droplets.

Carburettors use a related flow arrangement to introduce fuel into moving air. The teaching example concerns this arrangement; every spray bottle need not use the same mechanism.

A force depends on the pressure distribution around the object. The slogan “faster air means lower pressure” cannot establish that distribution by itself.

For passing-train and passing-car examples, first identify the compared air regions and the model assumptions. Do not treat vehicle speed alone as proof of a particular sideways force.

Aircraft lift involves a pressure distribution and downward deflection of air. Bernoulli's equation and Newton's laws describe compatible aspects of that flow.

Air above and below a wing is not required to arrive at the trailing edge together. That equal-transit-time explanation is incorrect.

Terminal velocity occurs when the forces balance, making acceleration zero. A dense sphere falling through a liquid experiences downward weight and upward buoyancy and drag.

For a small sphere in slow flow, Stokes' drag is 6πηrv. Here, r is sphere radius and v is speed relative to the liquid.

As speed increases, this drag increases. At terminal speed, weight equals buoyancy plus drag:

vₜ = 2r²(ρₛ − ρ𝒇)g / (9η)

ρₛ and ρ𝒇 are sphere and fluid densities. For fixed densities and viscosity, doubling radius makes this model's terminal speed four times larger.

The formula requires slow, creeping flow around the sphere and negligible wall effects. It cannot be applied to every falling object.

No. Surface tension concerns an interface, while viscosity concerns resistance within moving fluid. Surface tension has units of force per length, N/m.

A droplet's rounded shape illustrates surface effects. Resistance between sliding layers illustrates viscosity.

For a given liquid interface, surface tension generally decreases as temperature rises. This resembles the usual liquid-viscosity trend, but the two properties remain distinct.

Shape affects the pressure forces and friction acting on a moving body. Streamlining can reduce drag by improving the surrounding flow.

It does not eliminate viscosity or guarantee laminar flow at every speed. Flow behaviour depends on speed, fluid properties and object size as well as shape.

The National Defence Academy and Naval Academy (NDA and NA) questions below test two useful distinctions.

Position or particle? — NDA and NA II, 2016, Q103

The linked streamline-flow question compares successive particles at a fixed position. In steady flow, the velocity there remains unchanged.

This does not mean a moving particle's speed cannot change elsewhere. The narrowing-pipe example shows why that distinction matters.

Read the streamline-flow question.

Which property changes with temperature? — NDA and NA II, 2025, Q66

The linked question tests surface tension's usual temperature trend. For the same liquid interface, surface tension generally falls when temperature rises.

Do not extend that answer to gas viscosity. Gas viscosity generally follows the opposite temperature trend.

Read the surface-tension question.