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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==
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 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>Riedler, J., & Schneider, W. (1983). Viscous flow in corner regions with a moving wall and leakage of fluid. Acta Mechanica, 48(1-2), 95-102.</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{J_1(\theta)}{J_1(\alpha)}\right]\sin\theta + \frac{J_2(\theta)}{J_1(\alpha)}\cos\theta\right\} </math>
where
:<math>\begin{align}
J_1 &= \mathrm{sgn}(F) \int_0^\theta |F|^{1/n} \cos x\, dx,\\
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>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==
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