Impact of asymmetric zeta-potential on natural convection flow in a vertical microannulus with electroosmotic and Joule heating effects

Abstract

Analytical solutions of temperature distributions and flow formation of joint buoyancy and electroosmotic flow in a vertical microtube formed by two concentric microcylinders are presented. The central equations describing flow formations and thermal energy are offered and solved in closed-form in terms of modified Bessel’s function of first and second kinds. These solutions have significant application in predicting and analyzing flow formation and thermal behavior of Newtonian fluids in micropumps and microchips. Based on the exact solutions obtained, the effects of flow parameters are clearly explained with the use of line graphs. Based on the simulation of results using MATLAB, it is found that fluid temperature distributions and fluid velocity in the vertical microannulus could be enhanced by increasing the radius ratio of the concentric microcylinders.

Introduction

The study of electroosmotic flow (EOF) gains its relevance in micro geometries, due to the comparable channel width to the electric double layer length. This phenomenon is generated by the combined application of low electric voltage at the surfaces of the microchannel and the interaction between the walls and the charged fluid. A lot has been discussed over the years about this phenomenon owing to its significance in micromechanical systems (MEMS). Experimental existence of the phenomenon and theoretical background were respectively presented by Reuss ¹ and Probstein. ² Since then, many applicable extensions for this phenomenon have been carried out in the literature. Rice and Whitehead ³ in a narrow cylindrical capillary, discussed the significance of electrokinetic flow in the presence of low zeta potential. Kang et al. ⁴ revisited the work of Rice and Whitehead ³ in the presence of with high zeta potentials. The electrokinetic flow in ultrafine capillary slits was carried out by Burgreen and Nakache ⁵ while Debye and Hückel ⁶ offered a linearized form of the nonlinear Poisson equation governing the electric potential in the EDL for low electric potential. Also, Tsao ⁷ examined the EOF in an annulus, whereas, Wang et al. ⁸ gave an analytical solution of EOF in a semi-circular microchannel. In all these articles, the influence of buoyancy force induced by unequal heating at the surfaces of the microcylinders was not accounted for. Other related articles can be seen in Refs.^9–17

The natural convection flow (NCF) is a kind of flow generated as a consequence of unequal density induced by unequal wall temperature. This kind of heat transfer is significant in cooling nuclear reactors, automobiles and other devices at microlevel. The investigation of NCF in microgeometries extends its applicability in cooling microchips and micro-electro-mechanical systems (MEMS). Some of the earliest works on this phenomenon include the works of Elenbaas, ¹⁸ Ostrach, ¹⁹ Sparrow and Gregg ²⁰ and Bodoia and Osterle ²¹ where they respectively studied the NCF in parallel plates, with and without heat sources, in vertical plate with uniform heat flux and uniform temperature. Later, Aung ²² and Aung et al. ²³ analyzed respectively the fully developed and developing free convection flow between vertical flat plates. Since then, many articles on different physics of flows have been published. To mention a few, Weng and Chen ²⁴ studied the drag reduction of NCF in vertical microchannel while Weng and Chen ²⁵ investigated the NCF in a vertical microchannel. Jha and Oni ²⁶ studied the time-dependent NCF in a vertical annulus heated/cooled asymmetrically. Jha et al. ²⁷ scrutinized the mixed convection flow in a vertical microannulus with heat source/sink. They gave separate solutions for the heat source/sink cases and concluded that heat generating case has higher velocity and fluid temperature compared with the case of heat sink in the microannulus. Other related articles on NCF in an annular geometry include Refs.^28–33

In spite the contributions above, precise theory of electroosmotic flow in vertical microannulus with buoyancy force and asymmetric heat flux is still rare. Such solution has significant application in chemical separation at microlevel, which could be used in drug delivery to a targeted cell in the body. In view of this, the purpose of this article is to discuss the role of asymmetric zeta-potential on NCF in a vertical microannulus with electroosmotic and Joule heating effect. The novelty of this article is the presentation of analytical mathematical models to predict flow formation and skin-friction at the surfaces of the microcylinders of NCF in a vertical microannulus with electroosmotic effect.

Mathematical study

Consider the flow on an incompressible balance $(z:z)$ electrolyte, where $z$ is the valence of the electrolyte in a vertical microannulus induced by the combined influence of buoyancy (caused by density difference), electroosmotic force caused by the EDL at the interface between the microannulus surfaces and the fluid and an externally applied electric field as revealed in Figure 1. The outer and inner cylinders are of radii $k_2$ and $k_1$ respectively such that $k_2>k_1$ and the length of the cylinders is strictly greater than the width $(k_2-k_1)$ of the microannulus. Constant isofluxes, $q_1$ and $q_2$ are prescribed at the inner and outer cylinders respectively.

Figure 1.

Schematic diagram of the problem.

The flow is presumed to be hydrodynamically and thermally fully developed. Other assumptions made include:

• (i) The flow is laminar with equivalent Reynolds number less than unity.

• (ii) The charge in the EDL agrees with the Boltzmann distribution and the Debye length is strictly smaller than the annular gap.

• (iii) The EDL width is not affected by the temperature gradients.

• (iv) The temperature jump effect as well as the velocity slip are incorporated at the microannulus surfaces.

For fully developed flow, the flow becomes unidirectional ( $\overrightarrow{V}=0\overrightarrow{r}+0\overrightarrow{\vartheta }+u\left(r\right)\overrightarrow{z}$ ). Following, ³⁴ the governing electric potential, continuity, momentum and energy equations in dimensional form respectively become:

$$\frac{1}{r}\frac{d}{\mathrm{dr}}\left(r\frac{d{\psi }^{\prime }}{\mathrm{dr}}\right)=\frac{2F{z}^{\prime }{C}_0}{\epsilon \prime }\sinh \left(\frac{z^{\prime }F{\psi }^{\prime }}{R^{\prime }\hat{T}}\right)$$

(1)

$$\frac{\partial u}{\partial z}=0$$

(2)

$$\frac{\partial u}{\partial t}=\frac{\mu }{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\beta g{\rho }_0(T-T_0)+{\rho }_{e}E{\prime }_z$$

(3)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial z}=\frac{k^{\prime }}{{\rho }_0{C}_p}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\frac{{E^{\prime 2}}_z}{{\rho }_0{C}_p\sigma }{\cosh }^2}\\ \multicolumn{1}{c}{\left(\frac{\mathrm{zF}{\psi }^{\prime }}{R^{\prime }\hat{T}}\right)+\frac{\mu }{{\rho }_0{C}_p}{\left(\frac{\partial u}{\partial r}\right)}^2}\end{array}$$

(4)

subject to

$$\begin{array}{c}\multicolumn{1}{c}{{\psi }^{\prime }={\mathrm{\zeta }}_1,\frac{\partial T}{\partial r}=-\frac{{\mathrm{q}}_1}{k},u\left(r\right)=\alpha \prime \frac{\partial u}{\partial r}\mathrm{at}r=k_1}\\ \multicolumn{1}{c}{{\psi }^{\prime }={\mathrm{\zeta }}_2,\frac{\partial T}{\partial r}=\frac{{\mathrm{q}}_2}{k},u\left(r\right)=-\alpha \prime \frac{\partial u}{\partial r},\mathrm{at}r=k_2}\end{array}$$

(5)

where ${\psi }^{\prime }$ is the dimensional electric potential due to EDL, $F$ is the Faraday’s constant, “ $z^{\prime }$ is the valence number of ions in the solution,” ${\rho }_0$ is the density of the fluid to $T_0$ , $T_0$ is the ambient fluid temperature and $q_{i=1,2}$ are the prescribed heat fluxes at the surfaces of the microcylinders. Equation (1) is the Poisson-Boltzmann equation which describes the EDL distributions in the microannulus, the fourth term of equation (3) represents the electric body force, which is obtained from the combination of applied electric gradient and EDL potential. While the last two terms in the energy equation (4) signify the viscous dissipation and Joule heating terms respectively. Equation (2) on the other hand represents the continuity equation. All other physical quantities employed in equations (1)–(5) are well defined in nomenclature.

Introducing the following dimensionless quantities and parameters ¹³ :

$$\begin{array}{c}R=\frac{r-k_1}{w},Z=\frac{z}{w},w=k_2-k_1,T=\frac{\theta q_{2}w}{k}+T_{s1},\hfill \\ \multicolumn{1}{c}{U=\frac{u}{u_0},\alpha =\delta =\frac{\lambda }{w},\mathrm{Gr}=\frac{g\beta {\rho }_0{q}_2{w}^4}{\kappa \mu \nu },}\\ {\xi }_1^{*}=\frac{z^{\prime }F{\zeta }_1}{R^{\prime }{T}_0},{\xi }_2^{*}=\frac{z^{\prime }F{\zeta }_2}{R^{\prime }{T}_0},\Gamma =\frac{{\epsilon }^{\prime }{R}^{\prime }\hat{T}{\zeta }_1}{z^{\prime }F{\rho }_0{\nu }^2},\hfill \\ \multicolumn{1}{c}{E_z=\frac{{E^{\prime }}_{z}w}{{\zeta }_1},\eta =\frac{k_1}{k_2},s=\frac{{\zeta }_{1}w}{\sigma (T_1-T_0)},\psi =\frac{z^{\prime }F\psi \prime }{R^{\prime }{T}_0},}\\ \kappa =\frac{w}{{\lambda }_D},{\lambda }_D={\left[\frac{{\epsilon }^{\prime }{R}^{\prime }{T}_0}{2F^2{z}^{\prime 2}{C}_0}\right]}^{1/2},\mathrm{Ra}=\frac{{\rho }_0{c}_{p_0}{u}_{0}w}{k^{\prime }},\hfill \\ \multicolumn{1}{c}{{\mathrm{r}}_q=\frac{{\mathrm{q}}_1}{{\mathrm{q}}_2},\alpha =\frac{{\alpha }^{\prime }}{w}}\end{array}$$

(6)

Considering the steady, fully developed region and a very small EDL size relative to $w$ , so that equations (1), (3), and (4) on applying equation (6) respectively become:

$$\frac{1}{(\eta +(1-\eta )R)}\frac{d}{\mathrm{dR}}\left((\eta +(1-\eta )R)\frac{d\psi }{\mathrm{dR}}\right)-k^2\psi =0$$

(7)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{1}{(\eta +(1-\eta )R)}\frac{d}{\mathrm{dR}}\left((\eta +(1-\eta )R)\frac{\mathrm{dU}}{\mathrm{dR}}\right)}\\ \multicolumn{1}{c}{+\mathrm{Gr}\theta -E_z\Gamma k^2\psi =0}\end{array}$$

(8)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{RaU}\frac{\partial \theta }{\partial Z}=\frac{1}{(\eta +(1-\eta )R)}\frac{\partial }{\partial R}\left((\eta +(1-\eta )R)\frac{\partial \theta }{\partial R}\right)}\\ \multicolumn{1}{c}{+s{E_z}^2}\end{array}$$

(9)

Subject to

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{d}\mathrm{\theta }}{\mathrm{dR}}=-r_q,U\left(R\right)=\alpha \frac{\mathrm{dU}}{\mathrm{dR}}\mathrm{at}R=0}\\ \multicolumn{1}{c}{\frac{\mathrm{d}\mathrm{\theta }}{\mathrm{dR}}=1,U\left(R\right)=-\alpha \frac{\mathrm{dU}}{\mathrm{dR}},\mathrm{at}R=1}\end{array}$$

(10)

By using the transformation $\chi =\eta +(1-\eta )R$ , it is convenient to examine the influence of annular gap on flow formation and skin-friction at the surfaces of the microannulus. Equations (7)–(10) are transformed as follows ²⁴ :

$$\frac{1}{\chi }\frac{d}{d\chi }\left(\chi \frac{d\psi }{d\chi }\right)-{\kappa }^2\psi =0$$

(11)

$$\frac{1}{\chi }\frac{d}{\mathrm{dR}}\left(\chi \frac{\mathrm{dU}}{d\chi }\right)+\frac{\mathrm{Gr}\theta }{{(1-\eta )}^2}-\frac{E_z\Gamma {\kappa }^2\psi }{{(1-\eta )}^2}=0$$

(12)

$$\frac{\mathrm{RaU}}{{(1-\eta )}^2}\frac{\partial \theta }{\partial Z}=\frac{1}{\chi }\frac{\partial }{\partial \chi }\left(\chi \frac{\partial \theta }{\partial \chi }\right)+\frac{s{E_z}^2}{{(1-\eta )}^2}$$

(13)

Subject to:

$$\begin{array}{c}\multicolumn{1}{c}{\psi \left(\eta \right)={\xi }_1^{*},\frac{\mathrm{d}\mathrm{\theta }}{d\chi }=\frac{-r_q}{(1-\eta )},U\left(\chi \right)=\alpha (1-\eta )\frac{\mathrm{dU}}{d\chi }\mathrm{at}\chi =\eta }\\ \multicolumn{1}{c}{\psi \left(1\right)={\xi }_2^{*},\frac{\mathrm{d}\mathrm{\theta }}{d\chi }=\frac{1}{(1-\eta )},U\left(\chi \right)=-\alpha (1-\eta )\frac{\mathrm{dU}}{\mathrm{dR}},\mathrm{at}\chi =1}\end{array}$$

(14)

where $\kappa =\frac{k}{(1-\eta )}$ , ${\xi }_1^{*},{\xi }_2^{*}$ are the non-dimensional asymmetric zeta-potentials at the microannulus surfaces.

The equations above describe the mathematical equations governing electroosmotic NCF in a vertical microannulus with Joule heating effect. The term ${\mathrm{r}}_q$ explains the level of unevenness in heat fluxes whereas $\mathrm{\alpha }$ signifies the dimensionless velocity slip-length.

For fully developed flow, the expression $\frac{\partial \theta }{\partial Z}$ is a constant and is computed by multiplying through by $\chi$ , integrating equation (13) and substituting equation (14) as follows:

$$\frac{\partial \theta }{\partial Z}=\frac{1}{\mathrm{Ra}}\left[\frac{2(1+\eta r_q)+s{E_z}^2(1+\eta )}{(1+\eta )}\right]$$

(15)

So that equation (13) becomes:

$$\frac{1}{\chi }\frac{d}{d\chi }\left(\chi \frac{d\theta }{\partial \chi }\right)+\frac{s{E_z}^2}{{(1-\eta )}^2}=\frac{2(1+\eta r_q)+s{E_z}^2(1+\eta )U}{(1+\eta ){(1-\eta )}^2}$$

(16)

The solution of equation (11) with boundary condition (14) can be conveniently written in form of modified Bessel’s function of first and second kinds of order zero as (Olver et al. ³⁵ ):

$$\psi \left(\chi \right)=C_1{I}_0\left(\chi \kappa \right)+C_2{K}_0\left(\chi \kappa \right)$$

(17)

where $K_0$ and $I_0$ are the modified Bessel’s function of second and first kinds of order zero and $C_1$ and $C_2$ are constants obtained as:

$$\begin{array}{c}\multicolumn{1}{c}{C_1=\frac{{\xi }_2^{*}{K}_0\left(\eta \kappa \right)-{\xi }_1^{*}{K}_0\left(\kappa \right)}{I_0\left(\kappa \right)K_0\left(\eta \kappa \right)-I_0\left(\eta \kappa \right)K_0\left(\kappa \right)},}\\ \multicolumn{1}{c}{C_2=\frac{{\xi }_1^{*}{I}_0\left(\eta \kappa \right)-{\xi }_2^{*}{I}_0\left(\kappa \right)}{I_0\left(\kappa \right)K_0\left(\eta \kappa \right)-I_0\left(\eta \kappa \right)K_0\left(\kappa \right)}}\end{array}$$

(18)

Equations (12) and (16) are the mathematical models governing the velocity profile and temperature distributions in the microannulus respectively. These equations are coupled and linear, so that their solutions are obtained by first making $\theta$ subject of formula from equation (12) as follows:

$$\theta =\frac{{(1-\eta )}^2}{\mathrm{Gr}}[\frac{E_z\Gamma {\kappa }^2\psi }{{(1-\eta )}^2}-\frac{1}{\chi }\frac{d}{\mathrm{dR}}\left(\chi \frac{\mathrm{dU}}{d\chi }\right)]$$

(19)

So that on applying the operator $\frac{1}{\chi }\frac{d}{\mathrm{dR}}\left(\chi \frac{d}{d\chi }\right)$ to both sides of (19) and comparing with equation (16), we have:

$$\begin{array}{c}\multicolumn{1}{c}{\frac{1}{\chi }\frac{d}{\mathrm{dR}}\left(\chi \frac{d}{d\chi }\right)\frac{1}{\chi }\frac{d}{\mathrm{dR}}\left(\chi \frac{\mathrm{dU}}{d\chi }\right)}\\ \multicolumn{1}{c}{+\frac{\mathrm{Gr}[2(1+\eta r_q)+s{E_z}^2(1+\eta )]U}{(1+\eta ){(1-\eta )}^4}}\\ \multicolumn{1}{c}{=\frac{\mathrm{sGr}{E_z}^2}{{(1-\eta )}^4}+\frac{E_z\Gamma k^2\psi }{{(1-\eta )}^4}\frac{1}{\chi }\frac{d}{\mathrm{dR}}\left(\chi \frac{d\psi }{d\chi }\right)}\end{array}$$

(20)

The homogeneous part of equation (20) can be written as:

$$(L^4-m^4)U=0$$

(21)

where $L^2=\frac{1}{\chi }\frac{d}{\mathrm{dR}}\left(\chi \frac{d}{d\chi }\right)$ , and $m=\frac{i^{1/2}{\left\{\mathrm{Gr}[2(1+\eta r_q)+s{E_z}^2(1+\eta )]\right\}}^{1/4}}{{(1+\eta )}^{1/4}(1-\eta )}$ .

So that equation (21) becomes:

$$(\frac{1}{\chi }\frac{d}{\mathrm{dR}}\left(\chi \frac{\mathrm{dU}}{d\chi }\right)+m^{2}U)(\frac{1}{\chi }\frac{d}{\mathrm{dR}}\left(\chi \frac{\mathrm{dU}}{d\chi }\right)-m^{2}U)=0$$

(22)

Equation (22) is the combination of standard Bessel’s function and modified Bessel’s function. On solving equation (20) with boundary conditions (14), we obtained the velocity profile in closed form as:

$$\begin{array}{c}\multicolumn{1}{c}{U=C_3{I}_0\left(m\chi \right)+C_4{K}_0\left(m\chi \right)+C_5{J}_0\left(m\chi \right)+C_6{Y}_0\left(m\chi \right)}\\ \multicolumn{1}{c}{+\frac{E_z\Gamma {\mathrm{\kappa }}^4[C_1{I}_0\left(\chi \kappa \right)+C_2{K}_0\left(\chi \kappa \right)]}{{\mathrm{\kappa }}^4-{\mathrm{m}}^4}}\\ \multicolumn{1}{c}{-\frac{2s{E_z}^2(1+\eta )}{2(1+\eta r_q)+s{E_z}^2(1+\eta )}}\end{array}$$

(23)

By using equation (19), the temperature distributions in the microannulus is obtained as:

$$\begin{array}{c}\multicolumn{1}{c}{\theta =\frac{E_z\Gamma }{\mathrm{Gr}}[{\kappa }^2-\frac{{\kappa }^6{(1-\eta )}^2}{({\kappa }^4-{\mathrm{m}}^4)}][C_1{I}_0\left(\chi \kappa \right)+C_2{K}_0\left(\chi \kappa \right)]}\\ \multicolumn{1}{c}{-\frac{{(1-\eta )}^2{m}^2}{\mathrm{Gr}}[C_3{I}_0\left(m\chi \right)+C_4{K}_0\left(m\chi \right)}\\ \multicolumn{1}{c}{-C_5{J}_0\left(m\chi \right)-C_6{Y}_0\left(m\chi \right)]}\end{array}$$

(24)

By putting the boundary conditions in equations (14) on equations (23) and (24), the constants $C_3,C_4,C_5,$ and $C_6$ are obtained as follow; “

$$\begin{array}{c}\multicolumn{1}{c}{C_3=\frac{det\left(\begin{array}{cccc}{X}_5\hfill & X_2\hfill & X_3\hfill & X_4\hfill \\ \multicolumn{1}{c}{X_{10}}& X_7\hfill & X_8\hfill & X_9\hfill \\ \multicolumn{1}{c}{X_{16}}& X_{13}\hfill & X_{14}\hfill & X_{15}\hfill \\ \multicolumn{1}{c}{X_{21}}& X_{18}\hfill & X_{15}\hfill & X_{20}\hfill \end{array}\right)}{det\left(\begin{array}{cccc}{X}_1\hfill & X_2\hfill & X_3\hfill & X_4\hfill \\ \multicolumn{1}{c}{X_6}& X_7\hfill & X_8\hfill & X_9\hfill \\ \multicolumn{1}{c}{X_{12}}& X_{13}\hfill & X_{14}\hfill & X_{15}\hfill \\ \multicolumn{1}{c}{X_{17}}& X_{18}\hfill & X_{19}\hfill & X_{20}\hfill \end{array}\right)}{C}_4=\frac{det\left(\begin{array}{cccc}{X}_1\hfill & X_5\hfill & X_3\hfill & X_4\hfill \\ \multicolumn{1}{c}{X_6}& X_{10}\hfill & X_8\hfill & X_9\hfill \\ \multicolumn{1}{c}{X_{12}}& X_{16}\hfill & X_{14}\hfill & X_{15}\hfill \\ \multicolumn{1}{c}{X_{17}}& X_{21}\hfill & X_{15}\hfill & X_{20}\hfill \end{array}\right)}{det\left(\begin{array}{cccc}{X}_1\hfill & X_2\hfill & X_3\hfill & X_4\hfill \\ \multicolumn{1}{c}{X_6}& X_7\hfill & X_8\hfill & X_9\hfill \\ \multicolumn{1}{c}{X_{12}}& X_{13}\hfill & X_{14}\hfill & X_{15}\hfill \\ \multicolumn{1}{c}{X_{17}}& X_{18}\hfill & X_{19}\hfill & X_{20}\hfill \end{array}\right)}}\\ \multicolumn{1}{c}{C_5=\frac{det\left(\begin{array}{cccc}{X}_1\hfill & X_2\hfill & X_5\hfill & X_4\hfill \\ \multicolumn{1}{c}{X_6}& X_7\hfill & X_{10}\hfill & X_9\hfill \\ \multicolumn{1}{c}{X_{12}}& X_{13}\hfill & X_{16}\hfill & X_{15}\hfill \\ \multicolumn{1}{c}{X_{17}}& X_{18}\hfill & X_{21}\hfill & X_{20}\hfill \end{array}\right)}{det\left(\begin{array}{cccc}{X}_1\hfill & X_2\hfill & X_3\hfill & X_4\hfill \\ \multicolumn{1}{c}{X_6}& X_7\hfill & X_8\hfill & X_9\hfill \\ \multicolumn{1}{c}{X_{12}}& X_{13}\hfill & X_{14}\hfill & X_{15}\hfill \\ \multicolumn{1}{c}{X_{17}}& X_{18}\hfill & X_{19}\hfill & X_{20}\hfill \end{array}\right)}{C}_6=\frac{det\left(\begin{array}{cccc}{X}_1\hfill & X_2\hfill & X_3\hfill & X_5\hfill \\ \multicolumn{1}{c}{X_6}& X_7\hfill & X_8\hfill & X_{10}\hfill \\ \multicolumn{1}{c}{X_{12}}& X_{13}\hfill & X_{14}\hfill & X_{16}\hfill \\ \multicolumn{1}{c}{X_{17}}& X_{18}\hfill & X_{19}\hfill & X_{21}\hfill \end{array}\right)}{det\left(\begin{array}{cccc}{X}_1\hfill & X_2\hfill & X_3\hfill & X_4\hfill \\ \multicolumn{1}{c}{X_6}& X_7\hfill & X_8\hfill & X_9\hfill \\ \multicolumn{1}{c}{X_{12}}& X_{13}\hfill & X_{14}\hfill & X_{15}\hfill \\ \multicolumn{1}{c}{X_{17}}& X_{18}\hfill & X_{19}\hfill & X_{20}\hfill \end{array}\right)}}\end{array}$$

(25)

where det represents the determinant of the matrix and $X_{i\prime s};i=1,2,3,\dots$ are constants defined in Appendix.

In order to incorporate the temperature jump effect, following (Avci and Aydin ³⁶ ), the temperature distribution in the vertical microchannel is obtained as:

$$\begin{array}{c}\multicolumn{1}{c}{{\theta }^{*}\left(Y\right)=\frac{(T-T_1)}{q_2{D}_h/k}=\frac{E_z\Gamma }{\mathrm{Gr}}[{\kappa }^2-\frac{{\kappa }^6{(1-\eta )}^2}{({\mathrm{\kappa }}^4-{\mathrm{m}}^4)}]}\\ \multicolumn{1}{c}{[C_1{I}_0\left(\chi \kappa \right)+C_2{K}_0\left(\chi \kappa \right)]}\\ -\frac{{(1-\eta )}^2{m}^2}{\mathrm{Gr}}[C_3{I}_0\left(m\chi \right)+C_4{K}_0\left(m\chi \right)\hfill \\ \multicolumn{1}{c}{-C_5{J}_0\left(m\chi \right)-C_6{Y}_0\left(m\chi \right)]-{\beta }_t{r}_q}\end{array}$$

(26)

From the velocity solution obtained, the skin frictions at the micro-annular surfaces are given as:

$$\begin{array}{c}\multicolumn{1}{c}{{\frac{\mathrm{dU}}{d\chi }|}_{\chi =\eta }=(1-\eta )m[C_3{I}_1\left(m\eta \right)-C_4{K}_1\left(m\eta \right)}\\ \multicolumn{1}{c}{-C_5{J}_1\left(m\eta \right)-C_6{Y}_1\left(m\eta \right)]}\\ \multicolumn{1}{c}{+\frac{(1-\eta )E_z\Gamma {\kappa }^5}{({\kappa }^4-m^4)}\{C_1{I}_1\left(\kappa \eta \right)-C_2{K}_1\left(\kappa \eta \right)\}}\end{array}$$

(27)

$$\begin{array}{c}\multicolumn{1}{c}{{\frac{\mathrm{dU}}{d\chi }|}_{\chi =1}=(1-\eta )m[C_3{I}_1\left(m\right)-C_4{K}_1\left(m\right)-C_5{J}_1\left(m\right)}\\ \multicolumn{1}{c}{-C_6{Y}_1\left(m\right)]+\frac{(1-\eta )E_z\Gamma {\kappa }^5}{({\kappa }^4-m^4)}\{C_1{I}_1\left(\kappa \right)-C_2{K}_1\left(\kappa \right)\}}\end{array}$$

(28)

Also, the volumetric flow-rate in the microannulus is obtained as:

$$\begin{array}{l}Q=\frac{1}{{\left(1-\eta \right)}^2}\underset{\eta }{\overset{1}{{\displaystyle \int }}}\chi U(\chi )d\chi \\ =\frac{1}{{\left(1-\eta \right)}^2}\left\{\begin{array}{l}\frac{C_3}{m}\left[I_1\left(m\right)-\eta I_1\left(m\eta \right)\right]-\frac{C_4}{m}\left[K_1\left(m\right)-\eta K_1\left(m\eta \right)\right]-\\ \frac{C_5}{m}\left[J_1\left(m\right)-\eta J_1\left(m\eta \right)\right]-\frac{C_6}{m}\left[Y_1\left(m\right)-\eta Y_1\left(m\eta \right)\right]+\frac{E_z\Gamma {\kappa }^3}{\left({\kappa }^4-m^4\right)}\\ \left\{C_1\left[I_1\left(\kappa \right)-\eta I_1\left(\kappa \eta \right)\right]-C_2\left[K_1\left(\kappa \right)-\eta K_1\left(\kappa \eta \right)\right]\right\}-\frac{s{E}_z^2\left(1-{\eta }^2\right)\left(1+\eta \right)}{2\left(1+\eta r_q\right)+s{E}_z^2\left(1+\eta \right)}\end{array}\right\}\end{array}$$

(29)

Results and discussion

The discussion of results for electroosmotic NCF in a vertical microannulus with asymmetric zeta-potentials at the surfaces of the microcylinders in the existence of Joule heating is carried out in the section. The solutions gotten showed that the sundry parameters influencing flow formations are ratio of radii $(\eta )$ which is inversely proportional to annular gap, slip-parameter $(\alpha )$ , Debye Hückel parameter $(\kappa )$ which is inversely proportional to the EDL size, Joule heating parameter $(s)$ which is proportioanl to the thermal energy, ratio of heat fluxes $(r_q)$ , magnitude of electroosmotic force $(\Gamma )$ , and the Grashof number $\left(\mathrm{Gr}\right)$ which defines the strength of the bouyant force. These parameters have significant applications in engineering designs and understanding the physics of flow formation. Therefore, it is expedient to understand how these parameters affect flow formation and minimization of skin-friction in the system. To ascertain this objectives, line graphs are generated for the analytical solutions obtained using Matlab software. Except otherwise stated, we have use the values of $0<\eta <1$ , with fixed value of $\eta =0.5$ , $0\le \alpha \le 0.1$ to represent the no-slip $(\alpha =0.0)$ and slip $(\alpha =0.1)$ domain, $0<\kappa \le 30$ , $0<s\le 5$ , $0\le r_q\le 1$ , $0<\mathrm{Gr}\le 50$ .

Figures 2 and 3 respectively depict the fluid temperature and velocity profile in the vertical microannulus as functions of radius ratio and asymmetry in wall zeta-potential for constant value of other leading parameters. It is noticeable that fluid temperature as well as velocity respond noticeably to change in $\eta$ and ${\xi }_2^{*}$ . In fact, fluid temperature and velocity both increase with increase in $\eta$ regardless the value of ${\xi }_2^{*}$ throughout the microannulus. From both figures, the maximum temperature as well as velocity are accomplished for the instance of symmetry zeta-potential at the surfaces of the microannulus regardless the values of $\eta$ . This is expected, because the annular gap varies inversely with $\eta$ , which sequentially increases the thermal energy within the small annular gap and thereby increasing the overall fluid temperature and velocity throughout the microannulus.

Figure 2.

Temperature distributions for different values of $\eta$ and ${\xi }_2^{*}$ .

Figure 3.

Velocity profile for different values of $\eta$ and ${\xi }_2^{*}$ .

In addition, while the parabolic nature of fluid velocity is preserved, we found a nearly constant temperature throughout the microannulus.

It is pertinent to also understand the role of buoyancy term on temperature distributions as well as velocity profile in the microannulus. To this end, Figures 4 and 5 respectively represent the temperature distributions and fluid velocity as functions of Grashof number $(\mathrm{Gr})$ and ${\xi }_2^{*}$ . It is established from Figure 4 that the impact of buoyancy term is to reduce fluid temperature throughout the microannulus. On the other hand (Figure 5), velocity decreases as $\mathrm{Gr}$ increases, around the center of the annulus but increases along the inner surface of the outer cylinder. From both figures, it is obvious that the maximum temperature and velocity can be obtained by subjecting the surfaces of the microannulus to equal zeta-potentials. In addition, for the situation of unequal zeta-potentials, two points of inflection (turning points) are observed of which at these points, fluid velocity is independent of $\mathrm{Gr}$ .

Figure 4.

Temperature distributions for different values of $\mathrm{Gr}$ and ${\xi }_2^{*}$ .

Figure 5.

Velocity profile for different values of $\mathrm{Gr}$ and ${\xi }_2^{*}$ .

For proper investigation of the impact of sundry parameters on volumetric flow-rate, Figures 6 to 8 depict the effect of various leading parameters on the amount of fluids passing through the microannulus (volumetric flow-rate). This phenomenon has significant importance in optimization of flow rate in design of micropumps and drug targeting. Figure 6 revealed that the maximum volumetric flow-rate can be achieved by exposing the microannulus surfaces to equal wall zeta-potentials. Also, flow-rate is found to first increase for small values of $\kappa$ (corresponding to large EDL) and then gradually decreases until flow formation transforms from combined buoyancy-electroosmotic flow to purely buoyancy driven flow. Throughout the microannulus, the volumetric flow-rate is found to decline with increase in $\eta$ regardless the values of ${\xi }_2^{*}$ and $\kappa$ . This could be attributed to the fact that increasing $\eta$ decreases the annular gap, which then distorted the free motion of flow formation. Furthermore, a relook at this figure shows that as $\kappa \to \infty$ , the volumetric flow-rate is independent of $\eta ,{\xi }_2^{*}$ and $\kappa$ . This is physically true as increase in $\kappa$ corresponds to a smaller EDL, which gradually diminishes the electroosmotic flow effect as $\kappa \to \infty$ .

Figure 6.

Volumetric flow-rate for different values of $\eta ,\kappa$ , and ${\xi }_2^{*}$ .

Figure 7.

Volumetric flow-rate for different values of $\kappa ,\mathrm{\Gamma }$ , and $s$ .

Figure 8.

Volumetric flow-rate for different values of $\kappa ,\mathrm{Gr}$ , and $r_q$ .

Figure 7 also discussed the volumetric flow-rate as functions of $\Gamma$ and $s$ in the microannulus. It is found that volumetric flow-rate rises with surge in Joule heating parameter. In addition, the maximum volumetric flow-rate is achieved for the case of electroosmotic flow. It is worthy to state that $\Gamma =0$ signifies the absence of electroosmotic effect while $\Gamma =1$ represents the presence of electroosmotic phenomenon with unity magnitude. This result shows that the volumetric flow-rate could have been wrongly computed should the EDL phenomenon at the surfaces of the microcylinders (which is often present for flows in microgeometries in the presence of electric field) had not been considered.

Figure 8 portrays the influence of $\mathrm{Gr}$ and $r_q$ “on the volumetric flow-rate. This figure demonstrated that volumetric flow rate decreases with increase in $\mathrm{Gr}$ and $r_q$ for $\kappa <10$ while a uniform flow rate is found for higher values of $\kappa$ .

Figure 9 shows the effect of unequal wall zeta-potentials, EDL size and ratio of radiuses on skin-frictions at the outer surface of inner cylinder. This figure discloses that the skin-friction for the symmetric wall zeta potential $({\xi }_2^{*}=1.0)$ are significantly different from those of asymmetric wall zeta potential, regardless of the annular gap or EDL size. Also, the skin-frictions at this surface is seen to rise with growth in radius ratio and EDL size. This is because increase radius ratio results to a decrease in annular gap, which increases the shear stress at the surfaces of the cylinders therefore increasing skin-friction.

Figure 9.

Skin-friction at the wall $R=0$ for different values of ${\xi }_2^{*},\kappa ,$ and $\eta$ .

Figure 10 depicts the combined impact of slip parameter, magnitude of electroosmotic force and radius ratio on skin-friction at the outer surface of inner cylinder. It is observed from this figure that the role of electroosmotic force is to reduce skin-friction at this surface regardless of the flow regime. Also, the slip flow (which is a closed to real life physics of flow in microannulus) is found to further has a lower skin-friction compared to the no-slip case. This outcome clearly demonstrated that the skin-friction at this surface would have been overestimated, if the surface slip-parameter was not put into consideration.

Figure 10.

Skin-friction at the wall $R=0$ for different values of $\alpha ,\Gamma$ , and $\eta$ .

Figure 11 displays the role of $\mathrm{Gr}$ and $r_q$ on skin-friction at the wall $R=0$ . It is noticed from this figure that both the buoyancy parameter as well as ratio of heat fluxes (which defines the magnitude of asymmetric wall heat fluxes) respond sparingly to skin-friction. It is also found that the skin-friction is a decreasing function $\mathrm{Gr}$ regardless the value of $r_q$ .

Figure 11.

Skin-friction at the wall $R=0$ for different values of $\mathrm{Gr},r_q$ , and $\eta$ .

It is noteworthy to mention that in the absence of electroosmotic force and joule heating effect, this work corresponds to the work of Avci and Aydin ³³ when the radius ratio $\eta$ approaches 1. Also, the finding contained here reduces to the work of Oni and Jha ¹³ once the Joule heating effect is relaxed and the geometry is transformed from microannulus to microchannel.

Conclusion

Analytical solution for electroosmotic NCF in a vertical microannulus with asymmetric zeta-potential and Joule heating is presented in this section. The equation of motion, electric potential, temperature distributions are derived and solved exactly. The influences of relevant parameters are presented using line graphs generated by MATLAB. The major conclusions are:

1. Fluid velocity, temperature, skin-friction and volumetric flow rate respond sharply to asymmetry in zeta-potential.

2. The volumetric flow rate as well as the skin-friction would have been erroneously calculated should the EDL as well as the slip parameters had not been put into consideration.

3. Fluid temperature and velocity profiles in the vertical microannulus increase with increase in radius ratio.

4. The role of electroosmotic force is to increase volumetric flow-rate and decrease skin-friction.