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{{Short description|Type of two-dimensional corner flow}}
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>{{cite journal |last=Taylor
==Flow description==
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
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
:<math>\nabla^4 \psi =0, \quad u_r = \frac 1 r \frac{\partial\psi}{\partial\theta}, \quad
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
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</math>
==Solution==
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<ref>{{cite book |last=Acheson |first=David J. |title=Elementary Fluid Dynamics |publisher=Oxford University Press |year=1990 |isbn=0-19-859660-X }}</ref>
:<math>f(\theta) = \frac{1}{\alpha^2 - \sin^2\alpha} [\theta \sin \alpha \sin (\alpha-\theta) - \alpha(\alpha-\theta) \sin\theta]</math>
Therefore, the velocity field is
:<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>
==Stresses on the scraper==
[[File:Taylor.
The tangential stress and the normal stress on the scraper due to pressure and viscous forces are
:<math>\sigma_t = \frac{2\mu U}{r} \frac{\sin\alpha-\alpha\cos\alpha}{\alpha^2 - \sin^2\alpha}, \quad \sigma_n =\frac{2\mu U}{r} \frac{\alpha\sin\alpha}{\alpha^2 - \sin^2\alpha} </math>
The same scraper stress if resolved according to Cartesian coordinates (parallel and perpendicular to the lower plate i.e. <math>\sigma_x = -\sigma_t \
:<math>\sigma_x = \frac{2\mu U}{r} \frac{\alpha-\sin\alpha\cos\alpha}{\alpha^2 - \sin^2\alpha}, \quad \sigma_y =\frac{2\mu U}{r} \frac{\sin^2\alpha}{\alpha^2 - \sin^2\alpha} </math>
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
The stress in the direction parallel to the lower wall decreases as <math>\alpha</math> increases, and reaches its minimum value <math>\sigma_x = 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>\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==
Since scraping applications are important for [[non-Newtonian fluid]] (for example, scraping paint, nail polish, cream, butter, honey, etc.,), it is essential to consider this case. The analysis was carried out by J. Riedler and [[Wilhelm Schneider (engineer)|Wilhelm Schneider]] in 1983 and they were able to obtain [[self-similar solution]]s for [[power-law fluid]]s satisfying the relation for the [[apparent viscosity]]<ref>{{cite journal |last1=Riedler |first1=J. |last2=Schneider |first2=W. |year=1983 |title=Viscous flow in corner regions with a moving wall and leakage of fluid |journal=Acta Mechanica |volume=48 |issue=1–2 |pages=95–102 |doi=10.1007/BF01178500 |s2cid=119661999 }}</ref>
:<math>\mu = m_z\left\{4\left[\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial \psi}{\partial \theta}\right)\right]^2 + \left[\frac{1}{r^2} \frac{\partial^2\psi}{\partial \theta^2} - r \frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial r}\right)\right]^2\right\}^{(n-1)/2}</math>
where <math>m_z</math> and <math>n</math> are constants. The solution for the streamfunction of the flow created by the plate moving towards right is given by
:<math>\psi = Ur\left\{\left[1-\frac{\mathcal J_1(\theta)}{\mathcal J_1(\alpha)}\right]\sin\theta + \frac{\mathcal J_2(\theta)}{\mathcal J_1(\alpha)}\cos\theta\right\} </math>
where
:<math>\begin{align}
\mathcal J_1 &= \mathrm{sgn}(F) \int_0^\theta |F|^{1/n} \cos x\, dx,\\
\mathcal J_2 &= \mathrm{sgn}(F) \int_0^\theta |F|^{1/n} \sin x\, dx
\end{align}
</math>
and
:<math>\begin{align}
F = \sin(\sqrt{n(2-n)}x-C)\quad \text{if}\, n<2,\\
F= \sqrt{x-C}\qquad \qquad \qquad \quad\text{if}\, n=2,\\
F=\sinh(\sqrt{n(n-2)}x-C)\quad \text{if}\, n>2
\end{align}
</math>
where <math>C</math> is the root of <math>\mathcal J_2(\alpha)=0</math>. It can be verified that this solution reduces to that of Taylor's for Newtonian fluids, i.e., when <math>n=1</math>.
==References==
{{Reflist|30em}}
[[Category:Fluid dynamics]]
[[Category:Flow regimes]]
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