Illustrated Physics
physics, drawn
Series

Viscosity — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The straight line between two plates. A layer of water 10 mm deep with its top plate drawn along at 1 m/s. In the steady state the velocity is a straight line from zero at the fixed surface to the plate's speed at the moving one, and the stress needed to keep it going is μ times that slope: 0.100 Pa. The fluid at each wall is at rest with respect to it, which is an experimental fact rather than a consequence of anything above.

    Momentum going sideways

    Viscosity is usually described as friction between layers of fluid, which gets the effect right and the mechanism wrong. What is actually happening is that momentum is being conducted across the flow — by the same equation, with the same solutions, as a drop of ink spreading.

    part 1 · fluids
  2. The parabola in a pipe, and what it integrates to. Steady flow in a round pipe: the velocity is a parabola, zero at the wall and greatest on the axis, and its average over the cross-section is 0.500 of the peak — exactly a half, by integration. Because the profile scales with r² and the area with r² as well, the flow goes as the fourth power of the radius: widening a pipe from 1 to 2 mm multiplies it by 16.

    The fourth power in a pipe

    Halve a pipe's radius and the flow through it falls to a sixteenth. The exponent is four rather than two, because narrowing a pipe both removes cross-section and slows what is left — and one law with that exponent in it governs a blood vessel, a hypodermic needle and a water main.

    part 2 · fluids
  3. One word, two mechanisms, opposite signs. Viscosity on a logarithmic axis against temperature over the range 280 to 360 kelvin, where gases and liquids can both be measured. The gases rise and the liquids fall, and the two families are separated by three decades of magnitude as well as by sign. The logarithmic slopes at the middle of the range are 0.76 for air, 0.69 for helium, -6.13 for water, -5.41 for ethanol, -23.00 for glycerol, so the steepest liquid responds 30 times more strongly than the gas and in the other direction. Nothing about the word viscosity requires this: what is being measured in both cases is the ratio of a shear stress to a shear rate, and that definition says nothing about what carries the momentum. In a gas it is molecules in free flight, so heating speeds up the carriers; in a liquid the molecules are permanently in contact and what has to happen is one of them getting past its neighbours, so heating removes an obstacle rather than adding a carrier. The obvious question this raises is what a dense gas near its critical point does, where neither picture holds, and the honest answer is that neither formula on this chart applies there at all.

    The thickness that goes both ways

    Heat a liquid and it thins; heat a gas and it thickens. The two are not a strong effect and a weak one but opposite signs, differing by a factor of thirty in size as well — and the word viscosity names one measurement made on two mechanisms that have almost nothing in common.

    part 3 · fluids
  4. The motion that arrives late and does not go far. A flat plate sliding back and forth in its own plane, with the fluid above it drawn at 0°, 60°, 120°, 180° of the cycle. The speed at depth y is exp(−y/δ) times a cosine whose phase lags by y/δ, and both halves are checked: the largest speed reached at any depth follows exp(−y/δ) to a part in a thousand million, and the fluid one skin depth out is fastest a full radian after the plate is. Two skin depths out the motion is an eighth of the plate's and a quarter of a cycle behind; four skin depths out there is essentially nothing. Alternating shear does not diffuse away without limit — it fills a fixed depth and stops.

    The shear that only reaches so far

    Viscosity carries momentum sideways without limit when the driving is steady. Reverse the driving and it stops: the motion fills a depth set by the viscosity and the frequency, arrives there late, and beyond that depth the fluid does not know the wall exists.

    part 4 · fluids
  5. The four vortices a standing wave leaves behind. Streamlines of the steady flow that a standing sound wave sets up in a channel, over half an acoustic wavelength, with the horizontal axis in units of the wave's own phase and the vertical axis scaled to the channel. The sound itself is a back-and-forth motion that averages to nothing; this is what does not average to nothing. Four closed cells fill each wavelength, two above the centreline and two below, turning in opposite senses, with the fluid moving along the walls toward the velocity nodes and back along the centre. The boundary layer that generates all of it is 69 micrometres thick, which is 0.7 per cent of the channel and is thinner than the width of a line in this drawing. The cells are not in the layer; they fill the channel.

    The drift a sound leaves behind

    A sound wave moves fluid back and forth and puts it back where it started. Over many cycles it does not: a steady circulation appears, four cells to a wavelength, driven entirely from inside a boundary layer seventy micrometres thick. Its speed contains the sound speed and the amplitude, and it contains no viscosity at all — so making the fluid thinner does not make the drift weaker.

    part 5 · fluids
  6. The pressure at which an oil becomes a glass. Viscosity against pressure for 4 liquids on a logarithmic axis, from Barus's rule with the pressure coefficient each one actually has. The rule is an exponential, so a gigapascal multiplies an ordinary oil's viscosity by a hundred million or more, and the curves cross the line conventionally taken to mark a glass — a million million pascal-seconds — at 1.37 GPa for a mineral oil, 0.98 GPa for a traction fluid. The vertical line is the peak pressure inside the loaded contact this figure is about, 1.32 gigapascals, computed from the Hertz solution for that geometry and load. Water is drawn for contrast: its pressure coefficient is thirty times smaller, it never approaches a glass over this range, and that is why it is useless as a lubricant in a rolling contact however clean it is. Each curve is drawn solid up to the glass line and dashed above it, because past that point the material is not a liquid and its viscosity is not what decides how it shears — the exponential continues, and the substance it was written for does not.

    The oil that is a glass for a quarter of a millisecond

    Every other account of viscosity here varies the temperature. Pressure does something larger and in the same direction for every liquid: viscosity rises exponentially, by a factor of ten to the eight or more at the gigapascal inside a loaded gear tooth. That is not a curiosity — it is the only reason there is a film there at all. Remove the pressure dependence from the calculation and the predicted film is five nanometres, under the roughness, and the surfaces touch.

    part 6 · fluids
  7. 14 decades of fluid, and a floor none of them reaches. The ratio of viscosity to entropy density for 8 substances, in units of the proposed lower bound, on a logarithmic axis. The quantity is a viscosity divided by how much entropy a cubic metre of the substance holds, and it has the dimensions of Planck's constant over Boltzmann's — so a bound on it is a statement with no material properties in it at all. The span here is a factor of 1.6e+14, from pitch at 20 °C at the top to the quark–gluon plasma at the bottom, which sits a factor of 1.6 above the floor. The values marked as computed are worked out from a viscosity and a tabulated entropy; the rest are quoted from the compilations, because a minimum along an isobar is an inference from many measurements rather than a single one.

    Whether a fluid can be made arbitrarily thin

    Viscosity has no units anybody would call fundamental, and nothing obviously stops it being as small as you like. Divide it by the entropy in a cubic metre and the units become Planck's constant over Boltzmann's — and a conjecture from 2005 says that ratio has a floor. Across fourteen decades of ordinary fluid nothing has been measured below it, the closest thing to it is the hottest matter ever made, and a kinetic-theory argument reaches the same number from the opposite direction.

    part 7 · fluids
  8. The flow past a thread takes thousands of widths to recover. The speed of the fluid beside a body, at right angles to the stream, as a fraction of the free stream, against distance from the body's axis in body radii, in slow viscous flow. Past a sphere, from Stokes's solution, the flow is back to 90 per cent of the stream by 7.5 radii whatever the Reynolds number. Past a cylinder, from Lamb's inner solution, the speed grows only as the logarithm of distance, with a slope set by the Reynolds number: it reaches 90 per cent of the stream at 29, 232, 1,843 and 1.5 × 10⁴ radii for Re = 10⁻¹, 10⁻², 10⁻³ and 10⁻⁴ in turn. Each cylinder curve is drawn out to where the inner solution reaches the stream speed, beyond which Oseen's outer flow takes over. Each tenfold drop in the Reynolds number pushes the recovery ten times further out, because what finally stops the disturbance is inertia, at a distance of order ν/U.

    The drag a distant wall decides

    A sphere moving slowly through a liquid has a drag that belongs to the sphere: six pi times the viscosity, the radius and the speed, and the liquid a few radii away hardly knows it is there. An infinitely long cylinder has no such drag. The equations of slow viscous flow have no solution past it at all — Stokes found this in 1851 — because a force spread over circles instead of spheres never finishes spreading. What rescues the cylinder is whatever distant thing supplies a largest length, and its drag carries the logarithm of that length for ever after.

    part 8 · fluids
  9. A rod in a shear flow turns in fits and starts. The angle between a rigid rod's axis and the flow direction, unwrapped, against the strain of a steady shear flow (shear rate times time), for spheroids of aspect ratio 1, 5, 20 lying in the plane of the shear, each starting along the flow. A sphere (r = 1) turns steadily at half the shear rate. A rod spends most of its time nearly aligned with the flow, turning slowly, and then flips through half a turn quickly; at r = 5 it is within 10° of the flow direction 46 per cent of the time, at r = 20, 82 per cent. Each flip is half of Jeffery's period, π(r + 1/r) units of strain.

    The rod that tumbles for ever in a stream

    Drop a short rod into honey being sheared between two plates and it does not settle into line with the flow. It lies nearly along the stream for a long time, then flips end over end in a moment, then lies along it again, and it keeps doing that for as long as the shear lasts. George Jeffery worked out the motion in 1922: every orientation belongs to a closed orbit, every orbit takes the same time, and the flow has no way to choose between them — so a suspension of rods never reaches a steady state, and its thickness rises and falls in time with the tumbling.

    part 9 · fluids
  10. The suction that holds two wet plates together. The force needed to pull apart two discs 10 cm across, starting 10 micrometres apart with water between them, at constant opening speeds of 1, 10 and 100 micrometres a second, against time; the dashed line is the largest suction the water can transmit, ½πR²pₐₜₘ = 398 N, reached when the pressure at the centre falls to zero and the water cavitates. At 1 µm/s the force starts at 29.5 N and falls as the cube of the widening gap, to 10.7 N after four seconds, when the gap is 14 micrometres. At 100 µm/s the viscous law would demand 2.9 kN; the water cannot pull that hard, and the force sits on the cavitation limit until the gap has opened enough for the viscous suction to fall below it.

    The wet plates that hold on when pulled

    Lift a wet glass plate straight up off another and it resists with a force far beyond its weight; slide it sideways and it comes away at a touch. The grip is not surface tension. It is the water that must rush into the widening gap, and the suction it takes to draw it in grows as the inverse cube of the gap — until the water tears open and the atmosphere sets the limit. The same cube makes a flat base float on a wet table where a ball would settle, and gives a micro-machined accelerometer exactly the damping it needs from a few micrometres of air.

    part 10 · fluids
  11. Nine drops in eighty-four years. The time between successive drops of the pitch in Thomas Parnell's funnel, cut open in 1930, against the year each drop fell: 8.2, 7.2, 8.1, 8.3, 8.7, 9.3, 12.4, 13.4 years. The first seven intervals average 8.3 years and scatter by about a year either way; after the building was air-conditioned in the late 1980s the next two took 12.4 and 13.4, averaging 12.9. Read as a viscometer through the funnel's geometry, the early rate gives a viscosity of about 2.3 × 10⁸ Pa·s — some two hundred billion times water's.

    The drop that takes a decade

    In 1927 a physics professor in Brisbane poured heated pitch into a sealed funnel, waited three years for it to settle, and cut the stem. Nine drops have fallen since. The record is a viscometer read about once a decade, and it gives the pitch a viscosity two hundred billion times water's. It also records the building's air-conditioning: after 1988 the drops slowed from eight years apart to thirteen, and a fall of a degree and a half in the average temperature is enough to explain it. The same reasoning, applied to medieval window panes, settles an old story without guessing what glass does at room temperature.

    part 11 · fluids

All series