Taylor scraping flow: Difference between revisions

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In [[fluid dynamics]], '''Taylor scraping flow''' is a type of two-dimensional [[corner flow]] occurring when one of the wall is sliding over the other with constant velocity, named after [[G. I. Taylor]].<ref>Taylor, G. I. "Similarity solutions of hydrodynamic problems." Aeronautics and Astronautics 4 (1960): 214.</ref><ref>Taylor, G. I. "On scraping viscous fluid from a plane surface." Miszellangen der Angewandten Mechanik (Festschrift Walter Tollmien) (1962): 313–315.</ref><ref>Taylor, G. I. "Scientific Papers (edited by GK Bachelor)." (1958): 467.</ref>.

==Flow description==

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Consider a plane wall located at <math>\theta=0</math> in the cylindrical coordinates <math>(r,\theta)</math>, moving with a constant velocity <math>U</math> towards the left. Consider an another plane wall(scraper) , at an inclined position, making an angle <math>\alpha</math> from the positive <math>x</math> direction and let the point of intersection be at <math>r=0</math>. This description is equivalent to moving the scraper towards right with velocity <math>U</math>. It should be noted that the problem is singular at <math>r=0</math> because at the origin, the velocities are discontinuous, thus the velocity gradient is infinite there.

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]]. [[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_r = -\frac{\partial\psi}{\partial r}</math>

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==Solution<ref>Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.</ref><ref>Pozrikidis, Costas, and Joel H. Ferziger. "Introduction to theoretical and computational fluid dynamics." (1997): 72–74.</ref>==

Attempting a [[Separation of variables|separable]] solution of the form <math>\psi =U r f(\theta)</math> reduces the problem to

:<math>f^{iv} + 2 f'' + f =0</math>

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:<math>f(0)=0,\ f'(0)=-1, \ f(\alpha)=0, \ f'(\alpha)=0</math>

The solution is

:<math>f(\theta) = \frac{1}{\alpha^2 - \sin^2\alpha} [\theta \sin \alpha \sin (\alpha-\theta) - \alpha(\alpha-\theta) \sin\theta]</math>

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:<math>\nabla p = \mu \nabla^2\mathbf{v}, \quad p(r,\infty)=p_\infty</math>

which gives,

:<math>p(r,\theta) - p_\infty = \frac{2\mu U}{r} \frac{\alpha\sin\theta+\sin\alpha\sin(\alpha-\theta)}{\alpha^2 - \sin^2\alpha} </math>

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==References==

{{Reflist}}

[[Category:Fluid dynamics]]