Viscosity
Viscosity measures how strongly a fluid resists shear: neighboring layers moving at different speeds drag on one another. It is the reason honey flows more reluctantly than water and why real pipe flow loses mechanical energy.
Dynamic viscosity and shear
τ = η du/dy
τ is shear stress, η dynamic viscosity and du/dy the velocity gradient perpendicular to the flow.
This linear relation defines a Newtonian fluid. The SI unit of dynamic viscosity is Pa s.
Dynamic viscosity η should not be confused with kinematic viscosity ν = η/ρ, which divides out density and has SI unit m² s⁻¹. Both appear in fluid mechanics, but they answer different questions.
Laminar flow in a tube
For steady laminar flow of a Newtonian fluid through a circular tube, Poiseuille's law gives:
Q = πΔp r⁴/(8ηL)
The fourth power of radius is the striking feature: a modest change in tube radius can strongly change flow rate. The law assumes laminar flow, a rigid circular tube and no-slip at the wall.
Equivalently, for a fixed flow rate, a more viscous fluid or a narrower tube requires a larger pressure drop. The pressure energy is dissipated into internal energy by viscous friction, which is why real fluid motion is not perfectly reversible.
Why pipe radius matters so much
For laminar Poiseuille flow with the same pressure drop, length and viscosity, Q ∝ r⁴.
Viscous drag at low Reynolds number
For a small sphere moving slowly enough that creeping-flow conditions apply, Stokes' law gives Fd = 6πηrv. The drag is proportional to speed rather than speed squared.
A falling sphere can reach terminal speed when weight, buoyancy and viscous drag balance.
Whether viscous or inertial effects dominate is summarized by the Reynolds number, typically Re = ρvD/η. Low Re favors smooth, viscosity-dominated flow; larger Re makes inertial effects and turbulence increasingly important. The transition value depends on geometry and disturbances, so it is not a universal single threshold.
Temperature changes viscosity
For most liquids, viscosity decreases as temperature rises because molecular rearrangements occur more readily. For gases, viscosity generally increases with temperature because faster molecules transport momentum more effectively between layers.
This opposite temperature trend is a useful reminder that viscosity is not simply “thickness.” It is a transport property arising from molecular interactions and momentum exchange, and its microscopic origin differs between dense liquids and dilute gases.
Not every fluid is Newtonian
In non-Newtonian fluids, the relation between shear stress and shear rate is not a simple constant η. Ketchup and some polymer solutions can shear-thin; concentrated suspensions can shear-thicken. “Viscosity” can then depend on the imposed shear rate and on flow history.
Before applying Poiseuille or Stokes formulas, check that the flow regime and fluid behavior fit the assumptions.
Worked examples
1. Shear stress
A Newtonian fluid has η = 0.80 Pa s and a velocity gradient du/dy = 3.0 s⁻¹. Find the shear stress.
Solution
τ = η(du/dy) = 0.80 × 3.0 = 2.4 Pa.
2. Changing tube radius
In ideal laminar Poiseuille flow, the tube radius doubles while pressure drop, length and viscosity stay fixed. By what factor does flow rate change?
Solution
Q ∝ r⁴, so Q'/Q = 2⁴ = 16.
3. Stokes drag
A small sphere of radius 1.0 mm moves slowly through a fluid of viscosity 0.50 Pa s at 0.020 m s⁻¹. Estimate the Stokes drag.
Solution
Fd = 6πηrv = 6π(0.50)(0.0010)(0.020) ≈ 1.88 × 10⁻⁴ N. This model assumes creeping flow around a sphere.