Taylor scraping flow: Difference between revisions

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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 (edited|editor-first=G. by GKK. |editor-last=Bachelor)." (|year=1958): |page=467. }}</ref>

==Flow description==

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:<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 \sincos\alpha + \sigma_n \cossin\alpha, \ \sigma_y = -\sigma_t \cossin\alpha + \sigma_n \sincos\alpha</math>) are

:<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>

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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 |lastlast1=Riedler |firstfirst1=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=1-21–2 |pages=95-10295–102 |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==