Taylor scraping flow: Difference between revisions

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Revision as of 19:21, 6 July 2018

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Taylor noticed that the inertial terms are negligible as long as the region of interest is within <math>r\ll\nu/U</math>( or, equivalently [[Reynolds number]] <math>Re = Ur/\nu<<1</math>), thus within the region the flow is essentially a [[Stokes flow]]. For example, [[George Batchelor]]<ref>Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.</ref> gives a typical value for lubricating oil with velocity <math>U=10\text{ cm}/\text{s}</math> as <math>r\ll0.4\text{ cm}</math>. Then for two-dimensional planar problem, the equation is

:<math>\nabla^4 \psi =0, \quad u_r = \frac 1 r \frac{\partial\psi}{\partial\theta}, \quad u_ru_\theta = -\frac{\partial\psi}{\partial r}</math>

where <math>\mathbf{v}=(u_r,u_\theta)</math> is the velocity field and <math>\psi</math> is the [[stream function]]. The boundary conditions are