Taylor scraping flow: Difference between revisions

Revision as of 16:24, 25 September 2018

Revision as of 22:23, 23 March 2019

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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 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<< \ll 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_\theta = -\frac{\partial\psi}{\partial r}</math>

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As noted earlier, all the stresses 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, which depends on the geometry of the contact. The stresses are shown in the figure as given in the Taylor's original paper.

The stress in the direction parallel to the lower wall decreases as <math>\alpha</math> increases, and reaches its minimum value <math>\sigma_x = \frac{2\mu U}{/r}</math> at <math>\alpha=\pi</math>. Taylor says ''The most interesting and perhaps unexpected feature of the calculations is that <math>\sigma_y</math> does not change sign in the range <math>0<\alpha<\pi</math>. In the range <math>\frac{\pi}{/2}<\alpha<\pi</math> the contribution to <math>\sigma_y</math> due to normal stress is of opposite sign to that due to tangential stress, but the latter is the greater. The palette knives used by artists for removing paint from their palettes are very flexible scrapers. They can therefore only be used at such an angle that <math>\sigma_n</math> is small and as will be seen in the figure this occurs only when <math>\alpha</math> is nearly <math>180^\circ</math>. In fact artists instinctively hold their palette knives in this position.'' Further he adds ''A plasterer on the other hand holds a smoothing tool so that <math>\alpha</math> is small. In that way he can get the large values of <math>\sigma_y/\sigma_x</math> which are needed in forcing plaster from protuberances to hollows.''

==Scraping a power-law fluid==