Fluid Mechanics by Eric Lauga

Author:Eric Lauga [Lauga, Eric]
Language: eng
Format: epub
ISBN: 9780192566737
Publisher: OUP Oxford
Published: 2022-05-04T00:00:00+00:00

Womersley number

Many other dimensionless numbers can be used to characterize the variety of ways in which fluids can move. For example, instead of being steady, the flow in Reynolds’ pipe could be made to oscillate in time with frequency f. A situation where time-periodic oscillations are significant is the flow in the circulation, where blood is periodically pumped by the beating heart in our network of veins and arteries. In that case, the dimensions of the frequency [ f ] = 1/T provide a different method to cancel out the dimensions of time appearing in the Reynolds number. Since the ratio between density and viscosity has dimensions [ρ/μ] = T/(L2), we could instead make the dimension of time disappear by considering the product ρf/μ, which has dimensions [ρf/μ] = 1/(L2). In a pipe of diameter D, the term ρD2f/μ is therefore dimensionless. In the context of blood flow, this leads to the dimensionless Womersley number

(10)

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