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Adding short description: "Type of two-dimensional corner flow" |
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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 |first=G. I. |title=Similarity solutions of hydrodynamic problems |journal=Aeronautics and Astronautics |volume=4 |year=1960 |page=214 }}</ref><ref>{{cite book |last=Taylor |first=G. I. |chapter=On scraping viscous fluid from a plane surface |title=Miszellangen der Angewandten Mechanik |series=Festschrift Walter Tollmien |year=1962 |pages=313–315 }}</ref><ref>{{cite book |last=Taylor |first=G. I. |title=Scientific Papers |editor-first=G. K. |editor-last=Bachelor |year=1958 |page=467 }}</ref>
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==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>
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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>
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</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==
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