Physics · Mechanics · Grade 10

Viscosity and flow

Why does a ball sink slowly in honey, while water in a pipe suddenly “roars” into turbulence?

The situation

In winter honey thickens and barely runs off a spoon; heated, it flows like water. A small ball dropped into honey sinks slowly and steadily, while in water it drops in an instant. Open a tap a little and the water comes out as a clear, smooth thread; open it more and it suddenly turns into a noisy, swirling flow. Behind a bridge pier the river water forms eddies, and a spinning kicked ball curves away from its path. How does the “stickiness” of a fluid affect all this?

Topic

Viscosity, laminar and turbulent flow. A ball sinking in honey and flow in a water-supply pipe

Key ideas

Stokes' law
Poiseuille flow
Reynolds number

Formulas

F = η S \frac{Δ v}{Δ y}

Newton's law of internal friction: $η$ — the coefficient of viscosity (Pa·s).

F_{c} = 6 π η r v, η = \frac{2 r^{2} (ρ_{s h} - ρ_{s}) g}{9 v}

Stokes' law ( $R e ≪ 1$ ) and the formula for measuring viscosity from it.

Q = \frac{π R^{4} Δ p}{8 η l}

Poiseuille's formula: the rate of laminar flow in a pipe.

R e = \frac{ρ v d}{η}

The Reynolds number: the ratio of inertial to viscous forces. In a pipe the flow becomes turbulent at $R e ≳ 2300$ .

ρ (\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla) \vec{v}) = - \nabla p + η Δ \vec{v} + ρ \vec{g}

The Navier–Stokes equation (incompressible viscous fluid, $\nabla \cdot \vec{v} = 0$ ).

Theory

Viscosity — internal friction

Layers of fluid moving at different speeds drag on one another: the fast layer pushes the slow one forward, and the slow one brakes it. As a result momentum is transferred between the layers and a tangential force appears: $F = η S Δ v / Δ y$ . In flow between plates (Couette flow) the velocity changes linearly, and the force needed to pull the plate gives $η$ directly. The viscosity of water is ≈ 1 mPa·s, that of glycerol at 20 °C ≈ 1.4 Pa·s, and that of honey several times more. When liquids are heated, $η$ falls sharply (for gases, on the contrary, it rises).

Stokes' law and measuring viscosity

When a small ball moves slowly ( $R e ≪ 1$ ), the drag force is proportional to speed: $F_{c} = 6 π η r v$ . As the ball falls, its weight, the buoyant (Archimedes) force and the drag quickly balance — it falls at a constant speed: $v = 2 r^{2} (ρ_{s h} - ρ_{s}) g / 9 η$ . In the laboratory the time $t$ to cover the distance $l$ between the rings is measured, $v = l / t$ , and $η$ is calculated. Conditions: $R e$ is small, the ball is far from the walls of the vessel, and the speed has settled before the first ring. At higher speeds the drag grows faster (≈ $v^{2}$ , the “Free fall and air” experiment). Dry friction (static, sliding, rolling), on the other hand, hardly depends on speed — see the “Car braking” experiment.

Poiseuille flow

In laminar flow in a pipe the layer next to the wall is at rest, while on the axis the speed is greatest, 2 times the mean speed — the profile is a parabola. The flow rate $Q = π R^{4} Δ p / 8 η l$ depends on the fourth power of the radius: if the radius is halved, the flow rate falls 16 times. There are similar exact solutions: between two plates (a linear or parabolic profile) and in a thin layer flowing down a slope ( $u = ρ g \sin α (h y - y^{2} / 2) / η$ , fastest at the surface).

Laminar and turbulent flow

In laminar flow the layers do not mix — dye is drawn out into a thin thread. When the speed increases and the Reynolds number $R e = ρ v d / η$ exceeds a critical value (≈ 2300 in a pipe), the flow becomes turbulent: irregular eddies mix the dye across the whole cross-section, the velocity profile flattens and the resistance rises sharply ( $Δ p \propto v^{1.75}$ ). $R e$ shows the ratio of the inertial force to the viscous force: honey is very viscous, so its flow is almost always laminar.

The effect of flow on a body, the Magnus effect

The flow pushes a body with a drag force; a wake of eddies is left behind the body. Behind a cylinder, from $R e \approx 50$ , eddies are shed alternately from the two sides — a Kármán vortex street; the shedding frequency is $f \approx 0.2 v / d$ (the Strouhal number). This is why wires hum in the wind. If a cylinder or ball is spinning, then on the side where its surface moves in the same direction as the flow the flow is faster and the pressure lower — a lift force across the flow appears (the Magnus effect, $F = ρ v Γ$ in an ideal fluid). The lift of a wing and Bernoulli's equation are covered in the “Aeroplane wing” experiment.

The Navier–Stokes equation and the momentum flux

The motion of a viscous fluid is described by the Navier–Stokes equation. It is the law of conservation of momentum: $\partial (ρ v_{i}) / \partial t = - \partial Π_{i k} / \partial x_{k}$ , where the momentum flux density tensor is $Π_{i k} = p δ_{i k} + ρ v_{i} v_{k} - σ_{i k}^{'}$ and $σ_{i k}^{'} = η (\partial v_{i} / \partial x_{k} + \partial v_{k} / \partial x_{i})$ are the viscous stresses. Couette, Poiseuille and slope flows are its exact solutions. The flow in the water tunnel is calculated with the lattice Boltzmann method: the distribution of particles moves from cell to cell and on average gives the Navier–Stokes equation (a two-dimensional model). Surface tension and capillarity are covered in the “Surface tension and capillarity” experiment.

Did you know?

In 1883 Osborne Reynolds injected dye into water in a glass pipe and showed laminar and turbulent flow; the dimensionless number that determines the flow regime was later named after him.
Jean Poiseuille was a physician and, to understand the flow in blood vessels, experimented with thin glass tubes. According to his law, if the radius of a tube decreases by only 10 %, the flow rate falls by about 34 % ( ${0.9}^{4} \approx 0.66$ ).

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Free fall and air — Grade 10
Jet propulsion and the rocket — Grade 10
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Lagrangian mechanics and chaos — Year 2
Hamiltonian mechanics — Year 2