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): 313-315.|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==

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 an another plane wall(scraper) , at an inclined position, making an angle <math>\alpha</math> from the positive <math>x</math> direction and let the point of intersection be at <math>r=0</math>. This description is equivalent to moving the scraper towards right with velocity <math>U</math>. It should be noted that theThe problem is singular at <math>r=0</math> because at the origin, the velocities are discontinuous, thus the velocity gradient is infinite there.

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 \ll 1</math>), thus within the region the flow is essentially a [[Stokes flow]]. For example, [[George Batchelor]]<ref>Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.</ref> gives a typical value for lubricating oil with velocity <math>U=10\text{ cm}/\text{s}</math> as <math>r<<0\ll0.4\text{ cm}</math>.<ref>{{cite book |last=Batchelor |first=George Keith |title=An Introduction to Fluid Dynamics |publisher=Cambridge University Press |year=2000 |isbn=0-521-66396-2 }}</ref> Then for two-dimensional planar problem, the equation is

:<math>\nabla^4 \psi =0, \quad u_r = \frac{ 1}{ r} \frac{\partial\psi}{\partial\theta}, \quad u_ru_\theta = -\frac{\partial\psi}{\partial r}</math>

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

Line 18 ⟶ 19:

</math>

==Solution==

==Solution<ref>Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.</ref><ref>Pozrikidis, Costas, and Joel H. Ferziger. "Introduction to theoretical and computational fluid dynamics." (1997): 72-74.</ref>==

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>

and thewith boundary conditions

:<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>

The solution is

:<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>

Pressure can be obtained bythrough integratingintegration of the followingmomentum equation

:<math>\nabla p = \mu \nabla^2\mathbf{v}, \quad p(r,\infty)=p_\infty</math>

which is 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>

==ForcesStresses on the scraper==

[[File:Taylor.svg|thumb|300px|Stresses on the scraper]]

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

:<math>F_x\sigma_t = \frac{2\mu U}{r} \frac{\sin\alpha-\sin\alpha\cos\alpha}{\alpha^2 - \sin^2\alpha}, \quad F_y\sigma_n =\frac{2\mu U}{r} \frac{\alpha\sin^2\alpha}{\alpha^2 - \sin^2\alpha} </math>

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

:The same scraper stress if resolved according to Cartesian coordinates (parallel and perpendicular to the lower plate i.e. <math>F_t\sigma_x = -\frac{2\mu U}{r}sigma_t \frac{\sin\alpha-\alpha\cos\alpha}{\alpha^2 -+ \sigma_n \sin^2\alpha}, \quad F_n\sigma_y = \frac{2\musigma_t U}{r} \frac{\alpha\sin\alpha}{\alpha^2 -+ \sin^2sigma_n \cos\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>

The same scraper force if resolved according to Cartesian coordinates (parallel and perpendicular to the lower plate i.e. <math>F_x = F_t \sin\alpha + F_n \cos\alpha, \ F_y = -F_t \cos\alpha + F_n \sin\alpha</math>) are

As noted earlier, all the forcesstresses 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 point of point, which depends on the geometry of the contact. The stresses are shown in the figure as given in the Taylor's original paper.

:<math>F_x = \frac{2\mu U}{r} \frac{\alpha-\sin\alpha\cos\alpha}{\alpha^2 - \sin^2\alpha}, \quad F_y =\frac{2\mu U}{r} \frac{\sin^2\alpha}{\alpha^2 - \sin^2\alpha} </math>

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."

As noted earlier, all the forces 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 point of point, which depends on the geometry of the contact.

==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]]

[[Category:Mathematics]]