Biorheology
Above: Reynold's classic experiment in which flow in a pipe is visualised by injecting dye, and in which is orderly and laminar at
low velocities (strictly low Re), becomes turbulent and chaotic at high velocities (strictly high Re).

Biorheology is the study of fluid flow in biological systems. Rheology is the study of fluid flow, and relies on the principles of fluid
mechanics - the physical properties of flowing fluids, a fluid being a liquid or a gas. Biorheology includes the study of swimming
and flight, of fluid flow in blood and in plants, the aerodynamics of seed dispersal in plants, the movement of gases in the
respiratory system, and the behaviour of cells exposed to flowing fluids.

One key concept is fluid
viscosity, a measure of how thick and sticky the fluid is.

Reynold's number (Re) is of fundamental importance in fluid mechanics. It is essentially the ratio of inertial to viscous forces and
is a dimensionless number (it has no units, it is just a number).

At very low Re, say much less than 1, viscosity dominates and we tend to have thick, sticky fluids flowing slowly, often by
creeping. This could include the flow of oil in a mechanism moving at low velocities (oil gets thinner or less viscous when warmed
and flowing quickly in a mechanical part that is fast moving) and coating processes (such as coating plastic tape with adhesive or
fabrics with PVC to simulate leather). In nature, Re becomes high on the microscopic scale: cells swimming or crawling in water
experience very low Re as water behaves as a highly viscous fluid on this tiny scale. It is as if a cell is swimming through treacle!

At high Re the viscosity becomes negligible and inertial forces due to the bulk movement of fluid becomes dominant. This occurs,
for example, at high flow velocities. Consider, for example, the flow of fluid past a cylinder perpendicular to the flow (assumed
stationary). Using a cylinder allows us to view a cross-section through a cylinder and consider the flow in 2D only (we pretend
that the cylinder is very long, so that we can ignore what happens at the ends of the cylinder).
Above: At low Re (less than about 20) the flow is
steady (it's pattern does not change with time), with
the fluid moving in orderly layers (lamina or sheets,
so-called laminar flow), but at high Re the flow
becomes increasingly disturbed. At Re = 0.01, the
flow is almost symmetrical, it is about the horizontal
axis, but is slightly asymmetric about the vertical axis.
At first a pair of stable (and still laminar) vortices
appear behind the cylinder (at Re of about 20) and
these increase in size as Re increases, until at about
Re of 100, one vortex detaches and gets carried
downstream, a new one grows in its place whilst the
other vortex detaches - the vortices detach in a
periodic and alternating fashion. The flow is no
longer steady, since it changes periodically over
time. At much higher Re the vortices become
irregular and detach in a more irregular manner, the
flow has become turbulent, unpredictable and chaotic
(and is definitely not laminar).

Note that a Re of infinity corresponds to a viscosity of
zero and the flow is totally laminar and symmetrical -
we have gone full circle, however, no real fluid has
zero viscosity though many have very low viscosity
and assuming a viscosity of zero (non-viscous or
inviscid fluid) often simplifies analysis, so long as Re
is not so high as to make the other effects dependent
on viscosity, such as vortices and turbulence,
relevant.

Left and below: the flow of inviscid fluid across a
cylinder can be readily modeled in Mathcad (see:
ref). Here we calculate the
stream function (psi) or
streamline function, which gives us the streamlines
which we plot below.
Above: the streamlines for fluid flowing past a stationary cylinder. The cylinder has been modeled as uniform horizontal flow
added to a
doublet (a pair of vortices rotating in opposite directions and the plot shows us these streamlines inside the cylinder
which are not real, but simply an artifact of our model).

Streamlines (values at which the stream function is constant) show us the pattern of flow. They show us the pathlines taken
by fluid particles in the case of steady flow (flow in which the pattern of flow does not change over time) since in this case
streamlines and pathlines coincide (ignoring diffusion). If we introduce particles, say tiny polystyrene beads that are neutrally
buoyant (they do not float or sink), into the fluid and photograph the particles in motion as the fluid flows, then we will have
short streaks on our photograph, corresponding to the paths taken by the particles. If we join up these trails with smooth
curves, then the resulting curves will be a good approximation of the streamlines. At any instant of time no fluid is crossing any
streamline and streamlines can not cross.

Technical note: Streamlines are the field lines of the velocity vector field and can be found from the stream function. The
stream function is a scalar field and the field lines or contours are the streamlines of the flow. The stream function is easier to
work with than the velocity field (a vector field). The change in the stream function between two points, say points P and Q, is
the volume flow rate of fluid passing across any (imaginary) curve PQ joining the two points. The volume flow rate between two
streamlines is constant and independent of where it is measured along those streamlines. All solid boundaries in contact with
the fluid are streamlines.
For those not interested in the mathematics, it is not essential to understand what follows, so simply examine the graph below:
Now if we calculate and plot the pressure around the cylinder, we find something very interesting:
Next we model the flow over a rotating
cylinder (or equivalently flow swirling
around a stationary cylinder). Again, we
are ignoring the effects of viscosity (a
good approximation for air or water at low
Re).

The plot is shown below. In this plot the
cylinder is rotating clockwise (red arrow)
and the flow is from left to right. Note how
now we have a flow pattern that is
asymmetric about the horizontal axis. The
relevance of this will soon become clear.

Technical note: This involves adding a
term to our stream function which is for a
line vortex. We can model stream
functions in this way: by adding together
elementary stream functions, because of
the principle of superposition. This is a
very powerful technique for modeling
complex flows.
Left: A (polar) plot of the pressure
acting on the cylinder around its
circumference from 0 to 360 degrees.
For the situation above, in which flow is
from left to right and the cylinder
rotates clockwise, the front of the
cylinder will be at 180 degrees which is
equivalent to the case in which flow is
from right to left and the cylinder
rotates counterclockwise, in which
case the front of the cylinder will be at
0 degrees.

In both these cases
the pressure is
higher beneath the cylinder
and
lowest above it.
This generates
aerodynamic (or hydrodynamic) lift
:
The cylinder is pushed upwards and
lifts off!

Reversing either the direction (sense)
of rotation of the cylinder, or the
direction of horizontal fluid flow
reverses the pressure distribution
(click thumbnail above) and generates
downward lift.

We seem to have discovered flight!
What we have found for our cylinder can be extended to an aerofoil (airfoil or wing), with its extended, tapered and arc-like
shape. A mathematical technique called conformal mapping allows us to obtain the pressure and flow patterns for an aerofoil.
(Imagine squashing and/or elongating both the cylinder and the streamlines, and then squashing the trailing edge more to make
it thinner, and we have the pattern of flow around a wing!

  • It is the circulation of air around an aerofoil that gives rise to lift.

The lift generated by an aerofoil is due to the lower air pressure above the wing, or higher pressure beneath it. An often
simplified explanation for this is that the air rushing over the top of the wing has further to travel because of the curvature of the
aerofoil and so speeds up, lowering the pressure. This is not actually correct. The air that separates at the front of the wing
does not meet up again exactly at the rear of the wing. The correct explanation for lift is the circulation of air around the wing.
First, look at the basic geometry of aerofoils (and their underwater equivalent, the hydrofoil):
Why does air circulate around an aerofoil?

When an aerofoil begins to move through the air, the air separates and flows across both surfaces of the wing. However, the
pointed tail causes problems, for fluid to turn the corner at the trailing edge, it would need infinite velocity! Instead what
happens is that a vortex is established behind the aerofoil, extending back behind it. (The reason for this can be explained in
terms of moving stagnation points, but I shall not explain that here). However, according to Kelvin's Theorem, the circulation
in an inviscid fluid is constant. More specifically the circulation around the aerofoil is constant and since there was no
circulation to begin with it is zero and must remain so. [In inviscid fluids, a vortex persists forever as the circulation is constant,
but in real fluids the viscosity dissipates the vortex eventually, as heat, sound and other forms of energy, by creating friction
and drag. However, as will be explained, we can still generate circulation in an inviscid fluid and so explain lift.] How then can
we get circulation to occur around the wing (without considering viscosity)? What we do is imagine drawing a box around the
aerofoil, at a distance far from the boundary, where viscosity is negligible, so that Kelvin's theorem (which only applies to
inviscid fluid) holds well. Now, the total circulation in this box must remain zero, since that is how it started and it can not
change. Then, when the trailing edge generates a vortex, to keep the total circulation equal to zero (to conserve circulation
and keep it constant) we have to have air circulating around the wing in the opposite direction to air in the trailing vortex.

  • When the aerofoil starts to move through the air, a vortex appears at its trailing edge (a starting vortex)
    and to conserve circulation, air begins to circulate around the aerofoil in the opposite sense to the vortex
    and this generates lift!
Thus, we have generated circulation and lift without considering viscosity. Note that the starting vortices occur when the
aerofoil accelerates and are not present in steady flight, however, circulation continues around the wing and vortices are
ultimately shed from the wing tip.


So far our models have only considered fluid flow and lift. Another important force acting on a wing is drag. In an inviscid
fluid, like those we have modeled, there is no drag, however, in all real fluids there will be some drag, caused by friction with
the fluid, which is due to its viscosity. All real fluids have some viscosity, even if it is often negligible.
Left: the principles we have developed can also be
applied to fish. Both the pectoral fins (the horizontal
fins at the front) and the caudal fin (tail fin) act as
hydrofoils to generate lift.

In animals the situation is much more complex to
that in aircraft, since the wings and fins often move.
Fish typically move all or part of their body in a sine
wave (S-shape) as they undulate from side-to-side.

In
anguiliform (eel-like) swimming, almost the
whole body follows this S-wave and so the whole
body generates thrust. This is a more primitive form
of swimming and is also seen in polychaete worms,
nematodes and water snakes.

In
carangiform swimming, it is mostly only the tail
that undulates. This is a more specialised form of
locomotion.

In
ostraciform swimming, only the very tail end is
flexible and the rest of the body is quite rigid, as in
the box fish.

There are many subclasses within these three
types, such as subcarangiform swimming.

In
labriform swimming, the thrust is provided not
from the tail (which may act simply as a rudder) but
by the pectoral fins, which flap much as in birds,
often tracing out a figure of 8. The pectoral fins can
then generate thrust by drag, sweeping the water
backwards and downwards in the power stroke (by
being angled vertical so as to maximise drag) and
so generate both forward thrust and lift, and by
rotating horizontal during the recovery stroke so as
to minimise drag and backward thrust. Alternatively,
the pectoral fins may remain horizontal and act as
gliding hydrofoils.
Left: swimming modes in fish.

In anguiliform swimming, as found in,
for example, the eel Anguila, sauries
and sandlances. Waves of undulation
generated at the front of the fish travel
rearward, growing in amplitude as they
do so. More than one wavelength is
present along the length of the body.
The alternating pulses of push
generated to the left and right side
cancel out to produce a net backwrd
push and forward thrust. The body
may be cylindrical or flattened from
side-to-side and elongated vertically
with dorsal and anal fins increasing the
surface area of the body to generate
more thrust. This is a slow simming
mode, with maximum speeds of around
1 m/s. Some eels can also crawl along
stretches of land in this snakelike
fashion and the elongated form of eels
may also help them squeeze through
narrow spaces.

Carangiform swimming is much faster,
reaching speeds of about 5 m/s. The
anterior part of the body moves lightle
(the yawing or movement from
side-to-side of the snout is greatly
reduced). Less than half a wavelength
of undulation is present, and this is
largely confined to the posterior third
of the body - the tail provides most of
the thrust and the caudal fin is typically
tall and short and forked or
crescent-shaped (lunate). The aspect
ratio of the caudal fin (hight/length) is
about 3.5. The sideways displacement
of the tail is less than in the eel.
Examples of carangiform swimmers are
herrings, sardines and carangids (e.g.
jacks).

Subcarangiform swimming is
intermediate between anguiliform and
carangiform and occurs in catfish,
trout, cod, goldfish and bass.
Ultrafast Swimmers

Thunniform swimming is the fastest mode and is found in tunas, marlins, sailfishes and some large sharks. Some 90% of
the thrust is generated by the caudal fin and the pectoral fins generate lift and may be greatly elongated, allowing the fish to
glide for intervals. Yawing of the snout is minimal and efficiency maximal. The muscles of the body are arranged so as to
transmit the force they generate to the caudal fin through tendons that extend from the muscle blocks along the narrow end
of the tail (caudal peduncle) to the fin. The other fins are depressable, folding into grooves for maximum streamlining and
rapidly erecting to affect rapid turns. Speeds may exceed 30 m/s.

Vortex Drag

So far we have looked at aerofoils and hrdofoils as 2D objects. However, in a real wing of finite length the flow pattern is more
complex. As fluid circulates around the wing it also moves up and down the wing in spirals. Eventually vortices are shed at the
wing tip as trailing vortices (to be slowly dissipated by fluid viscous effects). These wing-tip vortex trails may extend for
several miles behind a large aircraft (and pose a danger to small aircraft who must not fly too close behind larger aircraft).
The air inside the vortices cools and water may condense inside it, generating visible jet trails across the sky. Similar fin-tip
vortices occur in fish. These vortices are unavoidable and take energy to generate and so introduce a form of drag called
vortex drag. This drag is reduced, however, in longer wings and fins because the same lift can be generated by a weaker
circulation around a longer foil. These vortices become more significant in faster fish, which is why carangiform swimmers,
and more so thunniform swimmers, have caudal fins with large aspect ratios (reaching 6-10 in the caudal fins of thunniform
fish, the largest that can be maintained before the fin loses strength). In carangids and some tunas, for example, the gliding
pectoral fins can also be greatly elongated. Such fins minimise vortex drag for high-speed cruising.
Note the long span of the fins in the carangid and tuna and also the tendency for some of the
fins, e.g. of the swordfish, to be elliptical in plan, which also minimises vortex drag.

Redrawn from Bone et al., 1982: Biology of Fishes (Pub: Blackie and Son Ltd, Chapman and Hall).
Angle of Attack

Long pectoral fins are good for high-speed cruising, but so good for high manoeuvrability or rapid acceleration. Fish
requiring such agility have shorter pectoral fins and longer, shorter caudal fins. To generate lift, a foil must be held up at an
angle to the fluid stream, called the angle of attack. This generates and maintains circulation around the wing. Long foils are
more inclined to stall if the angle of attack is too high. This happens when a foil is inclined too steeply into the stream and the
fluid boundary detaches from the top of the foil as a turbulent layer. High angles of attack occur most often when
accelerating rapidly and so to manoeuvre rapidly a fish requires short pectoral fins and a caudal fin with a low aspect ratio.
As is often the case, each 'design' has its advantages and disadvantages and in biology organisms adopt different strategies
and so occupy different niches. The anguiliform mode is the more 'primitive' in the sense of being the most ancient, indeed
some worms swim in a similar fashion, however it is advantageous for the eel's mode of life. The more evolved thunniform
mode has enabled these fish to occupy a different niche, constantly cruising the open oceans at high-speed.

Profile Drag

It is well known that certain shapes are more aerodynamic. The same is true for objects moving in water - certain shapes are
more hydrodynamic and hence reduce what is called profile drag. Naively one might assume that the more slender an object,
the more aero/hydrodynamic it is. Certainly an arrow or spear travels well through the air, but submarines and torpedoes are
much fatter. This is because in water the most hydrodynamic shape is one with a width to length ratio of about 0.2 to 0.25.
Furthermore, drag is reduced if the widest point is about one-third along the length, from the front. This accounts for the
shape of whales, for example, who might not look aerodynamic, but certainly are hydrodynamic!
When fluid flows (horizontally) over a flat (horizontal) surface, molecules of the fluid stick reversibly to the surface, forming a
molecular layer of stationary fluid at the boundary. This is the no-slip boundary layer and is due to the stickiness or viscosity
of the fluid. Fluid molecules a bit further away are still transiently attracted to this stationary layer (the forces of attraction
between particles in a real fluid are temporary, constantly forming and breaking) and their flow velocity slows. The fluid is
under sheer and the velocity of successive layers of fluid, moving away from the boundary layer, gradually increases until
the mainstream velocity is reached. The section of fluid that has its velocity slowed to some degree by the surface is the
boundary layer and as such a boundary layer consists of sheets of fluid particles moving at ever greater velocity with
greater distance from the surface, it is a
laminar boundary layer. The solid boundary is dragging on the fluid (or
equivalently the fluid is dragging on the solid object) and this dissipates energy - it takes more energy to maintain the
movement of an object in a viscous fluid. The higher the viscosity, the greater this drag. This
skin friction drag, as it is
called, is due to forces acting parallel to the solid surface.

Streamlining reduces the turbulent wake behind a moving object. Turbulent wake results in large pressure differences
between the nose and tail and this causes
pressure drag. A turbulent wake occurs when pressure drag results from
changes in shape along the direction of flow is due to forces acting normal (perpendicular) to the surface. Since it depends
on the shape of the object, pressure drag is also called
form drag. The sum of skin friction drag and pressure (form) drag is
called
profile drag.

  • In a streamlined body, the shape is such that turbulent wake is small and confined to the tail of the object, and the
    main form of drag is skin friction.

  • In a bluff body, the fluid separates from the boundary of much of the object's length and a non-laminar turbulent
    boundary layer results. In this case most of the drag is due to pressure drag.

Streamlining clearly minimises drag, as does a laminar boundary layer. However, in practise it is very difficult to obtain a
laminar boundary layer along any significant length, since laminar boundary layers tend to be unstable and breakdown or
separate from the solid surface, after some distance along the surface, into turbulent boundary layers with a large increase
in total drag. Fish appear to have two strategies to minimise total drag: 1) they are streamlined; 2) they have roughened
skin, for example the skin of a shark is covered in small 'teeth' or
denticles. The second point seems counterintuitive: fish
have a roughened skin which increases turbulence in the boundary layer and so increases drag. However, this roughness
creates a shallow turbulent boundary layer around the whole animal, which prevents a much larger turbulent layer forming
as a result of pressure drag and so is thought to be the optimum compromise solution to minimise total drag (since relying
on a laminar boundary layer is not feasible in practice). Mucilage, secreted by the fish's skin in copious amounts, also
apparently reduces drag, at least in some fish.

Flight in Birds

Flight in birds, like swimming in fish, is complicated by movements of the foil. When gliding the wing behaves more like the
static aerofoils we have considered, but during flapping flight the wing traces out a sinuosid (sine wave). The wing base
(inner wing) and the body generate lift during both the upstroke and the downstroke. However, the forward thrust generated
on the downstroke is cancelled by the backthrust generated on the upstroke. In contrast, the wing tip (outer wing) maintains
an angle of attack during the downstroke, generating lift and forward thrust, and a zero angle of attack on the upstroke
9generating neither lift nor thrust). Thus the body and inner wing generates most of the lift, whilst the outer wing generates
the forward thrust. This is illustrated below:
Landing and Stalling

When the angle of attack exceeds a critical value (which depends upon the foil) the foil stalls. When a foil stalls, the flow
separates from the top of the foil, leaving turbulent vortices along the top of the wing. This flow separation begins at the
trailing edge and moves forward, reducing lift as it proceeds. When the foil stalls, the lift is no longer sufficient to raise the
foil and it plummets.

Stalling in mid-flight is clearly undesirable! However, landing at low speed requires the wings to stall. Most modern aircraft
land at high speed, with the wings in an unstalled position, and then use air-spoilers and reverse thrust to decelerate. Most
birds, however, land by a controlled stall. Pigeons land at low speed, and if you watch a pigeon land then you will see that
when the wings stall, any loose feathers held down by the air-flow suddenly pop-up.

It is important, however, that the pigeon delays stalling, thereby reducing the stalling speed, enabling a slower and gentler
landing. To achieve this, many birds, like the pigeon have a bastard wing or
alula. This looks like a small wing on the
leading edge of the main wing, about half-way along (located where the wing is most likely to stall). The alula pops out just
before landing. The alula functions like a slot in an aircraft wing.
The risk of stalling in landing or manoeuvering birds is also reduced by bending, opening out and twisting of the
feathers (which act like propeller blades) as illustrated above.

Both swimming in fish and flight in birds is so complex that it has so far defied complete mathematical analysis.
However, by reducing these movements into simpler components, as we have done, fluid mechanical principles do
illuminate many features of these complex modes of locomotion.

The power required for flight is considerable. A budgerigar has an optimum flight speed of about 35 km/h at which it
consumes about 22 ml oxygen per gram of weight per hour. Flying at slower and higher speeds, which is less efficient
for budgerigars, increases oxygen consumption to about 30 ml/g/h. Hovering flight in still air (that is not by flying
against a head wind) is particularly demanding and hovering hummingbirds consume about 42 ml oxygen per g per h.
In birds, the main chest flight muscles (such as the pectoralis major) may account for a quarter to a third of the
animal's body weight! It is often said that hummingbird flight muscle is the most powerful in the animal kingdom.
Looking at all muscles studied, this is true with one exception - hummingbird muscle has a power output equalled by
insect flight muscle.

Other swimming modes in fish

Some fish use their pectoral fins much like the wings of birds, moving them through a figure of 8 and generating lift
and/or actively sweeping back water to generate thrust. Pipefishes and seahorses hover in an extraordinary manner
by undulating their dorsal fins at high speed. Triggerfishes and filefishes swim by undulating their dorsal and anal fins,
moving forwards or backwards by reversing the direction of undulation (wave propagation). Skates and rays undulate
their pectoral fins, and some superimpose flapping 'flight' on top of this undulation (e.g. manta rays and eagle rays).

Flying Fish

Fish don't just swim. In fact it is doubtful whether some fish swim at all. Some are wormlike and live a life crawling
through narrow spaces (though such fish may be able to swim to some extent when they need to) and others
supplement swimming by walking on the seabed on their fins, or by climbing vegetation by use of their pectoral fins
(e.g.
Histrio histrio). Sea robins walk on their pectoral fins whilst batfishes walk as quintipeds 9on their 2 pectorals, 2
pelvics and their anal fin). Tongue soles (Cynoglossinae) walk in millipede fashion by means of undulating fins. Of
particular relevance here, however, are the flying fish. Many fish can glide by leaping out of the water, but true flight is
best developed in Exocoetidae or flying fish. These fish accelerate once free of the water by rapid flapping of the
caudal fin and the enlarged pectorals guide the fish and no doubt act as aerofoils and generate lift in addition to the
main body. Flying fish typically fly distances of 50 metres in this way, though flights of 400 m have been recorded, and
they gain 5 m height above the water's surface.

Flight in insects.

Coming soon: biorheology 2 - flow in tubes, blood and blood vessels.