Lubrication analysis of the viscous micro/nano pump with slip

Abstract

A viscous pump is a device such that a cylindrical rotor is eccentrically placed in a channel, so that the viscous resistance between the small and large gaps between the cylinder and the channel walls generate a net flow along the channel. Assuming that the gaps between the cylinder and the channel walls are small compared to the radius of the rotor, the hydrodynamic theory of lubrication may be utilized to study the viscous pump. Here lubrication theory is used to obtain an analytical solution which relates the flowrate, rotation rate, pressure drop and applied torque as functions of two geometric parameters for a viscous pump. This analysis differs from a previous similar study in two ways. Firstly, certain integrals are evaluated explicitly, and secondly the standard no-slip boundary condition of fluid mechanics has been replaced with the Navier boundary condition which allows a degree of tangential velocity slip on all solid boundaries. Comparison with the prior known solution shows that the solution obtained in this study predicts a slightly improved pump performance for the case of no-slip. For the case of slip, our results demonstrate that the performance of the pump is significantly improved.

Appendices

Appendix A: Integrals for slip solution

It is a simple matter to show that the equation for h ^±(x) is given by

$$h^{\pm}\left(x\right)\,=\,h_0^{\pm}+a-\sqrt{a^2-x^2},$$

(A.1)

hence we have

$$I_{n,k}^{\pm}\,=\,\int\limits_{-a}^{a} \frac{{\rm d}x}{\left[h_0^{\pm}+a+k\ell-\sqrt{a^2-x^2}\right]^n}.$$

(A.2)

Making the substitution x =  a sin  θ this may be written

$$I_{n,k}^{\pm}\,=\,\int\limits_{-\frac{\pi}{2}}^{\frac{\pi} {2}} \frac{a\,\cos\,\theta} {\left[h_0^{\pm}+a+k\ell-a\,\cos\,\theta\right]^n} {\rm d}\theta. $$

(A.3)

This integral is in a form which appears in Gradshteyn and Ryzhik (2000). In particular, using Formulae 2.553.3 and 2.554.1 we find that

$$I_{2,0}^{\pm} \,=\, \frac{2a}{h_0^{\pm}\left(h_0^{\pm}+2a\right)} \left[1+ \frac{2a}{\sqrt{h_0^{\pm}\left(h_0^{\pm}+2a\right)}}\tan^{-1}\left(\sqrt{\frac{h_0^{\pm}+2a} {h_0^{\pm}}}\right)\right]. $$

(A.4)

Similarly, using Formulae 2.553.3 and 2.554.2 it is easy to show that

$$I_{1,k}^{\pm}\,=\, \frac{4 \left(h_0^{\pm} + a + k \ell \right)}{\sqrt{\left(h_0^{\pm} + k \ell\right) \left(h_0^{\pm}+2a+k \ell \right)}} \tan^{-1}\left(\sqrt{\frac{h_0^{\pm} + 2a + k \ell}{h_0^{\pm} + k \ell}}\right) - \pi. $$

(A.5)

Appendix B: Integrals for no-slip solution

Here we need to evaluate I ^±_3,0 , I ^±_2,0 and I ^±_1,0 . We have already derived expressions for I ^±_2,0 and I ^±_1,0 . Using Formulae 2.553.3, 2.554.1 and 2.554.2 from Gradshteyn and Ryzhik (2000) we have

$$ I_{3,0}^{\pm} \,=\, \frac{a}{\left[h_0^{\pm}\left(h_0^{\pm}+2a\right)\right]^2} \left[\frac{2h_0^{\pm}\left(h_0^{\pm}+2a\right)+3a^2}{h_0^{\pm}+a}\right. \left.+ \frac{6a\left(h_0^{\pm}+a\right)}{\sqrt{h_0^{\pm}\left(h_0^{\pm}+2a\right)}}\tan^{- 1}\left(\sqrt{\frac{h_0^{\pm}+2a}{h_0^{\pm}}}\right)\right]. $$

(B.1)

Appendix C: Approximate integrals for no-slip solution

In Day and Stone (2000) the approximations

$$h^{\pm}\left(x\right)\approx h_0^{\pm}\left(1+ \frac{x^2} {2ah_0^{\pm}} \right),$$

(C.1)

and

$$I_{n,0}^{\pm}\approx\int\limits_{-\infty}^{\infty}\frac{{\rm d}x} {\left[h^{\pm}\left(x\right)\right]^n},$$

(C.2)

are employed. These approximations are justified since the lubrication theory is applicable for h ^± ≪ a and as such the flow properties in the gap vary on the scale (h ^± a) ≪ a. From Gradshteyn and Ryzhik (2000), Formula 3.241.4 we have¹

$$I_{n,0}^{\pm}\,=\, \frac{\sqrt{2\pi a}} {\left(h_0^{\pm}\right)^{n-\frac{1}{2}}} \frac{\Gamma\left(n-\frac{1} {2}\right)} {\Gamma\left(n\right)}, $$

(C.3)

which are obviously significantly simpler than the expressions obtained above. Substituting the dimensionless variables and the geometric parameters η and δ we have

$$I_{n,0}^{+}\,=\,\frac{\sqrt{2\pi}} {\left(\eta\delta\right)^{n-{{1}\over {2}}}} \frac{\Gamma\left(n-\frac{1}{2}\right)} {\Gamma\left(n\right)},$$

(C.4)

$$I_{n,0}^{-}\,=\,\frac{\sqrt{2\pi}} {\left[\left(1-\eta\right)\delta\right]^{n-\frac{1} {2}}} \frac{\Gamma\left(n-\frac{1}{2}\right)} {\Gamma\left(n\right)}.$$

(C.5)

Now, substituting the above expression into Eqs. (2.33), (2.34) and (2.35) and performing some tedious algebra it may be shown that we recover Eq. (3.16) of Day and Stone (2000).