Taylor scraping flow

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

${\displaystyle \sigma _{t}={\frac {2\mu U}{r}}{\frac {\sin \alpha -\alpha \cos \alpha }{\alpha ^{2}-\sin ^{2}\alpha }},\quad \sigma _{n}={\frac {2\mu U}{r}}{\frac {\alpha \sin \alpha }{\alpha ^{2}-\sin ^{2}\alpha }}}$

The same scraper stress if resolved according to Cartesian coordinates (parallel and perpendicular to the lower plate i.e. ${\displaystyle \sigma _{x}=-\sigma _{t}\cos \alpha +\sigma _{n}\sin \alpha ,\ \sigma _{y}=\sigma _{t}\sin \alpha +\sigma _{n}\cos \alpha }$) are

${\displaystyle \sigma _{x}={\frac {2\mu U}{r}}{\frac {\alpha -\sin \alpha \cos \alpha }{\alpha ^{2}-\sin ^{2}\alpha }},\quad \sigma _{y}={\frac {2\mu U}{r}}{\frac {\sin ^{2}\alpha }{\alpha ^{2}-\sin ^{2}\alpha }}}$

As noted earlier, all the stresses become infinite at ${\displaystyle r=0}$, 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 ${\displaystyle \alpha }$ increases, and reaches its minimum value ${\displaystyle \sigma _{x}=2\mu U/r}$ at ${\displaystyle \alpha =\pi }$. Taylor says: "The most interesting and perhaps unexpected feature of the calculations is that ${\displaystyle \sigma _{y}}$ does not change sign in the range ${\displaystyle 0<\alpha <\pi }$. In the range ${\displaystyle \pi /2<\alpha <\pi }$ the contribution to ${\displaystyle \sigma _{y}}$ 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 ${\displaystyle \sigma _{n}}$ is small and as will be seen in the figure this occurs only when ${\displaystyle \alpha }$ is nearly ${\displaystyle 180^{\circ }}$. 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 ${\displaystyle \alpha }$ is small. In that way he can get the large values of ${\displaystyle \sigma _{y}/\sigma _{x}}$ which are needed in forcing plaster from protuberances to hollows."

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 solutions for power-law fluids satisfying the relation for the apparent viscosity^[6]

${\displaystyle \mu =m_{z}\left\{4\left[{\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial \psi }{\partial \theta }}\right)\right]^{2}+\left[{\frac {1}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial \theta ^{2}}}-r{\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial }{\partial r}}\right)\right]^{2}\right\}^{(n-1)/2}}$

where ${\displaystyle m_{z}}$ and ${\displaystyle n}$ are constants. The solution for the streamfunction of the flow created by the plate moving towards right is given by

${\displaystyle \psi =Ur\left\{\left[1-{\frac {{\mathcal {J}}_{1}(\theta )}{{\mathcal {J}}_{1}(\alpha )}}\right]\sin \theta +{\frac {{\mathcal {J}}_{2}(\theta )}{{\mathcal {J}}_{1}(\alpha )}}\cos \theta \right\}}$

where

${\displaystyle {\begin{aligned}{\mathcal {J}}_{1}&=\mathrm {sgn} (F)\int _{0}^{\theta }|F|^{1/n}\cos x\,dx,\\{\mathcal {J}}_{2}&=\mathrm {sgn} (F)\int _{0}^{\theta }|F|^{1/n}\sin x\,dx\end{aligned}}}$

and

${\displaystyle {\begin{aligned}F=\sin({\sqrt {n(2-n)}}x-C)\quad {\text{if}}\,n<2,\\F={\sqrt {x-C}}\qquad \qquad \qquad \quad {\text{if}}\,n=2,\\F=\sinh({\sqrt {n(n-2)}}x-C)\quad {\text{if}}\,n>2\end{aligned}}}$

where ${\displaystyle C}$ is the root of ${\displaystyle {\mathcal {J}}_{2}(\alpha )=0}$. It can be verified that this solution reduces to that of Taylor's for Newtonian fluids, i.e., when ${\displaystyle n=1}$.