The Womersley number is a dimensionless number in biofluid mechanics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscous effects

Viscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

. It is named after John R. Womersley

John R. Womersley

John Ronald Womersley was a British mathematician who made important contributions to textiles, computing and arterial mechanics . He studied mathematics at Imperial College of Science and Technology and subsequently worked at the Shirley Institute Manchester...

 (1907–1958). The Womersley number is important in keeping dynamic similarity when scaling an experiment. An example of this is scaling up the vascular system for experimental study. The Womersley number is also important in determining the thickness of the boundary layer

Boundary layer

In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

 to see if entrance effects can be ignored.

Derivation

The Womersley number, usually denoted , can be written as



where is an appropriate length scale

Length scale

In physics, length scale is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple...

 (for example the radius of a pipe), is the angular frequency

Angular frequency

In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

 of the oscillations, and , , are the kinematic viscosity, density, and dynamic viscosity of the fluid, respectively. It can also be written in terms of the dimensionless Reynolds number (Re) and Strouhal number

Strouhal number

In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc Strouhal, a Czech physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind...

 (Sr):



The Womersley number arises in the solution of the linearized Navier Stokes equations for oscillatory flow (presumed to be laminar and incompressible) in a tube. It expresses the ratio of the transient or oscillatory inertia force to the shear force. When is small (1 or less), it means the frequency of pulsations is sufficiently low that a parabolic velocity profile has time to develop during each cycle, and the flow will be very nearly in phase with the pressure gradient, and will be given to a good approximation by Poiseuille's law, using the instantaneous pressure gradient. When is large (10 or more), it means the frequency of pulsations is sufficiently large that the velocity profile is relatively flat or plug-like, and the mean flow lags the pressure gradient by about 90 degrees.
Along with the Reynolds number, the Womersley number governs dynamic similarity.

The boundary layer thickness that is associated with the transient acceleration is related to the Womersley number. It is equal to inverse of the Womersley number. The Womersley number is also equal to the square root of the Stokes number

Stokes number

The Stokes number, named after Irish mathematician George Gabriel Stokes, is a dimensionless number corresponding to the behavior of particles suspended in a fluid flow...

.



where L is a characteristic length.

Biofluid Mechanics

In a flow distribution network that progresses from a large tube to many small tubes (e.g. a blood vessel network), the frequency, density, and dynamic viscosity are (usually) the same throughout the network, but the tube radii change. Therefore the Womersley number is large in large vessels and small in small vessels. As the vessel diameter decreases with each division the Womersley number soon becomes quite small. The Womersley numbers tend to 1 at the level of the terminal arteries. In the arterioles, capillaries, and venules the Womersley numbers are less than one. In these regions the inertia force becomes less important and the flow is determined by the balance of viscous stresses and the pressure gradient. This is called microcirculation

Microcirculation

The microcirculation is a term used to describe the small vessels in the vasculature which are embedded within organs and are responsible for the distribution of blood within tissues; as opposed to larger vessels in the macrocirculation which transport blood to and from the organs...

.

Some typical values for the Womersley number in the cardiovascular system for a canine at a heart rate of 2Hz are:

• Ascending Aorta -- 13.2
• Descending Aorta -- 11.5
• Abdominal Aorta -- 8
• Femoral Artery -- 3.5
• Carotid Artery -- 4.4
• Arterioles --.04
• Capillaries -- 0.005
• Venules -- 0.035
• Inferior Vena Cava -- 8.8
• Main Pulmonary Artery -- 15

It has been argued (West et al., 1997) that universal biological scaling laws (power-law relationships that describe variation of quantities such as metabolic rate, lifespan, length, etc., with body mass) are a consequence of the need for energy minimization, the fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

nature of vascular networks, and the crossover from high to low Womersley number flow as one progresses from large to small vessels.