Mathematical modelling of combined pressure driven and electrokinetic effect in a channel with induced magnetic field: An exact solution

Basant K. Jha; Michael O. Oni

doi:10.1016/j.jksus.2018.10.009

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Mathematical modelling of combined pressure driven and electrokinetic effect in a channel with induced magnetic field: An exact solution

Basant K. Jha, Michael O. Oni^⁎

Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

⁎Corresponding author. michaeloni29@yahoo.com (Michael O. Oni)

Received: 2017-10-16, Accepted: 2018-10-16, Published: 2018-10-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

This article presents exact solution for pressure driven flow formation of electrically conducting fluid in a parallel plate channel formed by two horizontal parallel plates with electrokinetic effects and induced magnetic field. Using the Poisson–Boltzmann, Navier-Stokes equations and induction equation, the governing electric potential, momentum, induced magnetic field and energy equations for the present article are presented and transformed to their corresponding dimensionless form using suitable parameters. The governing dimensionless equations are solved exactly and graphical representation are presented. During the cause of graphical illustration, it is found that the role of electrokinetic effects and Hartmann number is to decrease the electric potential, fluid velocity, induced magnetic field and fluid temperature. A special case is found and discussed when the value of Hartmann number equals the Debye-Hückel parameter. It is interesting to note that heat transfer is independent on governing parameters for large value of Debye-Hückel parameter.

Keywords

Electrokinetics

Induced magnetic field

Pressure driven

Exact solution

Channel

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Nomenclature

$b$: spacing between the parallel plates
$C_{0}$: concentration of ions in bulk fluid
$E_{x}$: electrostatic intensity
$F$: Faraday’s constant
$g$: acceleration due to gravity
$G_{2}$: dimensionless parameter
$H_{x}$: dimensional induced magnetic field
$\vec{H}$: vectorial induced magnetic field
$H$: dimensionless induced magnetic field
$J$: dimensionless induced current flux
$J_{θ}$: dimensionless induced current density
$M$: Hartmann number
$Nu$: Nusselt number

$P$: dimensionless pressure
$q$: heat flux
$q_{r}$: dimensionless heat flux
$R$: universal gas constant
$T$: dimensional temperature
$T_{ref}$: ref. temperature of the fluid
$u$: axial velocity
$u_{0}$: constant ref. velocity
$\bar{u}$: mean velocity
$U$: dimensionless axial velocity
$\vec{v}$: vectorial velocity profile
$z$: valence number of ions in the solution
$x, y$: axial and transverse coordinates respectively
$X$: dimensionless axial coordinate
$Y$: dimensionless transverse coordinate
$α$: dimensionless pressure gradient
$ε$: fluid permittivity
$ξ = ζ_{2} / ζ_{1}$: (dimensionless)
$ζ$: zeta $-$ potential (electrokinetic potential of the walls in the double layer)
$ϕ$: electrostatic potential
$Φ$: externally imposed electrostatic potential
$κ$: Debye-Huckel parameter
$λ_{D}$: Debye length
$ψ^{'}$: dimensional EDL potential
$ψ$: dimensionless EDL potential
$β$: thermal expansion coefficient
$k$: fluid thermal conductivity
$μ$: fluid dynamic viscosity
$μ_{e}$: magnetic permeability
$ρ$: fluid density
$σ$: electrical conductivity of the fluid
$τ$: skin-friction
$μ_{0}$: dynamic viscosity at $T = T_{0}$
$ν$: kinematic viscosity
$ρ_{f}$: fluid density
$ρ_{e}$: charge density
$θ$: dimensionless temperature

$0$: value at the wall $y = 0$
$1$: value at the wall $y = 1$
$e$: purely electrokinetic solution
$m$: bulk value
$p$: purely pressure driven flow
$s$: special case solution

1 Introduction

Electrokinetic phenomenon is significant in laboratory, industrial and engineering applications such as the removal of contaminants in soil and imposing electric ion on flow formation. This phenomenon involves the passing of low-voltage direct current electric field across the boundary on a fluid. Other applications can be found in medical field for cardiopulmonary resuscitation and development of battery cells.

When a liquid containing small amount of ions is brought in contact with a solid boundary, the like charges repel while the unlike charges attract. This situation leads to the formation of EDL (electric double layer) closed to the wall containing excess counter ions. Reuss (1809) and Probstein (1994) are the earliest scientists to discover the electrokinetic effects. Reuss (1809) in an experiment on porous clay found that particles dispersed in water migrate due to the constant application of electricity. Later, Helmholtz (1879) developed the EDL theory which relates the electric and flow parameters for electrokinetics. With increasing interest in understanding electrokinetic flows, many numerical investigations have been carried out. Patankar an Hu (1998) presented numerical simulation of microfluidic injection using electroosmotic forces through intersection of two channels. Ren and Li (Debye and Hückel, 1923) studied electrosomotic flows in microchannels with axially non-uniform zeta potentials and varying cross-sections. Yang and Li (1998) used the Debye-Hückel approximation (Ren and Li, 2001) to develop a numerical algorithm for elecrokinetically-driven Newtonian liquid flows. They concluded that for a liquid solution of low ionic concentration and a solid surface of high zeta potential the liquid flow in rectangular microchannels is significantly influenced by the presence of the EDL field and hence deviates from the flow characteristics described by classical fluid mechanics.

Many researches have been committed to study the combined effect of pressure and electrokinetic effects on flow formation in channel (Zade et al., 2007; Mukhopadhyay et al., 2009; Soong and Wang, 2003; Matin and Khan, 2016 ). Mukhopadhyay et al. (2009) examined the fully developed hydrodynamic and thermal transport in combined pressure and electrokinetically driven flow with asymmetric boundary condition. They established that both flow and heat transfer characteristics are significantly affected by the asymmetries in wall boundary conditions for both purely electroosmotic and combined pressure-driven and electroosmotic flow.

The study of electromagnetic induction has found many applications in design of electrical appliances such as inductors, transformers, alternators, electric motors and generators. The use of a magnetic field to control the motion of an electrically conducting fluid has been investigated under different conditions: with respect to geometry of the channel in which the fluid is flowing, the nature of the conducting fluid, source of the applied magnetic field, degree of ionization of the fluid and strength of the magnetic field. Rossow (1957) examined the flow of an incompressible boundary layer over a flat plate when the magnetic field is applied transversely to the direction of flow. He found that heat transfer is reduced by the application of magnetic field. Other important applications of electromagnetic and fluid interaction include, conversion of heat energy to electrical energy, ion propulsion studies, radio wave propagation in the ionosphere and controlled nuclear fusion, protection of internal surface of channels and nozzles from high temperature and high speed fluids (Ibrahim, 1967).

In the absence of induced magnetic field, (Jha and Oni, 2017; Jha et al., 2015) analysed the significance of Hartmann number on flow formation in different geometries. In the above sited articles, this assumption is only valid for very small magnetic Reynold number. To present a precise solution, the induced magnetic field should not be neglected. In view of this, Jha and Aina (2016) studied the role of induced magnetic field on MHD natural convection flow in a vertical microchannel formed by two electrically non-conducting infinite vertical parallel plates. Other relate articles are Jha and Sani (2013) and Gosh et al (2010). Other related works on combined pressure and electrokinetic effect in the presence of uniform magnetic field carried out both numerically and experimental studies can be seen in (Das and Mitra, 2012; Ganguly, 2015; Sarkar, 2014; Sarkar and Ganguly, 2018; Sarkar et al., 2017a, b, c ).

In all the mentioned articles above, to the best of authors’ findings, none of these articles have considered the induced magnetic field generated once the magnetic Reynolds number is high. This present article is aimed at investigating the combined role of induced magnetic field and electrokinetic in flow formation of an electrically conducting fluid in a conducting or non-conducting infinite parallel plates. The governing equations are developed and solve exactly, graphical and tabular represented are also presented to understand the role of basic governing parameters.

The novelty of the proposed work is to analyse the impact of induced magnetic field on flow formation and heat transfer in combined pressure and electrokinetically driven flow of conducting fluid in a horizontal channel. Furthermore, the role of conducting/ non-conducting channel walls on rate of heat transfer is extensively discussed. A similar problem is discussed by Mukhopadhyay et al. (2009) in the absence of induced magnetic field.

2 Mathematical formulation

Consider a laminar, fully developed forced convection flow of an electrically conducting, viscous, incompressible fluid in a conducting or non-conducting parallel plates channel with electrokinetic effects in the presence of transversely applied magnetic field. The $x -$ axis is taken parallel along the plates while the $y -$ axis is normal to it as depicted in Fig. 1. The following assumptions are made in order to obtain exact solution in this present article:

The flow is assumed to be driven by combined constant pressure gradient and external voltage gradient.
A uniform magnetic field strength $H_{0}$ is applied perpendicularly to the direction of flow
The plates are taken to be either conducting or non-conducting; Case I, both walls are non-conducting, Case II, the wall $y = 0$ is conducting while $y = b$ is non-conducting, Case III, the wall $y = 0$ is non conducting while $y = b$ is conducting.
The walls are negatively charged and the liquid contains an ideal solution of fully dissociated symmetric salt, the EDLs formed on the walls do not overlap, and the temperature variation over the channel cross section is negligible compared with the absolute temperature.
The charge distribution in the EDL follows Boltzmann distribution, hence the ion convection effects are negligible.
The wall potentials are considered low enough for Debye-Hückel linearization to be valid.
The external voltage is significantly higher than the flow induced voltage.
Except otherwise stated, all thermophysical parameters are assumed constant.

2.1

2.1 Electric potential distribution

Following above assumptions, the electrical potential distribution is obtained from Poison-Boltzmann equation (Mukhopadhyay et al., 2009) as:

(1)

\nabla^{2} ϕ = \frac{2 F z C_{0}}{ε} \sinh (\frac{zF ψ^{'}}{RT})

The potential $ϕ$ is due to combination of externally imposed field and EDL potential $ψ^{'}$ .

For fully developed flow, the external potential gradient is in the axial direction only, since the wall potentials are assumed low enough for Debye-Hückel linearization to be valid, Eq. (1) in dimensionless form becomes:

(2)

\frac{d^{2} ψ}{d Y^{2}} - κ^{2} ψ = 0

Subject to

(3)

ψ (0) = 1, ψ (1) = ξ

The details of this derivation can be found in (Mukhopadhyay et al., 2009; Matin and Khan, 2016)

3 Velocity and induced magnetic field

The momentum and induced magnetic field equations are obtained from the Navier-Stokes equations and induction equation respectively as:

(4)

ρ_{f} (\frac{\partial \vec{v}}{\partial t} + (\vec{v} ∙ \nabla) \vec{v}) = - \nabla p + μ \nabla^{2} \vec{v} + ρ_{e} E_{x} + μ_{e} (\nabla \times \vec{H}) \times \vec{H}

(5)

\frac{\partial \vec{H}}{\partial t} = \nabla \times (\vec{v} \times \vec{H}) + \frac{1}{σ μ_{e}} \nabla^{2} \vec{H}

Considering the steady fully developed flow ( $\vec{v} = u (y) \vec{i} + 0 \vec{j} + 0 \vec{k}$ ) and $(\vec{H} = H_{x} (y) \vec{i} + H_{0} \vec{j} + 0 \vec{k})$ in the presence of induced magnetic field, pressure gradient and electrokinetic effect, the governing momentum equation becomes:

(6)

μ \frac{d^{2} u}{d y^{2}} + μ_{e} H_{0} \frac{d H_{x}}{dy} - \frac{dp}{dx} + \frac{d^{2} ψ^{'}}{d y^{2}} ε E_{x} = 0

While the induced magnetic field equation is obtained from magnetic induction equation as (Gosh et al., 2010):

(7)

\frac{1}{μ_{e} σ} \frac{d^{2} H_{x}}{d y^{2}} + H_{0} \frac{du}{dy} = 0

Using suitable dimensionless parameters, Eqs. (6–7) become:

(8)

\frac{d^{2} U}{d Y^{2}} + M^{2} \frac{dH}{dY} - \frac{dP}{dX} + G_{2} ψ = 0

(9)

\frac{d^{2} H}{d Y^{2}} + \frac{dU}{dY} = 0

subject to the following dimensionless boundary conditions:

(10)

U (Y) = 0, A_{1} \frac{d H (Y)}{dY} + B_{1} H (Y) = 0, a t Y = 0

(11)

U (Y) = 0, A_{2} \frac{d H (Y)}{dY} + B_{2} H (Y) = 0, a t Y = 1

where the constant s

A_{1}, A_{2}, B_{1}

and

B_{2}

assume value zero or one to indicate the electrical conductivity or electrical non-conductivity of the walls (Jha and Aina, 2016; Jha and Sani, 2013; Gosh et al., 2010 ). For Case I (

B_{1} = B_{2} = 1

and

A_{1} = A_{2} = 0

) signifies that the both walls are electrically non-conducting, Case II (

A_{1} = B_{2} = 1

and

B_{1} = A_{2} = 0

) indicates that the lower wall situated at

Y = 0

is electrically conducting while the upper wall situated at

Y = 1

is electrically non-conducting, Case III (

A_{1} = B_{2} = 0

and

B_{1} = A_{2} = 1

) shows that the lower wall situated at

Y = 0

is electrically non-conducting while the upper wall situated at

Y = 1

is electrically conducting.

Since the flow is fully developed, it is usual to assume constant pressure gradient which is obtained from the fluid conservation:

(12)

\bar{U} = \underset{0}{\int^{1}} U (Y) d Y = 1

3.1

3.1 Energy equation

Due to the constant flux at the walls, the energy equation is obtained in dimensional form as (Mukhopadhyay et al., 2009)

(13)

u \frac{\partial T}{\partial x} = α \frac{\partial^{2} T}{\partial y^{2}} + \frac{S_{E}}{ρ C p}

Following (Mukhopadhyay et al., 2009), the energy equation in dimensionless form with the associated boundary conditions are given as:

(14)

\frac{d^{2} θ}{d Y^{2}} = \frac{U}{\bar{U}} (\frac{1 + q_{r}}{2} + S_{E}) - S_{E}

(15)

\frac{d θ (Y)}{dY} = - 1 a t Y = 0, a n d \frac{d θ (Y)}{dY} = q_{r} a t Y = 1

The following dimensionless parameters are used in this current article, and given as follow:

(16)

M^{2} = \frac{σ μ_{e}^{2} H_{0}^{2} b^{2}}{μ}, Y = \frac{y}{b}, X = \frac{x}{b}, U = \frac{u}{u_{0}}, θ = \frac{(T - T_{ref})}{q_{1} b / k}, P = \frac{p}{ρ u_{0}^{2}}, ξ = \frac{ζ_{2}}{ζ_{1}}, ψ = \frac{zF ψ^{,}}{RT}, κ = \frac{b}{λ_{D}}, λ_{D} = {[\frac{ε R T}{2 F^{2} Z^{2} C_{0}}]}^{1 / 2}, G_{2} = \frac{2 F^{2} Z^{2} C_{0} b^{2}}{ε R T}, H = \frac{H_{x}}{σ μ_{e} b H_{0} u_{0}}

It should be mentioned that in the absence of the magnetic field $(M = 0),$ all of the flow and heat transfer mathematical model reported above are consistent with those reported earlier by Mukhopadhyay et al. (2009).

3.2

3.2 Exact solution for the models

The electric potential, velocity profile, induced magnetic field, pressure gradient and temperature distributions of Eqs. (2, 8, 9, 12, 14) are solved with their corresponding boundary conditions respectively to obtain the following exact solutions:

(17)

ψ (Y) = \frac{[ξ \sinh (κ Y) + \sinh \{κ (1 - Y)\}]}{\sinh (κ)}

(18)

U (Y) = C_{3} \cosh (M Y) + C_{4} \sinh (M Y) - \frac{1}{M^{2}} \frac{dP}{dX} + \frac{G_{2} [ξ \sinh (κ Y) + \sinh \{κ (1 - Y)\}]}{(M^{2} - κ^{2}) \sinh (κ)}

(19)

H (Y) = \frac{Y}{M^{2}} \frac{dP}{dX} - \frac{\{C_{3} \cosh (M Y) + C_{4} \sinh (M Y)\}}{M} - \frac{G_{2} [ξ \cosh (κ Y) - \cosh \{κ (1 - Y)\}]}{κ (M^{2} - κ^{2}) \sinh (κ)} + C_{5} Y + C_{6}

(20)

\frac{dP}{dX} = \frac{D_{5} M + D_{2} \sinh (M) + D_{4} (\cosh (M) - 1)}{D_{1} (M - \sinh (M)) + D_{3} (1 - \cosh (M))}

(21)

θ (Y) = \frac{A}{\bar{U}} [\frac{\{C_{3} \cosh (M Y) + C_{4} \sinh (M Y)\}}{M^{2}} - \frac{Y^{2}}{2 M^{2}} \frac{dP}{dX} + \frac{G_{2} [ξ \sinh (κ Y) + \sinh \{κ (1 - Y)\}]}{κ^{2} (M^{2} - K^{2}) \sinh (κ)}] - \frac{S_{E} Y^{2}}{2} + C_{7} Y + C_{8}

A = (\frac{1 + q_{r}}{2} + S_{E})

The dimensionless skin-friction on both walls are obtained from Eq. (18) and given by:

(22)

τ_{0} = {\frac{dU}{dY}|}_{Y = 0} = C_{4} M + \frac{G_{2} κ [ξ - \cosh (κ)]}{(M^{2} - κ^{2}) \sinh (κ)}

(23)

τ_{1} = - {\frac{dU}{dY}|}_{Y = 1} = - M \{C_{3} \cosh (M) + C_{4} \sinh (M)\} - \frac{G_{2} κ [ξ \cosh (κ) - 1]}{(M^{2} - κ^{2}) \sinh (κ)}

Two important parameters for flow formation with induced magnetics field are induced current density $(J_{θ})$ and induced flux density $(J)$ which are respectively obtained as:

(24)

J_{θ} = - \frac{dH}{dY} = U (Y) - C_{5}

(25)

J = \underset{0}{\int^{1}} J_{θ} d Y = 1 - C_{5}

Another significant quantity in fluid and thermodynamics is the heat transfer coefficient. The heat transfer between the heated plats and the electrically conducting fluid is represented by Nusselt number and obtained as:

(26)

{Nu}_{0} = \frac{{\frac{d θ}{dY}|}_{Y = 0}}{θ_{m} - θ_{1}}, {Nu}_{1} = - \frac{{\frac{d θ}{dY}|}_{Y = 1}}{θ_{m} - θ_{2}}

where

θ_{m}

is the bulk temperature defined as:

(27)

θ_{m} = \frac{\underset{0}{\int^{1}} U (Y) θ (Y) d Y}{\underset{0}{\int^{1}} U (Y) d Y} = \sum_{i = 1}^{24} E_{i}

where

{C_{i}^{'} s, D}_{i}^{'} s a n d E_{i}^{'} s

are constants defined in appendix

4 Results and discussion

This article investigate the combined role of transversely applied magnetic field and electrokinetic effect on pressure driven flow in a parallel plates channel with induced magnetic field.

4.1

4.1 Special cases

From the solutions obtained from Eqs. (18) – (23), a discontinuity point is observed when the Hartmann number $(M)$ is equal to the Debye-Hückel parameter $(κ)$ . These quantities are possible to have the same magnitude in real life. Hence in such scenario, the exact solutions obtained in Eqs. (18) – (23) become invalid. To remove this discontinuity, we substitute $M = κ$ in the dimensionless Eqs. (8) – (15) and resolve. Although a degeneracy occur and is removed by using suitable complementary solution for the non-homogeneous part. On solving, the following expressions are obtained for velocity, pressure gradient and induced magnetic field:

(28)

U_{s} (Y) = C_{9} \cosh (κ Y) + C_{10} \sinh (κ Y) - \frac{1}{κ^{2}} \frac{dP}{dX} - \frac{G_{2} Y}{2 κ} [C_{1} \sinh (κ Y) + C_{2} \cosh (κ Y)]

(29)

{\frac{dP}{dX}}_{s} = \frac{(2 κ^{2} - 2 κ^{2} D_{13}) κ s i n h (κ) - G_{2} [C_{2} c o s h (κ) + C_{1} s i n h (κ)] (c o s h (κ) - 1) κ s i n h (κ)}{2 \{\sinh (κ) (\sinh (κ) - κ) - {(1 - \cosh (κ))}^{2}\}}

(30)

H_{s} (Y) = \frac{Y}{κ^{2}} \frac{dP}{dX} - \frac{1}{κ} [C_{9} \sinh (κ Y) + C_{10} \cosh (κ Y)] + \frac{G_{2}}{2 κ^{3}} [C_{1} \{κ^{2} \cosh (κ Y) - \sinh (κ Y)\} + C_{2} \{κ^{2} \sinh (κ Y) - \cosh (κ Y)\}] + C_{11} Y + C_{12}

From the velocity and induced magnetic field, all other parameters such as temperature, skin-friction, induced flux density and Nusselt number can be obtained when the value of Hartmann number is equal to the Debye-Hückel parameter.

For purely forced convection flow, i.e. $G_{2} = 0$ , from Eqs. (18–23) the velocity, pressure gradient and induced magnetic field reduce to:

(31)

U_{p} (Y) = \frac{1}{M^{2}} {\frac{dP}{dX}}_{p} \{\cosh (M Y) + [c o s e c h (M) - \coth (M)] \sinh (M Y) - 1\}

(32)

H_{p} (Y) = \frac{Y}{M^{2}} {\frac{dP}{dX}}_{p} - \frac{\{C_{3} \cosh (M Y) + C_{4} \sinh (M Y)\}}{M} + C_{15} Y + C_{16}

(33)

{\frac{dP}{dX}}_{p} = \frac{1}{D_{1} (\sinh (M) - M) - D_{3} (1 - \cosh (M))}

(34)

θ_{p} (Y) = \frac{A}{\bar{U}} \frac{1}{M^{2}} {\frac{dP}{dX}}_{p} [\frac{\{\cosh (M Y) + [c o s e c h (M) - \coth (M)] \sinh (M Y)\}}{M^{2}} - \frac{Y^{2}}{2}] - \frac{S_{E} Y^{2}}{2} + C_{17} Y + C_{18}

In similar manner, for purely electrokinetic flow, the pressure gradient becomes zero $(\frac{dP}{dX} = 0)$ and the fluid is driven by purely asymmetric external voltage gradient and hence the velocity and induced magnetic field becomes:

(35)

U_{e} (Y) = \frac{G_{2}}{(κ^{2} - M^{2})} \{\cosh (M Y) + [ξ c o s e c h (M) - \coth (M)] \sinh (M Y) - \frac{[ξ \sinh (κ Y) + \sinh \{κ (1 - Y)\}]}{\sinh (κ)}\}

(36)

H_{e} (Y) = \frac{G_{2}}{(M^{2} - κ^{2})} \{\cosh (M Y) + [ξ c o s e c h (M) - \coth (M)] \sinh (M Y) + \frac{[ξ \cosh (κ Y) - \cosh \{κ (1 - Y)\}]}{κ \sinh (κ)}\} + C_{19} Y + C_{20}

To have a clearer understanding on flow formation in the channel, graphs and tabular representations are carried out on fluid velocity, induced magnetic field, temperature distribution, skin-friction, induced current density and Nusselt number. Throughout the current research, the Debye-Hückel parameter is taken over the range of $0 \leq κ \leq 30$ to capture various physical situations ranging from small EDL to large EDL, $0 \leq M \leq 10$ for Hartmann number, the asymmetric external voltage gradient called zeta potential is assumed over $0 \leq ξ \leq 1$ . Three cases of induced magnetic field are considered; case I, when both walls are non-conducting and has caption (a) in figures, case II, the wall $Y = 0$ is electrically conducting and has caption (b) in figures while for case III the wall $Y = 1$ is electrically conducting with caption (c) in figures. It has been stated in literature that mathematical solution cannot be obtained for the case when both walls are simultaneously conducting; this is because of non-existence of solution for this case.

Fig. 2 presents the combined role of Debye-Hückel parameter $(κ)$ and dimensionless zeta potential $(ξ)$ on dimensionless electrostatic potential in the channel. It is found that the EDL potential is higher at the walls due to the supply of external voltage at the walls but decreases with increase in $κ$ . This is due to the fact that increasing $κ$ decreases the Debye-length (EDL size) which in turn decrease EDL potential. A careful look at this figure shows that the electrostatic potential is zero for large value of $κ$ around the centre of the channel; where EDL is minimum.

Electrostatic potential for different values of κ a n d ξ .

Fig. 3 depicts the effect of $κ$ and Hartmann number $(M)$ on dimensionless fluid velocity in the channel. It is clear from this figure that fluid motion increases with increase in $κ$ at the region close to the plates, while the reverse trend is noticed at the center of the channel. This can be explained by externally applied voltage gradient at the walls which increases the kinetic energy at the EDL and hence increase fluid velocity at the walls. From the same figure, similar trend is also followed for variation of Hartmann number on fluid velocity in the channel. It is interesting to note that two points of inflexion occur in channel which can be attributed to the constantly applied pressure gradient in the channel.

Fig. 3

Velocity profile for different values of

κ a n d M

ξ = 0.5

On the other hand, Fig. 4 shows the impact of zeta potential on dimensionless fluid velocity in the channel. It is obvious that fluid velocity is highest at the center of the channel for purely asymmetric wall voltage gradient $(ξ = 0)$ . Contrary to previous figure, a point of connection is observed which can be attributed to the variation of zeta potential whose location in the channel is strictly dependent on $κ$ . It is good to state that at this point, fluid velocity is independent on $ξ$ .

Fig. 4

Velocity profile for different values of

κ a n d ξ

M = 6.0

Figs. 5a, 5b and 5c portray dimensionless induced magnetic field when both walls are non-conducting, the wall $Y = 0$ only is conducting and the wall $Y = 1$ only is conducting respectively for different values of $κ$ at fixed value of $M$ and $ξ$ . Dimensionless induced magnetic field is observe to monotonically decrease with increase in electrokinetic parameter $(κ)$ in the channel for all cases for $0 \leq Y \leq 0.5$ while the reverse situation occurs for $Y > 0.5$ . This can be explained due to the fact that $κ$ is inversely proportion to EDL length and therefore decreases the induced magnetic field. For all cases, a point of intersection is noticed at the center of the channel and at this point, induced magnetic field is independent of electrokinetic parameter $(κ)$ . A critical look at Figs. 5b and 5c suggests that they are converses of each other. Therefore the magnitude of induced magnetic field is the same irrespective of the electrically conducting plates.

Induced magnetic field for different values of κ at M = 6.0 , ξ = 1.0 .

Figs. 6a and 6b present the combined role of Hartmann number $(M)$ and zeta potential $(ξ)$ of dimensionless induced magnetic field in the channel for the three cases. For the case when both walls are electrically non-conducting (case I), the generated magnetic field increases with increase in zeta potential and Hartmann number but decreases along the plates for $Y > 0.2$ .

Induced magnetic field for different values of. M a n d ξ = 1.0 a t κ = 5.0 .

From Fig. 6b on the other hand, induced magnetic field decreases along the plates monotonically from $Y = 0$ to $Y = 1$ but it is evidence to note that the role of magnetic field and zeta potential is to increase dimensionless induced magnetic field. This is attributed to the fact that increase in Hartmann number $(M)$ increases the magnetic field strength which in turn enhances induced magnetic field. For brevity, the behaviour of case II and III are similar and hence Fig. 6c is neglected.

The effects of Debye-Hückel parameter $(κ)$ and Hartmann number $(M)$ on dimensionless induced current density is illustrated in Figs. 7a and 7b for case I and II respectively. This figures show that the highest induced current density is obtained at the center of the channel when $κ$ is very small and the case when one of the wall is conducting. It is obvious from both figures that induced current density is enhanced at the walls with increase in $κ$ and $M$ .

Induced current density for different values of. κ a n d M a t ξ = 1.0 .

Fig. 8 gives a contour variation of temperature distributions in the channel as a function of $κ$ at fixed value of $M$ and $ξ$ . It is observed that dimensionless temperature distribution decreases with increase in $κ$ . This phenomena is expected as continuous increase of $κ$ continually decreases EDL length and hence reduces the temperature distribution in the channel. A clearer view of variation of temperature distribution with $M a n d κ$ is presented in Fig. 9. This figure shows that the maximum fluid temperature are achieved at the walls where the external voltage gradient is applied. In addition, fluid temperature decreases with increase in $M$ and the lowest temperature distribution is found at the center of the channel.

Temperature distribution θ - θ ( 1 ) for different values of κ at M = 10 , ξ = 1.0 , q r = 1.0 , S E = 0.5 .

Temperature distribution for different values of κ and M a t ξ = 1.0 , q r = 1.0 , S E = 0.5 .

Fig. 10 displays temperature distributions as a function of zeta potential and wall heat flux at fixed value of $M$ and $κ$ . It is understandable from this figure that the role of zeta potential is to increase temperature distribution at the region closer to the plate $Y = 0$ while the converse result is obtained for other region in the channel. Also, fluid emperature decreases with increase in heat flux variation at the region $Y = 0$ whereas heat flux parameter has little significant at other region for $q_{r} = - 0.5$ .

Temperature distribution for different values q r and ξ a t M = 2.0 , κ = 5.0 , S E = 0.5 .

One of the important parameter in the study of fluid formation in an enclosed surface is the skin friction. It is defined as the drag force to which fluid hits the walls of the parallel plates. Figs. 11 and 12 reveal variation of skin friction with Debye-Hückel parameter $(κ)$ , zeta potential $(ξ)$ and Hartmann number $(M)$ at the lower and upper plates respectively. In both figures, skin-friction is found to increase with increase in $M$ and $κ$ . This is due to the fact that increasing Hartmann number increases magnetic field strength which in turn increase collision between the plate surfaces and the fluid.

Skin-friction for different values M , κ and ξ a t Y = 0 .

Skin-friction for different values M , κ and ξ a t Y = 1 .

Fig. 11 further shows that the role of zeta potential at this surface is to decrease skin-friction while the reverse trend is observed at the wall $Y = 1$ .

Over the years, there have been intense interest in understanding the rate of heat transfer between fluid and surfaces of solid object. This is due to its industrial and domestic applications. Figs. 13, 14 and 15 represent the rate of heat transfer represented by Nusselt number as a function of governing parameters. Fig. 13 depicts the significance of Debye-Hückel parameter $(κ)$ for symmetric and asymmetric wall zeta potential $(ξ)$ at both surfaces of the channel in the presence of induced magnetic field. For small values of $κ$ , the Nusselt numbers at both walls are very sensitive to both $ξ$ and $κ$ . But as $κ \to \infty$ , The Nusselt numbers at the walls become independent of $ξ$ and $κ$ regardless of the surface considered. In addition, the Nusselt number is seen to first increase with $κ$ and then reaches its peak and the start decreasing for small interval of $κ$ but attained a stable value for high values of $κ$ regardless of the asymetric in heat flux. This behaviour is observed to happen for any value of zeta potential. As explained in (Mukhopadhyay et al., 2009; Seth and Singh, 2016), this scenario is due to variation of the difference between wall and bulk fluid temperatures.

Nusselt number for different values of κ and ξ at M = 5.0 , q r = 1.0 , S E = 2.0 .

Nusselt number for different values of q r a n d κ at M = 5.0 , ξ = 1.0 , S E = 2.0 .

Similar trend is obtained for variation of Nusselt number with heat flux parameters at the wall $Y = 0$ in Fig. 14. It has been established from literature (Mukhopadhyay et al., 2009; Maynes and Webb, 2004) that Nusselt number for electrokinetic flow has a non-monotone pattern with negative value due to the fact that for some configurations, where the forward and reverse flows are equivalent, leading to insignificant bulk flow, the smallness of the mean velocity leads to a very high mean temperature, as it is obvious from the definition of bulk mean temperature.

Fig. 15 on the other hand shows a monotone behaviour for variation of Nusselt number with heat flux parameter $(q_{r})$ in the presence of induced magnetic field at the wall $Y = 1$ . It is found that Nusselt number decreases with increase $κ$ for the case of symmetric wall heat flux and increases otherwise until a uniform heat transfer coefficient is attained for large value of $κ$ . This behaviour can be attributed to the fact that for large value of $κ$ , a small EDL is setup and the temperature difference between the fluid and the heated wall becomes uniform which in turn results to a zero Nusselt number regardless the value of $q_{r}$ .

Table 1 presents numerical computations for induced current flux $(J)$ and Nusselt number at the surfaces of the plates. It is obvious that induced current flux is independent on Hartmann number or electrokinetic effect when one of the wall is electrically conducting. In addition, the negative value of Nusselt number can be attributed to the fact that the fluid considered is a heat generating one, this implies the fluid generates more heat than the heat supplied at the walls and has negative heat transfer coefficient. Similar behaviour is found in (Mukhopadhyay et al., 2009). It is good to state that numerical comparison of (Mukhopadhyay et al., 2009) cannot be made with this present work. This is because the pressure gradient was prescribed in their work; which is not a valid assumption for fully developed flow formation. In the current article, the constant pressure gradient is calculated using the conservation of mass in order to present a more scientific and physical situation.

Table 1 Numerical computations for induced current flux

(J)

and Nusselt number at

ξ = 0.5

S_{E} = 2.0

q_{r} = 1.0

				$J$ (Induced current flux)
$M$	$κ$	${Nu}_{0}$	${Nu}_{1}$	$A_{1} = A_{2} = 0, B_{1} = B_{2} = 1$ (Case I)	$A_{1} = B_{2} = 1, A_{2} = B_{1} = 0$ (Case II)	$A_{1} = B_{2} = 0, A_{2} = B_{1} = 1$ (Case III)
0.5	2.0	0.0015	0.0011	−7.1054e−15	1.0000	1.0000
	5.0	−0.0038	−0.0078	0.0000	1.0000	1.0000
	10.0	−3.2196e−07	−3.2197e−07	−4.6185e−14	1.0000	1.0000
4.0	2.0	−0.0471	−0.0440	3.3307e−16	1.0000	1.0000
	5.0	2.8665e−04	2.8678e−04	−4.4409e−16	1.0000	1.0000
	10.0	7.5611e−06	7.5611e−06	−5.6399e−14	1.0000	1.0000
8.0	2.0	−0.0395	−0.0376	6.6613e−16	1.0000	1.0000
	5.0	−1.7924e−04	−1.7920e−04	−1.9096e−14	1.0000	1.0000
	10.0	1.7738e−07	1.7738e−07	−8.4377e−14	1.0000	1.0000

5 Conclusion

An exact solution is presented for pressure driven flow formation in a parallel channel plates with electrokinetic effects in the presence of induced magnetic field in this article. Using the Poisson–Boltzmann equation, Debye-Hückel linearization and Navier-Stokes equations, the electric potential, momentum, induced magnetic field and energy equations are derived and solved exactly. Graphical and tabular illustration are presented to see the effects of Debye-Hückel parameter, Hartmann number and zeta potential on fluid flow formation. The following are the major conclusions drawn:

The role of magnetic field and electrokinetics is to reduce electric potential, velocity, induced magnetic field and fluid temperature.
A degeneracy occur when the value of Debye-Hückel parameter equals the Hartmann number.
Skin-friction is enhanced with increase in Debye-Hückel parameter and Hartmann number
Nusselt number in the presence of electrokinetic has a non-monotone pattern

References

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Appendix

$C_{1} = \frac{ξ - \cosh (κ)}{\sinh (κ)}$ , $C_{2} = 1$ , $D_{1} = \frac{1}{M^{2}}$ , $D_{2} = \frac{G_{2}}{({κ^{2} - M}^{2})}$ , $D_{3} = D_{1} [c o s e c h (M) - \coth (M)]$ , $D_{4} = D_{2} [ξ c o s e c h (M) - \coth (M)]$ , $D_{5} = D_{2} \{\frac{(\cosh (κ) - ξ) (\cosh (κ) - 1)}{κ \sinh (κ)} - \frac{\sinh (κ)}{κ}\} - 1$ , $C_{3} = D_{1} \frac{dP}{dX} + D_{2}$ , $C_{4} = D_{3} \frac{dP}{dX} + D_{4}$ , $D_{6} = A_{1} (C_{3} - D_{1} \frac{dP}{dX} - D_{2}) + B_{1} \{\frac{C_{4}}{M} - \frac{D_{2} (ξ - \cosh (κ))}{κ \sinh (κ)}\}$ , $D_{7} = (A_{2} + B_{2})$ , $D_{8} = A_{2} (C_{3} \cosh (M) + C_{4} \sinh (M) - D_{1} \frac{dP}{dX} - D_{2} ξ) + B_{2} \{\frac{C_{3} \sinh (M)}{M} + \frac{C_{4} coshh (M)}{M} - D_{1} \frac{dP}{dX} - D_{2} (\frac{(ξ - \cosh (κ)) \cosh (κ)}{κ \sinh (κ)} + \frac{\sinh (κ)}{κ})\}$ , $C_{5} = \frac{D_{6} B_{2} - D_{8} B_{1}}{A_{1} B_{2} - B_{1} D_{7}}$ , $C_{6} = \frac{D_{8} A_{1} - D_{6} D_{7}}{A_{1} B_{2} - B_{1} D_{7}}$ , $D_{9} = (q_{r} - 1) - A \{\frac{C_{4} (1 + \cosh (M))}{M} + \frac{C_{4} \sinh (M)}{M} - \frac{D_{2} (ξ - \cosh (κ))}{κ \sinh (κ)} - D_{1} \frac{dP}{dX} - \frac{D_{2}}{κ} (\frac{(ξ - \cosh (κ))}{\sinh (κ)} + \sinh (κ))\} + S_{E}$ , $D_{10} = \frac{(1 - \cosh (κ))}{κ^{2} \sinh (κ)}$ , $D_{11} = \frac{G_{2} (C_{2} \cosh (κ) + C_{1} \sinh (κ))}{2 κ \sinh (κ)}$ , $D_{12} = [\frac{\sinh (κ)}{κ^{3}} - \frac{1}{κ^{2}}]$ , $D_{13} = \frac{G_{2} C_{2}}{2 κ} [\frac{\cosh (κ)}{κ^{2}} + \frac{1}{κ^{2}} - \frac{\sinh (κ)}{κ}] - \frac{G_{2} C_{1}}{2 κ} [\frac{\cosh (κ)}{κ} - \frac{\sinh (κ)}{κ^{2}}]$ , $D_{14} = \frac{\{\cosh (κ) - 1\}}{κ}$ , $D_{15} = B_{1} \{\frac{C_{10}}{κ} - \frac{G_{2} (κ^{2} C_{1} - C_{2})}{2 κ^{3}}\} - A_{1} \{\frac{G_{2} (κ^{2} C_{2} - C_{1})}{2 κ^{2}} - C_{9} - \frac{1}{κ^{2}} {\frac{dP}{dX}}_{s}\}$ , $D_{16} = A_{2} + B_{2}$ , $D_{17} = A_{2} [\frac{1}{κ^{2}} {\frac{dP}{dX}}_{s} + \{C_{9} \cosh (κ) + C_{10} \sinh (κ)\} - \frac{G_{2}}{2 κ^{2}} \{C_{1} (κ^{2} \sinh (κ) - \cosh (κ)) + C_{2} (κ^{2} \cosh (κ) - \sinh (κ))\}] + B_{2} [\frac{\{C_{9} \sinh (κ) + C_{10} \cosh (κ)\}}{κ} - \frac{1}{κ^{2}} {\frac{dP}{dX}}_{s} - \frac{G_{2}}{2 κ^{3}} \{C_{1} (κ^{2} coshh (κ) - \sinh (κ)) + C_{2} (κ^{2} \sinh (κ) - \cosh (κ))\}]$

$C_{7} = \frac{D_{9}}{2}$ , $C_{8} = 0$ , $C_{9} = \frac{1}{κ^{2}} {\frac{dP}{dX}}_{s}$ , $C_{10} = D_{10} {\frac{dP}{dX}}_{s} + D_{11}$ , $C_{11} = \frac{D_{17} B_{1} - B_{2} D_{15}}{B_{1} D_{16} - A_{1} B_{2}}$ , $C_{12} = \frac{D_{16} D_{15} - A_{1} D_{17}}{B_{1} D_{16} - A_{1} B_{2}}$ $E_{1} = \frac{A C_{3}^{2}}{M^{2}} (\frac{\sinh (2 M)}{4 M} + \frac{1}{2})$ , $E_{2} = \frac{A C_{3} C_{4}}{{4 M}^{3}} (\cosh (2 M) - 1)$ , $E_{3} = - \frac{A C_{3}}{M^{5}} \frac{dP}{dX} \sinh (M)$ , $E_{4} = \frac{A D_{2} C_{3}}{M^{2}} [\frac{(\cosh (κ) - ξ)}{\sinh (κ)} \{\frac{κ \cosh (κ) \cosh (M) - M s i n h (κ) \sinh (M)}{(κ^{2} - M^{2})} - \frac{κ}{(κ^{2} - M^{2})}\} - \{\frac{κ \sinh (κ) \cosh (M) - M c o s h (κ) \sinh (M)}{(κ^{2} - M^{2})}\}]$ , $E_{5} = \frac{A C_{3} C_{4}}{{4 M}^{3}} (\cosh (2 M) - 1)$ , $E_{6} = \frac{A C_{4}^{2}}{M^{2}} (\frac{\sinh (2 M)}{4 M} - \frac{1}{2})$ , $E_{7} = \frac{A C_{4}}{M^{5}} \frac{dP}{dX} (1 - \cosh (M))$ , $E_{8} = \frac{A D_{2} C_{4}}{M^{2}} [\frac{(\cosh (κ) - ξ)}{\sinh (κ)} \{\frac{κ \cosh (κ) \sinh (M) - M s i n h (κ) \cosh (M)}{(κ^{2} - M^{2})}\} - \{\frac{M - M c o s h (κ) \cosh (M) - - κ sinhh (κ) \sinh (M)}{(κ^{2} - M^{2})}\}]$ , $E_{9} = \frac{A C_{3}}{{2 M}^{2}} \frac{dP}{dX} \{\frac{2 M \cosh (M) - M^{2} \sinh (M) - 2 \sinh (M)}{M^{3}}\}$ , $E_{10} = \frac{A C_{4}}{{2 M}^{2}} \frac{dP}{dX} \{\frac{2 - 2 \cosh (M) - \cosh (M) + 2 M \sinh (M)}{M^{3}}\}$ , $E_{11} = \frac{1}{6 M^{2}} \frac{dP}{dX}$ , $E_{12} = \frac{A D_{2}}{2 M^{2} κ^{3}} \frac{dP}{dX} [\frac{(ξ - \cosh (κ))}{\sinh (κ)} \{2 \cosh (κ) + κ^{2} \cosh (κ) - 2 κ s i n h (κ) - 2\} + \{2 \sinh (κ) + κ^{2} \sinh (κ) - 2 κ c o s h (κ)\}]$ , $E_{13} = \frac{A D_{2} C_{3}}{κ^{2}} [\frac{(\cosh (κ) - ξ)}{\sinh (κ)} \{\frac{κ \cosh (κ) \cosh (M) - M s i n h (κ) \sinh (M)}{(κ^{2} - M^{2})} - \frac{κ}{(κ^{2} - M^{2})}\} + \{\frac{κ \sinh (κ) \cosh (M) - M c o s h (κ) \sinh (M)}{(M^{2} - κ^{2})}\}]$ , $E_{14} = \frac{A D_{2} C_{4}}{κ^{2}} [\frac{(\cosh (κ) - ξ)}{\sinh (κ)} \{\frac{κ \cosh (κ) \sinh (M) - M s i n h (κ) \cosh (M)}{(κ^{2} - M^{2})}\} + \{\frac{M - M c o s h (κ) \cosh (M) + κ \sinh (κ) \sinh (M)}{(M^{2} - κ^{2})}\}]$ , $E_{15} = \frac{A D_{2} D_{1}}{κ^{2}} \frac{dP}{dX} \{\frac{(ξ - \cosh (κ)) (\cosh (κ) - 1)}{κ \sinh (κ)} + \frac{\sinh (κ)}{κ}\}$ , $E_{16} = \frac{A D_{2}^{2}}{κ^{2}} [\frac{(ξ - \cosh (κ))}{\sinh (κ)} \{\frac{\sinh (2 κ)}{4 κ} - \frac{1}{2}\} + \frac{2 (ξ - \cosh (κ))}{4 κ \sinh (κ)} \{\cosh (2 κ) - 1\} + \{\frac{\sinh (2 κ)}{4 κ} - \frac{1}{2}\}]$ , $E_{17} = \frac{{S_{E} C}_{3}}{2} \{\frac{2 M \cosh (M) - 2 \sinh (M) - M^{2} \sinh (M)}{M^{3}}\}$ , $E_{18} = - \frac{{S_{E} C}_{4}}{2} \{\frac{2 - 2 \cosh (M) - M^{2} \cosh (M) + 2 M \sinh (M)}{M^{3}}\}$ , $E_{19} = \frac{S_{E}}{6 M^{2}} \frac{dP}{dX}$ , $E_{20} = \frac{S_{E} D_{2}}{2 κ^{3}} [\frac{(ξ - \cosh (κ))}{\sinh (κ)} \{2 \cosh (κ) + κ^{2} \cosh (κ) - 2 κ \sinh (κ) - 2\} + \{2 \sinh (κ) + κ^{2} \sinh (κ) - 2 κ \cosh (κ)\}]$ , $E_{21} = C_{3} C_{7} [\frac{\sinh (M)}{M} - \frac{\cosh (M)}{M^{2}} + \frac{1}{M^{2}}]$ , $E_{22} = C_{4} C_{7} [\frac{\cosh (M)}{M} - \frac{\sinh (M)}{M^{2}} + \frac{1}{M^{2}}]$ , $E_{23} = \frac{- C_{7}}{2 M^{2}} \frac{dP}{dX}$ , $E_{24} = C_{7} D_{2} [\frac{(\cosh (κ) - ξ)}{\sinh (κ)} \{\frac{\cosh (κ)}{κ} - \frac{\sinh (κ)}{κ^{2}}\} - \{\frac{\sinh (κ)}{κ} - \frac{\cosh (κ)}{κ^{2}} + \frac{1}{κ^{2}}\}]$ , $D_{18} = A_{1} (C_{13} - D_{1} \frac{dP}{dX}) + B_{1} \{\frac{C_{14}}{M}\}$ , $D_{20} = A_{2} (C_{13} \cosh (M) + C_{14} \sinh (M) - D_{1} \frac{dP}{dX} -) + B_{2} \{\frac{C_{13} \sinh (M)}{M} + \frac{C_{14} coshh (M)}{M} - D_{1} \frac{dP}{dX}\}$ , $C_{13} = \frac{1}{M^{2}} \frac{dP}{dX}$ , $C_{4} = D_{1} [c o s e c h (M) - \coth (M)] \frac{dP}{dX}$ , $C_{15} = \frac{D_{18} B_{2} - D_{7} B_{1}}{A_{1} B_{2} - B_{1} D_{7}}$ , $C_{16} = \frac{D_{20} A_{1} - D_{18} D_{7}}{A_{1} B_{2} - B_{1} D_{7}}$ , $D_{21} = (q_{r} - 1) - A \{\frac{C_{14} (1 + \cosh (M))}{M} + \frac{C_{14} \sinh (M)}{M} - D_{1} \frac{dP}{dX}\} + S_{E}$ , $C_{17} = \frac{D_{21}}{2}$ , $C_{18} = 0$

$D_{22} = B_{1} \{\frac{D_{2} [ξ c o s e c h (M) - \coth (M)]}{M} - \frac{D_{2} (ξ - \cosh (κ))}{κ \sinh (κ)}\}$ , $D_{23} = A_{2} (D_{2} \cosh (M) + D_{2} [ξ c o s e c h (M) - \coth (M)] \sinh (M) - D_{2} ξ) + B_{2} \{\frac{D_{2} \sinh (M)}{M} + \frac{D_{2} [ξ c o s e c h (M) - \coth (M)] coshh (M)}{M} - D_{2} (\frac{(ξ - \cosh (κ)) \cosh (κ)}{κ \sinh (κ)} + \frac{\sinh (κ)}{κ})\}$ ,

$C_{19} = \frac{D_{22} B_{2} - D_{23} B_{1}}{A_{1} B_{2} - B_{1} D_{7}}$ , $C_{20} = \frac{D_{23} A_{1} - D_{22} D_{7}}{A_{1} B_{2} - B_{1} D_{7}}$

Introduction

Mathematical formulation

Electric potential distribution

Velocity and induced magnetic field

Energy equation
Exact solution for the models

Results and discussion

Special cases

Conclusion

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[1] Reuss F.F., . Sur un nouvel effet de l'électricité galvanique. Mémoires de la Société Impériale des Naturalistes de Moscou. 1809;2:327-337.
[Google Scholar]

[2] Probstein R.F., . Physicochemical Hydrodynamics: an Introduction. New York: John Wiley & Sons Inc; 1994.

[3] Helmholtz H., . Studien über electrische Grenzschichten. Ann. Phys.. 1879;243:337-382.
[CrossRef] [Google Scholar]

[4] Patankar N.A., Hu H.H., . Numerical Simulation of Electroosmotic Flow. Anal. Chem.. 1998;70:1870-1881.
[Google Scholar]

[5] Ren L., Li D., . Electroosmotic flow in hetrogeneous microchannels. J. Colloid Interface Sci.. 2001;243:255-261.
[Google Scholar]

[6] Yang C., Li D., . Analysis of Electrokinetic Effects on the Liquid Flow in Rectangular Microchannels. J. Colloids Surfaces. 1998;143:339-353. 22
[Google Scholar]

[7] Debye P., Hückel E., . The theory of electrolytes. I. Lowering of freezing point and related phenomena. Physikalische Zeitschrift. 1923;24:185-206.
[Google Scholar]

[8] Zade A.Q., Manzari M.T., Hannani S.K., . An analytical solution for thermally fully developed combined pressure-electroosmotically driven flow in microchannels. Int. J. Heat Mass Transfer. 2007;50:1087-1096.
[Google Scholar]

[9] Mukhopadhyay A., Banerjee S., Gupta C., . Fully developed hydrodynamic and thermal transport in combined pressure and electrokinetically driven flow in a microchannel with asymmetric boundary conditions. Int. J. Heat Mass Transf.. 2009;52:2145-2154.
[Google Scholar]

[10] Soong C.Y., Wang S.H., . Theoretical analysis of electrokinetic flow and heat transfer in a microchannel under asymmetric boundary conditions. J. Colloid Interface Sci.. 2003;265:202-213.
[Google Scholar]

[11] Matin M.H., Khan W.A., . Electrokinetic effects on pressure driven flow of viscoelastic fluids in nanofluidic channels with Navier slip condition. J. Mol. Liq.. 2016;215:472-480.
[Google Scholar]

[12] Rossov V.J., . Flow of electrically conducting fluids over a flat plate in the presence of a transverse magnetic field. Report 1358 National Advesory Committee for Aeronautics 1957
[Google Scholar]

[13] Ibrahim, D.K., 1967. Effects of induced magnetic field on inviscid magnetohydrodynamics channel flow, Retrospective Thesis and Dissertations paper 3925.

[14] Jha B.K., Oni M.O., . Impact of mode of application of magnetic field on rate of heat transfer of rarefied gas flow in a microtube. Alexandria Engr. J. 2017
[CrossRef] [Google Scholar]

[15] Jha B.K., Aina B., Isa S., . Fully developed MHD natural convection flow in a vertical microchannel: an exact solution. J. King Saud Univ.-Sci.. 2015;27:253-259.
[Google Scholar]

[16] Jha B.K., Aina B., . Role of induced magnetic field on MHD natural convection flow in a vertical microchannel formed by two electrically non-conducting infinite vertical parallel plates. Alexandria Engr. J. 2016
[CrossRef] [Google Scholar]

[17] Jha B.K., Sani I., . Computational treatment of MHD of transient natural convection flow in a vertical channel due to symmetric heating in the presence of induced magnetic field. J. Phys. Soc. Jpn.. 2013;82:084401
[Google Scholar]

[18] Gosh S.K., Beg O.A., Zueco J., . Hydromagnetic free convection flow with induced magnetic field effects. Meccanica. 2010;14:175-185.
[Google Scholar]

[19] Das S., Mitra S.K., . Magnetohydrodynamics in narrow fluidic channels in presence of spatially non-uniform magnetic fields: framework for combined magnetohydrodynamic and magnetophoretic particle transport. Microfluid Nanofluid. 2012;13:799-807.
[CrossRef] [Google Scholar]

[20] Ganguly S., . Thermally developing combined electroosmotic and pressure-driven flow of nanofluids in a microchannel under the effect of magnetic field. Chem. Eng. Sci.. 2015;126:10-21.
[Google Scholar]

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Mathematical modelling of combined pressure driven and electrokinetic effect in a channel with induced magnetic field: An exact solution

Abstract

Keywords

Electrokinetics

Induced magnetic field

Pressure driven

Exact solution

Channel

Nomenclature

Subscripts

1 Introduction

2 Mathematical formulation

2.1 Electric potential distribution

3 Velocity and induced magnetic field

3.1 Energy equation

3.2 Exact solution for the models

4 Results and discussion

4.1 Special cases

5 Conclusion

References

Appendix

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