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The Physics of fluid flow-a tutorial

Created: 10/6/2005
Updated: 26/3/2018


Fluid is the term used to describe a liquid or a gas.

In our day-to-day practice, we encounter fluid flow and the consequences of alteration of fluid flow, in the gases and vapours our patients receive and the liquids we administer.

The FRCA exam is concerned with the types of flow that may be encountered and the effects that can be utilised for the patient’s good.

For the first part of this discussion, I will deal with flow patterns and then in the second part will discuss various other effects and principles.


What is flow?

“This is the mass of a substance (in this case a fluid), that passes a certain point in one second.” The units are Litres per second

Flow types

Flow can be described as laminar, turbulent or a mixture of both.

Laminar flow

This is a flow pattern where all the particles in the fluid follow the same line of flow. These lines of flow can be visualised as “sheets” and are known as streamlines. In the case of a tube these streamlines are a set of concentric tubes, the velocity of which increases the closer to the centre one measures.

This can be seen when a unit of blood is run after a crystalloid solution. Hold the giving set vertically and observe the initial “arrow-head” front of blood that flows down the tube. Ah Physics in motion!

Turbulent flow

In contrast to laminar flow, the particles in this case are moving in different directions to each other, hence the description turbulent. A good example of this is smoke rising from a cigarette. Ask a smoking friend to light up and place on in an ashtray. If there isn’t a breeze you can observe the straight plume of smoke rising in a laminar way until it breaks into fluffy turbulent flow some way up. Then berate your friend for not giving up smoking.

Pressure – flow relationships for the flow patterns

The important distinction between laminar and turbulent flow for anaesthetic practice is that laminar flow requires lower pressures for the same flow rate compared with turbulent flow. This means lower energy to get the same work done and if applied to respiration for example, lower work of breathing. Which explains in part why acute asthmatics have such difficulty breathing.

Figure 1a Graph of Q vs. P for Laminar flow

Figure 1b Graph of Q vs. P for Turbulent flow

This brings us on to some more definitions that lead us gracefully on to the (in)famous Hagen-Poiseuille equation.



“This is the resistance a fluid offers to the motion of a solid through it”

The units of viscosity are Pascal seconds (Pa.s)


“Is simply the mass of a substance occupying unit volume”

The units of density are kilos/metre3 (Kg.m-3)

The famous Hagen-Poiseuille equation

This is a favourite of examiners as the concepts can confuse the unprepared.

The Hagen-Poiseuille equation defines the flow through a tube and how this flow is affected by the attributes of the tube; the length and radius, and the attributes of the fluid; the viscosity. The equation only applies Newtonian fluids undergoing laminar flow through tubes.

Newtonian fluids are fluids where the viscosity of the fluid is constant regardless of the accelerating forces within the streamlines. In other words the viscosity doesn’t change with the flow rate. Water is an example of a Newtonian fluid, oil paint and blood are examples of a non-Newtonian fluid. In fact blood’s viscosity decreases as its flow increases.

The Hagen-Poiseuille equation

Where Q = Flow in Litres/second 
n = Viscosity in Pa.s
P = Pressure in Pascals
r = Radius of the tube in meters
l = Length of the tube in question in meters

Firstly, it therefore follows that flow is directly proportional to the pressure difference, and to the fourth power of the radius. In other words, if pressure goes up the flow increases. Also note however that the flow increases markedly as the radius increases.
This explains why a 1 mm airway narrowing in a child, whose airway may only be 5 mm at the maximum, impairs flow more severely than the same amount in an adult. For an adult airway of 10 mm the reduction in flow is sixteen times less than in the child’s case. The radius component also explains why an 18G cannula has less flow (76ml/min) than a 16G (172ml/min).

Secondly, flow is inversely proportional to viscosity and length. Consider applying the same pressure to syringe full of honey versus a syringe full of water and the viscosity part makes sense. Lastly look at the flow rate recorded on a shorter 18G I.V. cannula and compare it with the longer 18G on a 15cm CVP line. The shorter cannula has a greater flow rate despite the two having the same radius.

Finally please also note that viscosity only affects laminar flow. The density of the fluid only comes into play when flow is turbulent. This again explains the practice of sometimes using helium in oxygen for patients in severe brochospasm. The lower density of the helium reduces the chances of turbulent flow and so improves the efficiency of respiration and possibly the flow patterns within the lungs.

Reynold’s number

The discussion of density for fluid flow now takes us to Reynold’s number. Reynold’s number attempts to describe the point at which flow changes from laminar to turbulent, and the spectrum in-between.

The equation for Reynold’s number (Re) is

For numbers less than 2000, the flow through a tube tends to be laminar. Between 2000 to 4000 the flow pattern is a mix of the two, and above 4000 the flow is mainly turbulent.

Bernoulli’s principle

Having dealt with flow patterns and factors effecting flow, we can now consider other effects that flow produces. In this case we will look at the Bernoulli principle and how this gives rise to the Venturi effect.


“For a non-compressible, non-viscous fluid undergoing laminar flow, the sum of the pressure, kinetic and potential energies per unit volume remains a constant at all points along the line of flow”

Mathematically represented by

P + 1/2.p v2 + pgh = constant 

P = Pressure
g = Acceleration due to gravity (m/s2)
h = Height of the tube 
p = Density of liquid
v = Velocity of fluid

What does that mean exactly?

This is a perfect system so all the energy is conserved as either pressure energy, potential (or stored) energy, and the energy existing as flow. We assume no loss of energy through heat caused by friction within the fluid or caused by drag on the tube’s walls.

This means that if we alter the energy of one portion of the system, it has an effect on the rest of the system. So if the kinetic energy rises, the potential energy and pressure must fall.

When we apply the Bernoulli principle in our practice we can ignore the portion due to gravity to make life a little simpler. This isn’t a fudge as if we confine our thoughts to a horizontal system, the potential energy is the same, so will cancel out mathematically.

Figure 2 A tube with an orifice

Consider our tube with a narrowing in Figure 2. There are no leaks, so the volume of fluid at point A is the same at point C. Consequently the narrowing at point B means that the fluid has to speed up in order to fulfil this continuity. As an analogy, imagine you are sitting in a boat and it starts to gain water.
You have a cup and your sailing partner has a bucket. In order to bail out the same quantity of water you have to move quicker, which is analogous to you being the flow at point B. In so doing you have to work harder and so expend more energy.

Looking again at the equation

P + 1/2.pv2 = constant

This means that point B has an increase in velocity compared with point A or C, so the pressure falls at point B.

In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738.

The Venturi effect

This is the consequence of the Bernoulli principle described above.

The pressure drop induced by the increase in velocity of a fluid passing through a narrow orifice can be used to entrain air or a nebuliser solution for treating our patients.

Figure 3 Section of a Venturi mask

At point A 100% oxygen flows into the wider point B via a narrow orifice. Because of the narrowing the oxygen speeds up and the pressure drop at that point is below atmospheric pressure and room air is drawn to this low pressure point, hence diluting the 100% oxygen to the calibrated value set by the coloured nozzle. The nozzle has a varying aperture open to room air that sets the entrainment ratio and hence the inspired concentration given to the patient.

The Coanda effect

This effect was named after a Romanian aircraft designer Henri Coanda, after an aircraft he designed went up in flames as a consequence of this effect.

Essentially any fluid coming into contact with a curved surface will cling to this surface and alter its direction of flow. You can illustrate this to yourself by running a thin stream of water from a tap, and bringing the curved surface of a spoon to touch it. The water follows the surface of the spoon. It does so because the solid stationary surface of the spoon slows the layer in immediate contact. This has a drag effect on the other layers, in effect pulling them into the line of the curved surface.

The Coanda effect is said to explain the maldistribution of air in the pulmonary tree after a constricted portion of bronchiole, as the flow will stream along one fork of the division, leading to unequal distribution of gas flow.


I hope this short guide has been useful aid to the gas laws and fluid flow. Remember start with a definition, give equations where you can, and try and illustrate your points with examples.


AnaesthesiaUK would like to thank Dr Lliam Edger for writing this article.

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