
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 bloodflow 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. == Derivation == The Womersley number, usually denoted $\backslash alpha$, is defined by the relation $\backslash alpha^2\; =\; \backslash frac\}\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; \backslash ,\; ,$ 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 $\backslash alpha\; =\; L\; \backslash left(\; \backslash frac\; \backslash right)^\backslash frac\; \backslash ,\; .$ 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: $\backslash alpha\; =\; L\; \backslash left(\; \backslash frac\; \backslash right)^\backslash frac\; \backslash ,\; .$ 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 (Sr): $\backslash alpha\; =\; \backslash left(\; 2\backslash pi\backslash ,\; \backslash mathrm\; \backslash ,\; \backslash mathrm\; \backslash right)^\backslash ,\; .$ 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 $\backslash alpha$ 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 $\backslash alpha$ is large (10 or more), it means the frequency of pulsations is sufficiently large that the velocity profile is relatively flat or pluglike, 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 $\backslash delta$ that is associated with the transient acceleration is inversely related to the Womersley number. This can be seen by recognizing the Womersley number as the square root of the Stokes number. $\backslash delta\; =\; \backslash left(\; L/\backslash alpha\; \backslash right)=\; \backslash left(\; \backslash frac\backslash right),$ where ''L'' is a characteristic length. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Womersley number」の詳細全文を読む スポンサード リンク
