Viscosity and Motion Through Fluids

Viscosity is resistance to relative motion inside a fluid

Slide a plate across a film of oil, then imagine replacing the oil by honey while keeping the speed and film thickness unchanged. A more viscous liquid needs a greater pulling force. Neighbouring fluid layers exchange momentum and exert tangential forces opposing their relative motion. This internal friction converts organised mechanical energy into internal energy. It explains why a real fluid can lose pressure along a pipe even when its average speed does not change.

A fluid at rest does not sustain a static shear stress like an elastic solid. Once shear stress is applied, however, the fluid deforms continuously. Its response depends on how quickly it is deforming. For the Newtonian fluids used in the simple model, shear stress is proportional to the rate of shear strain, not the accumulated strain. Some everyday materials are not Newtonian; the constant-viscosity formulas below are not universal descriptions of every paste or biological fluid.

From moving plates to the viscosity formula

Put a liquid film of thickness l between two broad parallel plates of area A. Hold the bottom plate fixed and move the upper plate to the right at constant speed v. With the no-slip boundary condition, the fluid touching the bottom is at rest and the fluid touching the top moves at v. In the simple steady, uniform, no-pressure-gradient arrangement, intermediate velocities increase linearly with distance from the bottom plate.

In time Δt the top moves Δx = vΔt. The shear strain acquired by the film is Δx/l. Divide this by Δt to obtain the shear rate v/l, with unit s⁻¹. If τ is shear stress, define the coefficient of viscosity η by τ = η(v/l). A layer is pulled forward by faster material above and pulled back by slower material below. The viscous force on the moving plate opposes its motion.

In a stationary circular pipe the liquid touching the wall is approximately at rest, while liquid near the axis moves fastest. Different cylindrical layers move at different speeds. The cross-sectional mean speed is not the centreline speed. No slip at the wall does not imply that the entire pipe flow stops. A pressure difference maintains flow against the viscous resistance.

Temperature and representative viscosity values

These representative values show that glycerine is much more viscous than water, while air’s dynamic viscosity is much smaller. Real honey varies greatly with composition and temperature, and real blood has a more complex rheology than a single Newtonian value. The source describes blood as several times more viscous than water and states that their viscosity ratio remains approximately constant from 0°C to 37°C; treat that as its simplified comparison, not a universal biomedical calibration.

For ordinary liquids viscosity generally decreases as temperature rises: the molecules can rearrange more easily despite their attractions. In gases viscosity generally increases with temperature because faster random molecular motion transfers momentum more effectively between layers. These opposite trends concern viscosity, not density. Heating a liquid also usually reduces its surface tension, which is a different physical property.

Pipe resistance: the tool needed for the flow exercises

Consider fully developed, steady laminar flow of an incompressible Newtonian liquid through a straight circular tube of radius R and length L. Let ΔP be the inlet pressure minus outlet pressure at equal elevation. A coaxial cylindrical fluid core of radius r feels forward pressure force ΔP πr². The surrounding liquid exerts a resisting shear force τ(r)2πrL. Force balance gives τ(r) = ΔP r/(2L).

Speed decreases outward, so its radial derivative dv/dr is negative. Here a derivative means the local rate of speed change with radius. The positive shear magnitude is −η dv/dr. Substituting the force balance gives dv/dr = −ΔP r/(2ηL). Adding these small speed changes from the stationary wall, where v(R) = 0, inward gives a parabolic profile. This is a short mathematical bridge for the pipe-flow exercises rather than a new empirical assumption about ideal nonviscous flow.

To obtain Q, split the circular section into thin rings. A ring of radius r and width dr has area approximately 2πr dr and transports volume v(r)2πr dr each second. Summing these rings means integrating from 0 to R. The integrals of r and r³ are R²/2 and R⁴/4; inserting the profile gives πR⁴ΔP/(8ηL). The mean speed is half the centreline speed in this profile. These results do not require a separate ideal Bernoulli pressure drop; they explicitly describe viscous dissipation.

The fourth-power radius dependence is powerful. With viscosity, tube length and pressure difference fixed, halving the radius reduces Q by a factor of sixteen. Changing pressure changes Q only in direct proportion in the same model. This explains why syringe-needle bore can control flow much more strongly than modest changes in thumb pressure. A real injection also includes tissue resistance and possible non-Newtonian behaviour; the lesson is the strong geometric sensitivity.

Flow regime and the role of size

Small Reynolds number suggests strong viscous control and smooth laminar behaviour. In a long straight pipe the transition is commonly discussed around Re of a few thousand, with details depending on disturbances; this is not a universal boundary for all flows. For a similar shape and the same fluid, keeping the same Reynolds number makes characteristic speed inversely proportional to size. Thus a smaller wind-tunnel aircraft model generally requires a greater speed to reach a comparable transition condition. Other similarity requirements may matter in a real wind tunnel.

Stokes’ drag on a slowly moving sphere

A sphere moving through a fluid drags the adjacent fluid, which disturbs neighbouring layers. For a small rigid sphere moving sufficiently slowly through an effectively unbounded Newtonian fluid, Stokes’ law gives a drag opposite the relative velocity. The source states this law without deriving the numerical factor 6π; dimensional analysis alone cannot supply that factor. This low-Reynolds-number law is not a general drag formula for every sphere at every speed.

A raindrop or a pendulum moving through air experiences resistance, but a typical large raindrop may deform and have a Reynolds number too large for Stokes’ law. Use the spherical low-speed model for sufficiently small droplets or suitable laboratory spheres. A force proportional to v gives a convenient model only within its regime.

Terminal speed is a force balance, not the disappearance of forces

Release a dense sphere in a stationary viscous liquid. Weight acts downward, while buoyancy and drag act upward once it falls. At first drag is small and the sphere accelerates. Increasing speed increases drag. Eventually the upward forces can equal weight and acceleration becomes zero even though the sphere keeps moving. That constant limiting speed is terminal speed.

Buoyancy equals the weight of displaced fluid, a prerequisite force used here: for sphere volume V = 4πa³/3, it is ρ_f Vg. If ρ_s is sphere density, weight is ρ_s Vg. At terminal speed, subtract buoyancy from weight and equate the remainder to Stokes drag. Keeping separate density symbols prevents using the fluid density twice.

Terminal speed scales as a², the density difference and 1/η. Doubling radius gives four times the terminal speed only if the new motion still satisfies the Stokes assumptions. A neutral-density sphere does not settle simply because its mass is nonzero; buoyancy cancels its weight. Terminal motion has zero net force, not zero weight or zero drag.

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