The viscosity that belongs to the tube

The fluid that answers back found that for many fluids viscosity is not a number but a function of the shear rate, and the paste that forgets it was stirred added the time the fluid has been sheared. Both kept one assumption that is so natural it is rarely stated: that the fluid’s behaviour at a point depends on what is happening at that point — the local rate, the local history — and not on how far away the walls are. The fourth power in a pipe used that assumption to derive Poiseuille’s law, in which the flow through a pipe goes as the fourth power of its radius and a viscosity measured once serves for every pipe.

For blood in small vessels the assumption fails, and it fails for a reason that has nothing to do with the chemistry of blood. The fluid contains objects — red blood cells, flexible discs about eight micrometres across and two thick — that are comparable in size with the vessels they flow through. A fluid with an internal length behaves differently in pipes near that length, and the result is a viscosity that belongs as much to the tube as to the blood. It was measured by Robin Fåhræus and Torsten Lindqvist in 1931, and it is one reason the heart can drive blood through capillaries at all.

A viscosity that falls as the tube narrows #

The experiment is simple in principle: drive blood through glass tubes of different diameters at a known pressure difference, measure the flow, and ask what viscosity a uniform liquid would need to give that flow by Poiseuille’s law. That is the tube’s apparent viscosity.

The figure plots a fit, made by Axel Pries and colleagues in 1992, to glass-tube measurements from many laboratories. In tubes wider than about half a millimetre blood at the normal red-cell fraction is about 3.2 times as viscous as its plasma, and the value does not change with the diameter, as a fluid property should not. Below that it falls: to about 2.4 times at a tenth of a millimetre, 1.6 at twenty micrometres, and 1.25 at a diameter of about seven micrometres, less than the size of a red cell. Then it turns sharply upward. At three micrometres the apparent viscosity is 5.5 times plasma’s, because a cell eight micrometres across can pass only by folding itself into a tube narrower than half its own diameter.

The fall is the Fåhræus–Lindqvist effect, and it is large. Between the artery and the capillary the blood becomes more than twice as easy to push, for the same blood.

A sleeve of plasma at the wall #

The explanation is visible under a microscope. In a small vessel, flowing red cells are not spread evenly across it: they crowd towards the middle and leave a layer next to the wall, a couple of micrometres thick, that contains only plasma.

The figure uses the simplest model of that arrangement: a core of blood with a single viscosity, here four times plasma’s, inside a sleeve of plasma of fixed thickness, with Poiseuille flow in each. The velocity is continuous at the boundary, and so is the shear stress, so the plasma, being four times less viscous, shears four times as fast for the same stress. A layer that is only a sixth of the radius carries a disproportionate share of the velocity difference between wall and centre, and the core moves with a flatter, blunted profile, riding on the sleeve. The flow through the tube is what a uniform liquid 1.57 times as viscous as plasma would pass, far less than the core’s four.

Why the cells leave the wall is a question of fluid mechanics rather than biology. A deformable particle in a shear flow near a wall is pushed away from it by a lift force: the flow around the particle is not symmetric, because the wall restricts it on one side, and the particle’s deformation makes the asymmetry push outward. Red cells tumble and tank-tread as they are sheared, and they migrate inwards until the lift is balanced by collisions with other cells in the crowded core. The balance leaves a layer whose thickness depends on the flow rate and the red-cell fraction but only weakly on the size of the tube — a few micrometres in vessels from ten micrometres to hundreds.

A fixed thickness in a shrinking tube #

That weak dependence on size is the whole of the effect.

The two-layer model gives the apparent viscosity in closed form: plasma’s viscosity divided by one minus the fourth power of the core’s share of the radius times one minus the ratio of plasma to core viscosity. The fourth power comes from Poiseuille’s law, which weights the outermost part of a pipe most heavily because that is where the shearing, and the resistance, is concentrated. A sleeve a two-hundredth of the radius wide changes the flow through a millimetre tube by five per cent. The same sleeve in a twenty-micrometre tube is a quarter of the radius and removes most of the resistance. The figure plots the model for three sleeve thicknesses: every curve has the same shape, sliding towards smaller tubes as the sleeve thins, and every one runs from the core’s viscosity in wide tubes towards plasma’s in narrow ones.

This is the general form of a fluid property ceasing to be one. A viscosity belongs to a fluid when every internal length of the fluid is small compared with the scales of the flow. The viscosity that does not care how much gas there is found gas viscosity independent of density because the mean free path is small compared with the vessel; when it is not, in a rarefied gas or a very narrow channel, the gas slips at the wall and its apparent viscosity falls with the channel’s size, which is the same effect with a molecule’s free path in place of a red cell’s sleeve. Here the internal length is the sleeve’s thickness, set by the size and deformability of the cells.

Fewer cells inside than come out #

The same sleeve has a second consequence, which Fåhræus found two years before the viscosity measurements, in 1929.

The blood leaving a narrow tube has the same proportion of red cells as the blood entering it; nothing is lost. But a snapshot of the tube’s contents at any moment shows fewer cells than that proportion. The cells travel in the fast core and the plasma in the sleeve moves slowly, so the cells cross the tube quicker than the plasma does and spend less time inside it; to carry the same number of cells per second, fewer need be present at once. The figure plots the ratio of the two fractions: at the normal red-cell fraction it falls to 0.72 in tubes of about eleven micrometres, and further at lower fractions, before rising again in the narrowest tubes, where cells go through in single file.

The Fåhræus effect is the same physics as the viscosity drop, seen as a shortage of cells rather than as a lubricating layer, and it partly explains the other: a tube with fewer cells in it at any moment has a less viscous content. It also matters in the body directly. The red-cell fraction inside the microcirculation is markedly lower than in the large vessels, so the blood’s oxygen-carrying capacity inside the capillary beds is less than a sample from a vein suggests.

Cheaper cells in narrow vessels #

The red-cell fraction also matters less in narrow tubes.

In a wide tube adding red cells raises the viscosity steeply, as it does for any concentrated suspension: going from twenty to sixty per cent red cells nearly triples it. In a tube eight micrometres across the same change raises it by less than a third. There the cells travel in single file, each separated from the next by a slug of plasma and from the wall by a thin film, and a cell adds little resistance beyond the drag of its own film. The body uses this: the red-cell fraction can rise, as it does at altitude, without a proportionate cost in the smallest vessels, where most of the resistance of the circulation lies.

The resistance does not disappear. The fourth power in a pipe found that a pipe’s resistance grows as the inverse fourth power of its radius, so halving a vessel’s radius raises its resistance sixteenfold, and the Fåhræus–Lindqvist effect in that range reduces it by perhaps a factor of two. It softens the fourth power at the small end without removing it. The largest share of the pressure drop in the circulation is in the arterioles, a few tens of micrometres across, where the effect is already present and the fourth power is steep; and it is the arterioles whose radius the body adjusts to control flow.

Where particles go in a flowing pipe #

The migration of red cells away from the wall is one case of a general and surprisingly subtle phenomenon: particles carried in a flowing liquid do not stay where they are put across a pipe. In 1961 Gerhard Segré and Alex Silberberg sent rigid, neutrally buoyant spheres down a tube and found them gathering neither at the centre nor at the wall but in a ring at about six-tenths of the radius — the tubular pinch. Two lift forces balance there: one from the curvature of the velocity profile, pushing particles towards the wall, and one from the wall itself, pushing them away. The effect needs inertia and vanishes in the slowest flows, and it is now exploited in microfluidic devices that sort cells by size, each size focusing to its own position in a curved channel.

Red cells in small vessels behave differently because they deform. A flexible particle in shear takes up a shape that is not symmetric fore and aft, and the flow around that shape pushes it away from the wall even with no inertia at all. So red cells drift towards the centre, not to a ring, and a mixture of deformable red cells and stiffer platelets and white cells separates: the stiffer cells are pushed out of the crowded core towards the wall, where platelets are needed to seal a damaged vessel and white cells to leave it. The big one comes to the top found grains of different sizes separating in a shaken or poured granular mixture; the separation of blood cells by stiffness in a shear flow is a liquid cousin of it, and the body relies on it.

It also explains why the core of the two-layer model is more viscous than blood in a wide tube. The cells missing from the wall layer are not lost; they are in the core, which therefore holds them at a higher concentration than the blood as a whole, and the suspension that seizes when pushed hard found how steeply a suspension’s viscosity rises with its concentration. The core’s four times plasma’s viscosity, against the whole blood’s 3.2, is that concentration showing.

Why narrower tubes finally cost more #

Below about six micrometres the trend reverses, and the reason is the cells’ size rather than their distribution. A red cell is a biconcave disc about eight micrometres across, with a surface area about forty per cent more than a sphere of the same volume would need. That excess area is what lets it fold into a bullet or a slipper shape and pass through a capillary narrower than itself, and it sets a limit: a cell cannot pass through a cylinder so narrow that its volume, at that area, would not fit. The limiting diameter is about 2.7 micrometres. Approaching it, the cells press against the wall through a thinning lubricating film, and the apparent viscosity climbs steeply.

A cell that has lost deformability — through age, through the rigidity of the sickle-shaped cells in sickle-cell disease, or in malaria, where the parasite stiffens the cell it lives in — cannot fold and blocks vessels far wider than the limit. The minimum of the curve and its steep rise to the left are therefore a measurement of how flexible the cells are, and changes in them are part of what goes wrong in those diseases.

Why a single viscosity usually works #

It is worth being clear about why the ordinary picture holds in the first place, since the Fåhræus–Lindqvist effect is its failure. Viscosity is the rate at which a fluid carries momentum sideways, from faster layers to slower ones — momentum going sideways — and it is a property of the fluid when the thing carrying the momentum, a molecule or a particle, moves only a short distance compared with the scale over which the flow changes. For water, the relevant distance is molecular, a fraction of a nanometre, and water’s viscosity holds in any channel wider than a few nanometres. The shear that only reaches so far found the other place the ordinary picture has a length in it — the depth to which an oscillating wall’s motion diffuses — but that length belongs to the flow and its frequency, not to the fluid.

A suspension has a new length, the size of its particles and the layers they leave near walls, and it is large. For blood it is micrometres, the size of the smallest vessels, so the continuum description breaks down exactly where much of the circulation’s resistance lies. For sand in water or fibres in paper pulp it is millimetres, and the same effects appear in pipes a few centimetres across. The rule is the one that ends any continuum description: it holds when the flow’s scale is large compared with the fluid’s internal lengths, and near the crossover the fluid’s properties start to depend on the container.

What the glass tube does not show #

The figures describe steady flow of blood in straight, smooth, rigid glass tubes, which is how the effect was discovered and measured. Living vessels are none of those things. Their inner surface is lined with a layer of long-chain molecules, the glycocalyx, a fraction of a micrometre thick, which slows the plasma next to the wall and raises the resistance of small vessels above the glass-tube values; measurements in living tissue give apparent viscosities in the smallest vessels two or three times those in glass. The vessels branch every few hundred micrometres, and at each branch the cells divide unevenly between the daughters, which alters the red-cell fraction from vessel to vessel through the network. And flow in small vessels is pulsatile and variable, while the sleeve takes a certain distance to form after each branch.

The two-layer model is a caricature even in glass. It treats the core as a uniform fluid with one viscosity, the sleeve as sharply bounded, and its thickness as fixed; real sleeves have a ragged edge that fluctuates as individual cells approach and leave the wall. It shows where the effect comes from and gets its size roughly right, which is its purpose, and the fits it is drawn against are fits to data, not a theory.

Still open: how thick the sleeve is, and why #

The thickness of the cell-free layer, and how it depends on the flow rate, the red-cell fraction and the deformability of the cells, is the quantity the whole effect turns on, and it is not predicted well from first principles. Simulations that follow thousands of deformable cells through a vessel now reproduce the Fåhræus and Fåhræus–Lindqvist effects quantitatively, and show that the balance between wall-induced lift and cell–cell collisions sets the layer, but they are expensive and depend on models of the cell membrane that are still being refined. How the glycocalyx changes the layer in living vessels, and how much of the difference between glass-tube and living-tissue resistances it accounts for, is actively measured and disputed.

The habit worth carrying away is to ask whether a fluid has a length of its own before trusting a viscosity measured somewhere else. Blood carries a length — the thickness of the cell-free sleeve its cells leave at a wall — and in a tube a few times that length the sleeve does much of the flowing, so the apparent viscosity falls by more than half from artery to capillary and then rises steeply where the cells must fold. A viscosity quoted without the size of the tube is, for such a fluid, only half a measurement.