Magnetohydrodynamic flow with slippage in an annular duct for microfluidic applications

Abstract

In this paper, we investigate theoretically the 3D laminar flow of an electrolyte in an annular duct driven by a Lorentz force. The duct is formed by two concentric electrically conducting cylinders limited by insulating bottom and top walls. A uniform magnetic field acts along the axial direction, while a potential difference is applied between the cylinders so that a radial electric current traverses the fluid. The interaction of the current and the magnetic field produces a Lorentz force that drives an azimuthal flow. The steady flow is solved using a Galerkin method with Bessel–Fourier series in the radial direction and trigonometric series along the vertical direction, allowing different combinations of slip conditions at the walls. The orthogonality of both series with the general boundary conditions of the third kind is used to find an analytic approximation. Velocity patterns and flow rates are explored by varying the aspect ratio of the duct and the gap between the cylinders, as well as the slippage at the walls. Results can provide useful information for optimization and design of annular microfluidic devices.

Appendix

1.1 Boundary conditions of third kind

Required boundary conditions can be applied to Bessel functions of order $\nu$; therefore,

$$\begin{aligned} \frac{v_{\nu ; m}}{r}\bigg |_{r=1} + ls_e\frac{d}{{\hbox {d}}r}\left( \frac{v_{\nu ; m}}{r}\right) \bigg |_{r=1}&=0 \end{aligned}$$

(70)

$$\begin{aligned} \frac{v_{\nu ; m}}{r}\bigg |_{r=\eta } - ls_i\frac{d}{{\hbox {d}}r}\left( \frac{v_{\nu ; m}}{r}\right) \bigg |_{r=\eta }&=0. \end{aligned}$$

(71)

In compact form

$$\begin{aligned} v_{\nu ; m}'(1)&=\left( 1-\frac{1}{ls_e}\right) v_{\nu ; m}(1) \end{aligned}$$

(72)

$$\begin{aligned} v_{\nu ; m}'(\eta )&=\left( \frac{1}{\eta }+\frac{1}{ls_i}\right) v_{\nu ; m}(\eta ). \end{aligned}$$

(73)

1.2 Orthogonality of Bessel functions with third kind boundary conditions

$$\begin{aligned} \frac{1}{r}\frac{d}{{\hbox {d}}r}\left( r\frac{{\hbox {d}} v_{\nu ; m}}{{\hbox {d}}r}\right) - \frac{\nu ^2}{r^2} v_{\nu ; m}&= -\alpha _m^2 v_{\nu ; m} \end{aligned}$$

(74)

$$\begin{aligned} \frac{1}{r}\frac{d}{{\hbox {d}}r}\left( r\frac{{\hbox {d}} v_{\nu ; n}}{{\hbox {d}}r}\right) - \frac{\nu ^2}{r^2} v_{\nu ; n}&= -\alpha _n^2 v_{\nu ; n}. \end{aligned}$$

(75)

After multiplying (74) by $r u_{\nu ; n}$ and (75) by $r v_{\nu ; m}$, then subtracting the second to the first resulting equation and integrating, we get:

$$\begin{aligned} \left[ r v_{\nu ; n} \frac{{\hbox {d}} v_{\nu ; m}}{{\hbox {d}}r}- r v_{\nu ; m} \frac{{\hbox {d}} v_{\nu ; n}}{{\hbox {d}}r}\right] \bigg |_{\eta }^{1}=\left( \alpha _n^2-\alpha _m^2\right) \int _\eta ^1 r v_{\nu ; m} v_{\nu ; n} {\hbox {d}}r. \end{aligned}$$

(76)

The equation can be expanded and expressed as

$$\left( \alpha _{n}^{2}-\alpha _{m}^{2}\right) \int _{\eta} ^{1} r v_{\nu ; m} v_{\nu ; n} {\hbox {d}}r= v_{\nu ; n}(1) v_{\nu ; m}'(1)-v_{\nu ; m}(1) v_{\nu ; n}'(1)$$

(77)

$$-\eta v_{\nu ; n} (\eta ) v_{\nu ; m}'(\eta ) + \eta u_{\nu ; m} (\eta ) v_{\nu ; n}'(\eta ).$$

(78)

If the $v_{\nu ; n}$ functions either satisfy the no-slip boundary conditions $v_{\nu ; m} (1)=v_{\nu ; m} (\eta )= v_{\nu ; n} (1 )= v_{\nu ; n} (\eta )=0$ or the boundary conditions (72) and (73), the orthogonality condition is satisfied and the right-hand side of Eq. (78) is zero. That is:

$$\begin{aligned} \left( \alpha _n^2-\alpha _m^2\right) \int _\eta ^1 r v_{\nu ; m} v_{\nu ; n} {\hbox {d}}r= 0. \end{aligned}$$

(79)

Another important result occurs when the eigenfunctions have the same eigenvalue, then multiplying the expanded form of Eq. (74) by $2r^2\frac{{\hbox {d}} v_{\nu ; m}}{{\hbox {d}}r}$:

$$2r^{2} v_{\nu ; m}' v_{\nu ; m}''+2r^{2}\left( v_{\nu ; m}''\right) ^{2}- 2\nu ^{2} v_{\nu ; m} v_{\nu ; m}' = -2\alpha _{m}^{2}r^{2} v_{\nu ; m} v_{\nu ; m}'$$

(80)

Using the following identities in the last expression:

$$\left[ \left( r v_{\nu ; m}'\right) ^{2}\right] '=2r^{2}v_{\nu ; m}'v_{\nu ; m}''+2r\left( v_{\nu ; m}'\right) ^{2}$$

(81)

$$\left[ \left( r v_{\nu ; m}\right) ^{2}\right] '=2r^{2}v_{\nu ; m}v_{\nu ; m}'+2r\left( v_{\nu ; m}\right) ^2$$

(82)

$$\left[ \left( r v_{\nu ; m}'\right) ^{2}+\left( \alpha _{m} r v_{\nu ; m}\right) ^{2}-\left( \nu v_{\nu ; m}\right) ^{2}\right] '=2\alpha _{m}^{2} r {v}_{\nu ; m}^{2}$$

(83)

$$\left[ r^{2}\left( v_{\nu ; m}'^{2}+\alpha _{m}^{2} v_{\nu ; m}^{2}\right) -\nu ^{2} v_{\nu ; m}^{2}\right] '=2\alpha _{m}^{2} r v_{\nu ; m}^{2}.$$

(84)

Changing both sides and integrating over the region of interest $[\eta ,1]$, then:

$$\begin{aligned} 2\alpha _m^2\int _\eta ^1 r v_{\nu ; m}^2 {\hbox {d}}r = \left[ r^2\left( v_{\nu ; m}'^2+\alpha _m^2 v_{\nu ; m}^2\right) -\nu ^2 v_{\nu ; m}^2\right] \big |_\eta ^1 \end{aligned}$$

(85)

Using again the no-slip boundary conditions $v_{\nu ; m} (1)=v_{\nu ; m} (\eta )=0$, then

$$\begin{aligned} \int _\eta ^1 r v_{\nu ; m}^2 {\hbox {d}}r = \frac{v_{\nu ; m}'^2(1)-\eta ^2v_{\nu ; m}'^2 (\eta )}{2\alpha _m^2}. \end{aligned}$$

(86)

For the slip boundary conditions (72) and (73), we have:

$$\begin{aligned} \int _\eta ^1 r v_{\nu ; m}^2 {\hbox {d}}r =\frac{\left[ \left( 1-\frac{1}{l_{s2}}\right) ^2+\alpha _m^2-\nu ^2\right] v^2_{\nu ; m}(1)-\left[ \left( 1+\frac{\eta }{l_{s1}}\right) ^2+\eta ^2\alpha _m^2-\nu ^2\right] v^2_{\nu ; m}(\eta )}{2\alpha _m^2} \end{aligned}$$

(87)

or

$$\begin{aligned} \int _\eta ^1 r v_{\nu ; m} r v_{\nu ; n} {\hbox {d}}r = S_n\delta _{mn} \end{aligned}$$

(88)

where

$$\begin{aligned} S_n=\frac{\left[ \left( 1-\frac{1}{ls_e}\right) ^2+\alpha _n^2-\nu ^2\right] v^2_{\nu ; n}(1)-\left[ \left( 1+\frac{\eta }{ls_i}\right) ^2+\eta ^2\alpha _n^2-\nu ^2\right] v^2_{\nu ; n}(\eta )}{2\alpha _n^2}. \end{aligned}$$

(89)