The Womersley number (α) is a dimensionless number in biofluid mechanics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscous effects. It is named after John R. Womersley (1907–1958) for his work with blood flow in arteries. 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 to see if entrance effects can be ignored.
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Derivation
The Womersley number, usually denoted
where L is an appropriate length scale (for example the radius of a pipe), ω is the angular frequency of the oscillations, and ν, ρ, μ are the kinematic viscosity, density, and dynamic viscosity of the fluid, respectively. The Womersley number is normally written in the powerless form
In the cardiovascular system, the pulsation frequency decreases as the blood is distanced from the origin of pulsation, the heart. However, the Womersley number, like many characteristic numbers, defines a system by order of magnitude (OoM). The pulsation frequency maintains a single OoM throughout the body (<1 s^-1) and is square rooted in the Womersley equation, reducing the OoM further. Therefore, the frequency change in blood flow does not affect the characteristics defined by the Womersley number.
Characteristic length, or in the case of blood flow, the diameter of the vessel, is a defining characteristic of a system and often the driving factor of characteristic numbers. Since the vessel diameters in the body differ up to three OoM, the Womersley number will depend predominantly on diameter. That being said, using standard values for frequency, viscosity and density, the Womersley number of human blood flow can be estimated as follows:
Below is a list of estimated Womersley numbers in different human blood vessels:
It can also be written in terms of the dimensionless Reynolds number (Re) and Strouhal number (St):
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
The boundary layer thickness
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.
Some typical values for the Womersley number in the cardiovascular system for a canine at a heart rate of 2 Hz are:
It has been argued 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 nature of vascular networks, and the crossover from high to low Womersley number flow as one progresses from large to small vessels.