Taylor scraping flow: Difference between revisions

Revision as of 14:43, 8 May 2017

Revision as of 16:27, 8 May 2017

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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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:<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>

==ForcesStresses on the scraper==

The tangential forcestress and the normal forcestress on the scraper due to pressure forces and viscous forces are

:<math>F_t\sigma_t = \frac{2\mu U}{r} \frac{\sin\alpha-\alpha\cos\alpha}{\alpha^2 - \sin^2\alpha}, \quad F_n\sigma_n =\frac{2\mu U}{r} \frac{\alpha\sin\alpha}{\alpha^2 - \sin^2\alpha} </math>

The same scraper forcestress if resolved according to Cartesian coordinates (parallel and perpendicular to the lower plate i.e. <math>F_x\sigma_x = F_t\sigma_t \sin\alpha + F_n\sigma_n \cos\alpha, \ F_y\sigma_y = -F_t\sigma_t \cos\alpha + F_n\sigma_n \sin\alpha</math>) are

:<math>F_x\sigma_x = \frac{2\mu U}{r} \frac{\alpha-\sin\alpha\cos\alpha}{\alpha^2 - \sin^2\alpha}, \quad F_y\sigma_y =\frac{2\mu U}{r} \frac{\sin^2\alpha}{\alpha^2 - \sin^2\alpha} </math>

As noted earlier, all the forcesstresses become infinite at <math>r=0</math>, because the velocity gradient is infinite there. In real life, there will be a huge pressure at the point of point, which depends on the geometry of the contact.

==References==