Parametric investigation of the Nernst–Planck model and Maxwell’s equations for a viscous fluid between squeezing plates

Abstract

The Poisson–Boltzmann equation is derived from the assumption of thermodynamic equilibrium where the ionic distribution is not affected by fluid flow. Although this is a reasonable assumption for steady electroosmotic flow through straight micro-channels, there are some important cases where convective transport of ions has nontrivial effects. In these cases, it is necessary to adopt the Nernst–Planck equation instead of the Poisson–Boltzmann equation to model the internal electric field. The modeled system of equations is transformed by similarity transformation to derive the equations of flow field, electric potential, electrokinetic force, entropy generation, and energy equation. The Parametric Continuation Method (PCM) is used to solve the system of ordinary differential equations. It is concluded that decrease in the mass diffusion decreases the anion distribution from lower to upper plate. The Batchelor number decreases the strength of magnetic field. Entropy generation and the Bejan number are maximum near the two plates because of the maximum disorderness due to plate movements and have minimum value in the fluid’s center. Also the Eckert number increases viscous heating, which causes the entropy production in the vicinity of the two plates to increase.

1 Introduction

In the analysis of electroosmotic flows, the internal electric potential is usually modeled by the Poisson–Boltzmann equation. The Poisson–Boltzmann equation is derived from the assumption of thermodynamic equilibrium where the ionic distribution is not affected by fluid flow. Although this is a reasonable assumption for steady electroosmotic flows through straight micro-channels, there are some important cases where convective transport of ions has nontrivial effects. In these cases, it is necessary to adopt the Nernst–Planck equation instead of the Poisson–Boltzmann equation to model the internal electric field. Hu [1] studied the steady electro-kinetic flow in squeezing channels in a T-shaped configuration. The nonlinear Poisson–Boltzmann equation for variable electro-kinetic radius and surface potential was numerically obtained. It was found that fluid flow can be controlled by applying a potential at each reservoir connected to the end of a channel. The inter-facial electro-kinetic effects on fluid flow through a micro-channel between parallel plates was studied by Mala [2]. Experiments were conducted to investigate the effects of electric double layer on flow characteristics with different potassium chloride concentrations in water and with different plate materials. The fluid flow through a micro-channel was also studied by Arulanumdum [3] through numerical methods. This investigation was bounded to the micro-channel flow in the fully developed region. The entry region effects were neglected and the charge density was assumed to be in the Boltzmann equilibrium distribution. Yang [4] investigated the entry flow effect induced by an applied electrical potential through micro-channel between parallel plates. The Nernst–Planck equation governing the ionic concentration distribution was numerically solved using a finite difference method.

Ferrohydrodynamics (FHD) deals with the mechanics of fluid motions influenced by strong forces of magnetic polarization. Magnetic fluids have several applications of heat transfer through ferrofluids. One such phenomenon is liquid-cooled speakers which involves small bulk quantities of the ferrofluid to conduct heat away from the speaker coils [1]. This innovation increases the amplifying power of the coil, and hence it leads to the loudspeaker producing high-fidelity sound. Another use of magnetic fluids is to bring the drugs to a targeted site in a human body, a magnetic field can pilot the path of a drop of ferrofluid in the human body [2]. Mutua [5] studied Stokes problem of a convective flow past a vertical infinite plate in a rotating system in the presence of variable magnetic field. They concluded that some or all of the parameters affect the fluid velocity and temperature. Consequently, their effect alters the rate of heat transfer and skin friction along the axes. Increase in magnetic parameter M and the Eckert number leads to an increase in the velocity profiles for both free convection cooling and heating at the plate. Seth [3] studied MHD flow and heat transfer along a porous flat plate with mass transfer and found that the fluid velocity component increased with an increasing value of time and Hall parameter, but decreased owing to an increasing value of transpiration parameter and magnetic field parameter. Victor [4] studied unsteady MHD free convection couette flow between two vertical permeable plates in the presence of thermal radiation using Galerkin’s finite element method. It was found that the radiation parameter and the Prandtl number have a greater effect on the temperature than on the velocity. On the other hand, the magnetic parameter and the Grashof number have no effect on the fluid temperature. Gunakala [6] investigated unsteady MHD couette flow between two infinite parallel porous plates in an inclined magnetic field with heat transfer. The lower plate was considered to be porous and stationary. He found out that an increase in the magnetic number led to a decrease in the velocity of the fluid. Similarly the effect of magnetic field and heat transfer have also been studied in detail by [7,8,9,10,11,12,13,14,15,16,17].

The existing information on the topic witnesses that the electric potential and magnetic field dependent entropy generation of a viscous fluid between squeezing plates has never been reported and is the very first study in the literature. In the following sections, the problem is formulated, analyzed, and discussed through graphs and tables.

2 Formulation of the problem

Consider the axisymmetric and fully developed pressure driven flow of an incompressible elecroviscous fluid between the gap of the squeezing plates as shown in Fig. 1. The parallel plates are separated by a distance $a(t) = l\sqrt{1 - \beta t}$, where l is the representative length equivalent to the plates separation at $t = 0$. Furthermore, we assume that fluid contains symmetric anions (+) and cations (−) with valencies $z_{+} = -z_{-} = z = 1$, equal diffusivities $D_{+} = D _{-} = D$ and that the bulk ion concentration of each ionic species is $n_{o}$. Electro-kinetic flows of fluid containing ionic species are described by the equation of continuity and the Navier–Stokes equations with an electrical body force term. These flow field equations are coupled with the Poisson equation relating the electrical potential to the charge distribution, and Nernst–Planck equations for conservation of each ion species. The induced magnetic field $(B_{x}, B_{y}, 0)$ in the fluid is generated by the applied magnetic field defined as

$$ H_{x} = \frac{\beta rM_{o}}{\mu _{2}(1 - \beta t)}, \qquad H_{y} = \frac{rN_{o}}{\mu _{2}(1 - \beta t)}, $$

where $M_{o}$ and $N_{o}$ are used to dimensionless $H_{x}$, $H_{y}$ and $\mu _{1}$, $\mu _{2}$ are the magnetic permeabilities of outside and inside media between the two plates respectively. Following experimental study, the above parameters of magnetic field are zero on the lower plate [18].

Figure 1

Geometry of the problem

The electro-kinematic flow equations have been nondimensionalized by introducing d, ϑ̅, $\frac{d}{\overline{\vartheta }}$, $n_{o}$, and $\frac{k_{b}T}{ze}$ as scaling variables for length, velocity, time, ion number density, and electrical potential, respectively, so the governing equations become as follows.

• Continuity equation [7]:

  $$ \vec{\nabla }.\vec{u} = 0; $$

  (1)

• Momentum equation with electro-kinetic effect [7,8,9,10,11]:

  $$ \frac{\partial \vec{u}}{\partial t} + (\vec{u}.\vec{\nabla }) \vec{u} = - \frac{1}{\rho } \vec{\nabla }\vec{p} + \nu \vec{\nabla }^{2} \vec{u} - \delta \bigl(n^{+} - n^{-} \bigr)\vec{\nabla }\vec{ U} + \vec{C}; $$

  (2)

• Magnetic induction equation [16, 17]

  $$ \frac{\partial \vec{C}}{\partial t} = \vec{\nabla } \times ( \vec{u} \times \breve{C} ) + \frac{1}{\sigma \mu _{e}} \vec{\nabla }^{2} \vec{C}; $$

  (3)

• Poisson equation:

  $$ \vec{\nabla }^{2}\vec{U} = \frac{-1}{2}K_{1}^{2} \bigl(n^{+} - n^{-}\bigr); $$

  (4)

• The Nernst–Plank equations [3,4,5,6, 18,19,20,21,22,23,24,25,26]:

  $$\begin{aligned}& \frac{\partial n^{+}}{\partial t} + \vec{\nabla }.\bigl(\vec{u} n^{+} \bigr) = \frac{ \nu }{S_{c}} \bigl(\vec{\nabla }^{2}n^{+} + \vec{\nabla }.\bigl(n^{+} \vec{\nabla }\vec{ U}\bigr) \bigr), \end{aligned}$$

  (5)
  $$\begin{aligned}& \frac{\partial n^{-}}{\partial t} + \vec{\nabla }.\bigl(\vec{u} n^{-} \bigr) = \frac{ \nu }{S_{c}} \bigl(\vec{\nabla } ^{2}n^{-} + \vec{\nabla }.\bigl(n^{-} \nabla U\bigr) \bigr); \end{aligned}$$

  (6)

• Equations of energy [12,13,14,15,16,17]:

  $$ \rho C_{p} \biggl(\frac{\partial \bar{T}}{\partial t} + \vec{u} . \vec{ \nabla } T \biggr) = \kappa \vec{\nabla }^{2} \vec{T} + Trac (T.L ), $$

  (7)

where $\aleph _{e}$ is the Reynolds number, $S_{ch}$ is the Schmidt number, D is ion diffusivity, z is valency, e is elementary charge, $k_{B}$ is the Boltzmann constant, T is absolute temperature, U is local total electrical potential induced by ions, $\epsilon _{o}$ is the permittivity of vacuum, ϵ is a dielectric constant, d is the distance between disks, $\varsigma _{o}$ is the surface electric potential at a reference position, $\kappa _{d}$ is an inverse Debye constant, B is fixed for a given fluid at a constant temperature, and $n^{+}$, $n^{-}$ are the number per unit volume of anions and cations, respectively.

3 Boundary conditions

The boundary conditions are chosen as follows:

$$ \begin{aligned} &u = v = 0, \qquad U = 0, \qquad n^{+} = c_{1}, \qquad n^{-} = 0, \\ &\qquad \beta _{0} = 0, \qquad b = 0, \qquad T = 0 \quad \text{at } y = 0, \\ & u = 0, \qquad v = -\frac{\beta l}{2\sqrt{1 - \beta t}}, \qquad U = \frac{x}{l \sqrt{1-\beta t}}, \qquad n^{+} = 0, \qquad n^{-} = \frac{n_{0}}{1 - \beta t}, \\ & \quad\quad \beta _{0} = \frac{\mu }{l \sqrt{1 - \beta t}}, \qquad b = \frac{x \mu }{l^{2} (1-\beta t)}, \qquad T = T_{h} \quad \text{at } y = h(t), \end{aligned} $$

(8)

where $n_{w}$ is the outward unit vector normal to the disks surface, $s_{cd}$ is the dimensionless surface charge density, and Ψ is the total electric potential.

The following similarity transformations [16, 17] are chosen for reducing the partial differential Eqs. (1)–(7) to a system of ordinary differential equations:

$$ \begin{aligned} & u = \frac{\beta x}{2(1 - \beta t)}f'( \eta ), \qquad v = -\frac{\beta l}{\sqrt{1 - \beta t}}f(\eta ), \qquad \beta _{0} = \frac{\mu }{l \sqrt{1 - \beta t}} M(\eta ), \\ & b = \frac{x \mu }{l^{2} (1-\beta t)} N(\eta ), \qquad n^{+} = \frac{n_{0}}{1 - \beta t}H(\eta ), \\ & n^{-} = \frac{n_{0}}{1 - \beta t}K(\eta ), \qquad U = \frac{x P(\eta )}{l \sqrt{1-\beta t}}, \qquad T = T_{h} \theta (\eta ), \quad \text{where } \eta = \frac{y}{l \sqrt{1 - \delta t}}. \end{aligned} $$

(9)

Equation (1) is identically satisfied and Eqs. (2)–(7) take the following form:

$$\begin{aligned}& f'''' = S \bigl(2f'' + \eta f''' + f'f'' - ff''' \bigr) \\& \hphantom{f'''' =}{} - B K_{1} ^{2} \sqrt{\delta } \bigl[\alpha _{2} \bigl(P'H - P'K \bigr) + \alpha _{1} \bigl(P'H + PH' - P'K- PK' \bigr) \bigr] \\& \hphantom{f'''' =}{} - 2M'f'' - Mf''' - f''N - 2f'N' - S^{2}Bt \bigl(Mf' + \eta M'f' \bigr) - BtS \bigl(ff'N + f^{\prime 2} M \bigr) \\& \hphantom{f'''' =}{} - S^{2}Bt \bigl(2fN + \eta fN' \bigr) + BtS \bigl(ff''M + ff' M' + ff'N + f^{2}N' \bigr), \end{aligned}$$

(10)

$$\begin{aligned}& N'' = S^{2}Bt \bigl( 2N + \eta N' \bigr) - BtS \bigl( Mf'' + f'M' + f'N + fN' \bigr), \end{aligned}$$

(11)

$$\begin{aligned}& M'' = S^{2}Bt \bigl( M + \eta M' \bigr) + BtS \bigl( fN + f'M \bigr), \end{aligned}$$

(12)

$$\begin{aligned}& P'' = - \frac{1}{2} K_{2}^{2} \sqrt{\delta } (H - K ), \end{aligned}$$

(13)

$$\begin{aligned}& H'' = SS_{c} \bigl( 2H + \eta H' - fH' \bigr) - \frac{1}{\sqrt{ \delta }} \biggl[ P'H' - \frac{1}{2} K_{2}^{2} \sqrt{\delta } \bigl(H ^{2} - KH \bigr) \biggr], \end{aligned}$$

(14)

$$\begin{aligned}& K'' = SS_{c} \bigl( 2K + \eta K' - fK' \bigr) - \frac{1}{\sqrt{ \delta }} \biggl[ P'K' - \frac{1}{2} K_{2}^{2} \sqrt{\delta } \bigl(KH - K^{2} \bigr) \biggr], \end{aligned}$$

(15)

$$\begin{aligned}& \theta '' = \mathit{Pr} S \bigl(\eta \theta ' - f \theta ' \bigr) + \mathit{Pr} \mathit{Ec} \bigl(4 \delta f^{\prime 2} + f^{\prime\prime 2} \bigr), \end{aligned}$$

(16)

and the boundary conditions are reduced to

$$ \begin{aligned} &f(0) = 0, \qquad f'(0) = 0, \qquad P(0) = 0, \qquad H(0) = c_{1}, \qquad K(0) = 0, \\ &N(0) = 0, \quad\quad M(0) = 0, \qquad \theta (0) = 0 \\ & f(1) = 0.5, \qquad f'(1) = 0, \qquad P(1) = 1, \qquad H(1) = 0, \qquad K(1) = 1, \\ &N(1) = 1, \quad\quad M(1) = 1, \qquad \theta (1) = 1, \end{aligned} $$

(17)

where $S = \frac{\beta l^{2}}{2\nu }$ is the squeeze number, $\mathit{Sc} = \frac{\mu }{\rho D}$ is the Schmidt number, $Bt = \sigma \mu _{0}\nu$ is the Batchelor number, $P_{r} = \frac{\mu C_{p}}{\kappa }$ is the Prandtl number, $\mathit{Ec} = \frac{1}{C_{p} T_{h}} [\frac{\beta x}{2 (1 - \beta t )} ]^{2}$ is the Eckert number, $\delta = \frac{l^{2} (1-\beta t )}{x^{2}}$, $B = \frac{ \rho k^{2} T^{2} \varepsilon _{0} \varepsilon }{2 z^{2} e^{2} \mu ^{2}}$, $K_{1}^{2} = \frac{2 z^{2} e^{2} n_{0}}{\varepsilon _{0} \varepsilon k _{b} T}$, $K_{2}^{2} = \frac{2 z^{2} e^{2} l^{2} n_{0}}{\varepsilon _{0} \varepsilon k_{b} T}$, $\alpha _{1} = \frac{2 u n_{0}}{\beta \rho }$, $\alpha _{1} = \frac{2 x u n_{0}}{\beta \rho }$, and $\lambda = \frac{l^{2}}{\mu \mu _{0}}$.

4 Entropy generation

The volumetric rate of local entropy generation, in the case of the existence of a magnetic field, can be expressed in the following form [27]:

$$ N_{s} = \frac{\kappa }{T^{2}_{\infty }} (\nabla T )^{2} + \frac{T: \nabla V}{T_{\infty }} + \frac{1}{T_{\infty }} \bigl[ (J-QC ) (E+V\times C ) \bigr], $$

(18)

which can be expressed in its dimensionless form by the following expression:

$$ \begin{aligned}[b] N_{g}& = \frac{\kappa }{T^{2}_{\infty } l^{2}} \biggl[ \biggl( \frac{ \partial T}{\partial x} \biggr)^{2} + \biggl(\frac{\partial T}{\partial y} \biggr)^{2} \biggr] + \frac{\mu }{T_{\infty }} \biggl[4 \biggl( \frac{ \partial \vec{u}}{\partial x} \biggr)^{2} + \biggl(\frac{\partial \vec{u}}{\partial y} + \frac{\partial \vec{v}}{\partial x} \biggr)^{2} \biggr] \\ &\quad{} + \frac{1}{T_{\infty }} \sigma \bigl[ u^{2} B^{2}_{0} - 2 uvbB _{0} + v^{2} b^{2} \bigr]. \end{aligned} $$

(19)

After transformation it can be written as

$$ \mathit{Ns} = \theta ^{\prime 2} + \varOmega \mathit{Pr} \mathit{Ec} \bigl(4 \delta f^{\prime 2} + f^{\prime\prime 2} \bigr) + \frac{\varOmega \mathit{Pr} \mathit{Ec} Bt \lambda }{S} \bigl(f^{\prime 2} M^{2} + f ^{2} N^{2} + 2 f f' M N \bigr), $$

(20)

where $T_{\infty }$ is a reference temperature, κ is the thermal conductivity, Q is the electric charge density, J is the electric current, E is the electric field, V is the velocity vector, the entropy generation rate is $\mathit{Ns} = \frac{N_{g}}{N_{g0}}$, the characteristic entropy generation rate is $N_{g0} = \frac{k_{nf} T _{H}}{T_{\infty }^{2} a^{2}}$, the dimensionless temperature difference is $\varOmega = \frac{T_{\infty }}{T_{H}}$, respectively. Therefore the local entropy generation in a dimensionless form can be written as

$$ \mathit{Ns} = N_{H} + N_{f} + N_{mf}, $$

(21)

where the local entropy generation due to heat transfer is $N_{H}$, the local entropy generation due to fluid friction is $N_{f}$, and the local entropy generation due to magnetic field is $N_{mf}$. In order to get the idea of relative importance of heat transfer effects and viscous effects, the Bejan number is defined:

$$ \mathit{Be} = \frac{N_{H}}{\mathit{Ns}} = \frac{\theta ^{\prime 2}}{\theta ^{\prime 2} + \varOmega \mathit{Pr} \mathit{Ec} (4 \delta f^{\prime 2} + f^{\prime\prime 2} ) + \frac{\varOmega \mathit{Pr} \mathit{Ec} Bt \lambda }{S} (f^{\prime 2} M^{2} + f^{2} N^{2} + 2 f f' M N )}. $$

(22)

In the absence of magnetic field ($G = H = 0$), Eqs. (14)–(16) were solved.

Physical quantities of interest are the skin friction coefficient and the Nusselt number which are defined as follows:

$$ C_{f} = \frac{\mu (\frac{\partial u}{\partial y} )_{y=h(t)}}{ \rho \upsilon ^{2}}, \qquad N_{u} = \frac{-lk (\frac{\partial T}{\partial y} )_{y=h(t)}}{kT _{h}}. $$

(23)

It terms of Eq. (13), we obtain

$$ \frac{2\nu (1-\beta t)^{\frac{3}{2}}}{\beta x} C_{f} = f''(1), \qquad (1-\beta t)^{\frac{1}{2}} N_{u} = -\theta '(1). $$

(24)

5 Analytic solution by parametric continuation method

Application of the parametric continuation method to nonlinear equations (10)–(16) with boundary conditions in Eq. (17) and optimal choice of continuation parameter is applied in this section. The following algorithm is presented as a sequence of steps to be followed for the application of this method.

• Canonical form of BVP as a first order ODE. To convert Eqs. (10)–(16) into a first order ODE, suppose the following:

  $$ \begin{aligned} &f = h_{1}, \qquad f' = h_{2}, \qquad f'' = h_{3}, \qquad f''' = h_{4}, \\ & N = h_{5}, \qquad N' = h_{6}, \qquad M = h_{7}, \qquad M' = h_{8}, \\ & P = h_{9}, \qquad P' = h_{10}, \qquad H = h_{11}, \qquad H' = h_{12}, \\ & K = h_{13}, \qquad K' = h_{14}, \qquad \theta = h_{15}, \qquad \theta ' = h_{16}. \end{aligned} $$

  (25)

  By putting these transformations in Eqs. (10)–(16), they become

  $$\begin{aligned}& \begin{aligned}[b] h'_{4} &= S (2h_{3} + \eta h_{4} + h_{2}h_{3} - h_{1}h_{4} ) \\ &\quad{}- B K_{1}^{2} \sqrt{ \delta } \bigl[\alpha _{2} (h_{10} h_{11} - h_{10} h_{13} ) + \alpha _{1} (h_{10} h_{11} + h_{9} h_{12} - h_{10} h_{13} - h_{9} h_{14} ) \bigr] \\ &\quad{}- 2 h_{8}h_{3} - h_{7}h _{4} - h_{3}h_{5} + 2h_{2}h_{6} \\ &\quad{} + S^{2}Bt (h_{2}h_{7} + \eta h_{2}h_{8} ) + BtS \bigl(h _{1}h_{2}h_{5} + h_{2}^{2}h_{7} \bigr) + S^{2}Bt (2h_{1}h_{5} + \eta h_{1}h_{6} ) \\ &\quad{} - BtS \bigl(h_{1}h_{3}h_{7} + h_{1}h_{2}h_{8} + h_{1}h_{2}h_{5} + h _{1}^{2}h_{6} \bigr), \end{aligned} \end{aligned}$$

  (26)
  $$\begin{aligned}& h'_{6} = S^{2} Bt (\eta h_{6} + 2h_{5} ) - BtS (h_{3}h _{7} + h_{2}h_{8} + h_{2}h_{5} + h_{1}h_{6} ), \end{aligned}$$

  (27)
  $$\begin{aligned}& h^{\prime }_{8} = S^{2} Bt (\eta h_{8} + h_{7} ) + Bt S (h _{1}h_{5} + h_{2}h_{7} ), \end{aligned}$$

  (28)
  $$\begin{aligned}& h^{\prime }_{10} = - \frac{1}{2} B K_{2}^{2} \sqrt{\delta } ( h_{11} - h_{13} ), \end{aligned}$$

  (29)
  $$\begin{aligned}& h'_{12} = \mathit{Sc} S (2h_{11} - \eta h_{12} - h_{1}h_{12} ) - \frac{1}{\sqrt{ \delta }} \biggl[h_{10}h_{12} - \frac{1}{2} B K_{2}^{2} \sqrt{\delta } \bigl(h_{11}^{2} - h_{11}h_{13} \bigr) \biggr], \end{aligned}$$

  (30)
  $$\begin{aligned}& h'_{14} = \mathit{Sc} S (2h_{13} - \eta h_{14} - h_{1}h_{14} ) - \frac{1}{\sqrt{ \delta }} \biggl[h_{10}h_{14} - \frac{1}{2} B K_{2}^{2} \sqrt{\delta } \bigl(h_{13}^{2} - h_{11}h_{13} \bigr) \biggr], \end{aligned}$$

  (31)
  $$\begin{aligned}& h'_{16} = \mathit{Pr} S (\eta h_{16} - h_{1}h_{16} ) + \mathit{Pr} \mathit{Ec} \bigl(4 \delta h_{2}^{2} + h_{3}^{2} \bigr) \end{aligned}$$

  (32)

  and the boundary conditions become

  $$ \begin{aligned} &h_{2}(0) = 0, \qquad h_{1}(0) = 0, \qquad h_{2}(1) = 0, \qquad h_{1}(1) = 1, \qquad h_{5}(0) = 0, \\& h_{5}(1) = 1,\quad\quad h_{7}(0) = 0, \qquad h_{7}(1) = 1, \qquad h_{9}(0) = 0, \qquad h_{9}(1) = 1, \\ &h_{11}(0) = c1, \qquad h_{11}(1) = 0, \quad\quad h_{13}(0) = 0, \qquad h_{13}(1) = 1, \\ & h_{15}(0) = 0, \qquad h_{15}(1) = 1. \end{aligned} $$

  (33)

• Introduction of a parameter p and imbedding of the obtained ODE in a p-parameter family. To obtain ODE in a p-parameter family, let us introduce the p-parameter in Eqs. (18)–(24), and so

  $$\begin{aligned}& \begin{aligned}[b] h'_{4} &= S \bigl[2h_{3} + \eta h_{4} + h_{2}h_{3} - h_{1} (h_{4} - 1 )p \bigr] \\ &\quad{}- B K_{1}^{2} \sqrt{\delta } \bigl[\alpha _{2} (h_{10} h_{11} - h_{10} h_{13} ) + \alpha _{1} (h_{10} h _{11} + h_{9} h_{12} - h_{10} h_{13} - h_{9} h_{14} ) \bigr] \\ &\quad{}- 2 h_{8}h_{3} - h_{7}h_{4} - h_{3}h_{5} + 2h_{2}h_{6} \\ &\quad{} + S^{2}Bt (h_{2}h_{7} + \eta h_{2}h_{8} ) + BtS \bigl(h _{1}h_{2}h_{5} + h_{2}^{2}h_{7} \bigr) + S^{2}Bt (2h_{1}h_{5} + \eta h_{1}h_{6} ) \\ &\quad{} - BtS \bigl(h_{1}h_{3}h_{7} + h_{1}h_{2}h_{8} + h_{1}h_{2}h_{5} + h _{1}^{2}h_{6} \bigr), \end{aligned} \end{aligned}$$

  (34)
  $$\begin{aligned}& h'_{6} = S^{2} Bt (\eta h_{6} + 2h_{5} ) - BtS \bigl[h_{3}h _{7} + h_{2}h_{8} + h_{2}h_{5} + h_{1} (h_{6} - 1 )p \bigr], \end{aligned}$$

  (35)
  $$\begin{aligned}& h^{\prime }_{8} = S^{2} Bt \bigl[\eta (h_{8} - 1 )p + h_{7} \bigr] + Bt S (h_{1}h_{5} + h_{2}h_{7} ), \end{aligned}$$

  (36)
  $$\begin{aligned}& h^{\prime }_{10} = - \frac{1}{2} B K_{2}^{2} \sqrt{\delta } \bigl[ h_{11} - h_{13} + h_{10} - (h_{10} - 1 )p \bigr], \end{aligned}$$

  (37)
  $$\begin{aligned}& \begin{aligned}[b] h'_{12} &= \mathit{Sc} S \bigl[2h_{11} - \eta h_{12} - h_{1} (h_{12} - 1 )p \bigr] \\&\quad{} - \frac{1}{\sqrt{\delta }} \biggl[h_{10}h_{12} - \frac{1}{2} B K_{2}^{2} \sqrt{\delta } \bigl(h_{11}^{2} - h_{11}h _{13} \bigr) \biggr],\end{aligned} \end{aligned}$$

  (38)
  $$\begin{aligned}& \begin{aligned}[b] h'_{14}& = \mathit{Sc} S \bigl[2h_{13} - \eta h_{14} - h_{1} (h_{14} - 1 )p \bigr] \\&\quad{} - \frac{1}{\sqrt{\delta }} \biggl[h_{10}h_{14} - \frac{1}{2} B K_{2}^{2} \sqrt{\delta } \bigl(h_{13}^{2} - h_{11}h _{13} \bigr) \biggr],\end{aligned} \end{aligned}$$

  (39)
  $$\begin{aligned}& h'_{16} = \mathit{Pr} S \bigl[\eta h_{16} - h_{1} (h_{16} -1 )p \bigr] + \mathit{Pr} \mathit{Ec} \bigl(4 \delta h_{2}^{2} + h_{3}^{2} \bigr). \end{aligned}$$

  (40)

• Differentiation by p leads to the following system with respect to sensitivities to the parametric p. Differentiate Eqs. (26)–(32) with respect to p

  $$ V'_{1} = A_{1}V_{1} + R_{1}, $$

  (41)

  where $A_{1}$ is a coefficient matrix, $R_{1}$ is a remainder, and $V_{1} = \frac{dh_{i}}{d\tau }$, $1\leq i \leq 16$.

• Application of the supposition principle and specifying the Cauchy problem for each component

  $$ V_{1} = aU + W_{1}. $$

  (42)

  Here U, $W_{1}$ are unknown vector functions. Solving the following two Cauchy problems for each component, we satisfy then automatically the original ODE

  $$ (aU + W_{1})^{\prime } = A_{1}(aU + W_{1}) + R_{1} $$

  (43)

  and left boundary conditions.

• Numerical solution of Cauchy problem. An implicit scheme is used for the solution of this problem as follows:

  $$\begin{aligned}& \frac{U^{i + 1} - U^{i}}{\triangle \eta } = A_{1}U^{i + 1}, \end{aligned}$$

  (44)
  $$\begin{aligned}& \frac{W^{i + 1} - W^{i}}{\triangle \eta } = A_{1}W^{i + 1} + R_{1}. \end{aligned}$$

  (45)

• Selection of corresponding blend coefficient. Since given boundary conditions are applied only for $h_{i}$, where $1 \leq i \leq 16$, when solving ODE for sensitivities, we need to apply $V_{2} = 0$, which in a matrix form looks as

  $$ J_{1} . V_{1} = 0 \quad \text{or} \quad J_{1} . (aU + W_{1}) = 0, $$

  (46)

  where $a = \frac{-J_{1} . W_{1}}{J_{1} . U}$.

6 Results and discussions

The mathematical formulation for the constitutive expressions of unsteady Newtonian fluid is employed to model the flow between the rectangular space of squeezing plates in the form of Eqs. (10)–(16) subject to the boundary conditions given in Eq. (17). These equations are solved and compared for numerical investigations through PCM and BVP4c. Parametric analysis is carried out for the dimensionless involved physical parameters. The effects of these parameters are shown in the form of Tables 1–2 and Figs. 2–13. The squeeze Reynolds number S is the ratio between the normal velocity of the upper plate and kinematic viscosity of the fluid. It is important to note that small or big values of S mean slow or rapid vertical velocity of the upper plate toward the lower plate. Positive values of S also mean that the upper plate is moving away from the lower plate, or the increase in the distance between the plates where negative values of S mean that the upper plate is moving toward the lower plate, or decrease of distance between the plates.

Figure 2

Profile of $f'(\eta )$ and $\theta (\eta )$ for different values of S and fixed values of $\mathit{Sc}=1.5$, $\delta =1.0$, $\alpha _{1}=1.2$, $\alpha _{2}=1.8$, $Bt = 0.3$, $B = 1.7$, $K_{1}=1.8$, $K_{2}=0.6$, $\mathit{Pr} =1.6$, $\mathit{Ec}=0.5$

Table 1 Comparison of the different values of $f''(\eta )$, $N'(\eta )$, $M'(\eta )$, $m'(\eta )$, $H'(\eta )$, $K'(\eta )$, and $\theta '(\eta )$ using the parametric continuation method approximation for the case $\mathit{Sc} = 1.5$, $\delta = 1.0$, $\alpha _{1} = 1.2$, $\alpha _{2} = 1.8$, $Bt = 0.3$, $B = 1.7$, $K_{1} = 1.8$, $K_{2} = 2.6$, $\mathit{Pr} =0.6$, $\mathit{Ec}=0.3$

Table 2 Comparison of the different values of $f''(\eta )$, $N'(\eta )$, $M'(\eta )$, $m'(\eta )$, $H'(\eta )$, $K'(\eta )$, and $\theta '(\eta )$ using the parametric continuation method approximation for the case $\mathit{Sc} = 1.5$, $\delta = 1.0$, $\alpha _{1} = 1.2$, $\alpha _{2} = 1.8$, $Bt = 0.3$, $B = 1.7$, $K_{1} = 1.8$, $K_{2} = 2.6$, $\mathit{Pr} =0.6$, $\mathit{Ec}=0.3$, $\varOmega =1.2$, $\lambda =0.3$

Figures 2 and 3 are made to depict the influence of squeeze Reynolds number S on the horizontal component of velocity field. It is clear from Fig. 2(a) that moving the upper plate toward the lower one pushes the fluid in horizontal direction. Increase in the normal velocity ($S = -0.1, -3.1, -10.1$) of the upper plate toward the lower plate is creating an opposing force to the vertical component, and so the fluid tends to move in horizontal direction. The effect of this normal velocity dominates the vertical velocity near the lower plate. However, as fluid crosses the central region, the vertical velocity overcomes the horizontal velocity and so $f'(\eta )$ starts decreasing. The effect of S is also investigated for temperature distribution as shown in Fig. 2(b). The temperature at upper plate at $\eta = 1$ is 1, and so by diffusion it is deceasing from layer to layer toward the lower plate. This decrease in temperature starts increasing after the central region due to decrease in fluid’s kinetic viscosity.

Figure 3

Profile of $N(\eta )$ and $M(\eta )$ for different values of S and fixed values of $\mathit{Sc} = 1.5$, $\delta =2.0$, $\alpha _{1}=1.2$, $\alpha _{2}=0.4$, $Bt = 0.3$, $B = 1.7$, $K_{1}=1.8$, $K_{2}=2.6$, $\mathit{Pr} =0.6$, $\mathit{Ec}=3.5$

The effect of S on magnetic field components $M(\eta )$ and $N(\eta )$ is shown in Fig. 3. An increase in the magnitude of squeeze Reynolds number S leads to decrease in kinetic viscosity of the fluid. This decrease in kinetic viscosity means an increase in the fluid’s density, which gives strength to the magnetic field from the lower to the upper plate as depicted in Fig. 3(a–b). Similarly this increase in density gives strength to the electric field because a strong magnetic field could be created due to strong electric field. Figure 4(a–b) clearly show that the cations distribution is getting stronger from the lower to the upper plate.

Figure 4

Profile of $H(\eta )$ and $K(\eta )$ for different values of S and fixed values of $\mathit{Sc}=0.5$, $\delta =2.0$, $\alpha _{1}=1.2$, $\alpha _{2}=0.4$, $Bt = 0.3$, $B = 1.7$, $K_{1}=1.8$, $K_{2}=2.6$, $\mathit{Pr} =0.6$, $\mathit{Ec}=3.5$

Figure 5 is made to investigate the effect of Batchelor number $B_{t}$ on magnetic field components. The Batchelor number $B_{t}$ for moderate values of the squeeze Reynolds number S is a measure of how easily the fluid slips from the magnetic field. When $B_{t}$ is very large, a small current will create a large induced magnetic field. When $B_{t}$ is very small, moderate-sized currents will only produce induced fields, which essentially have small perturbations on the applied field [28]. Clearly Fig. 5a shows that an increase in $B_{t}$ decreases the horizontal component of magnetic field. This is due to the fact that magnetic field effect is weaker near the lower plate which is getting stronger as fluid moves toward the upper plate.

Figure 5

Profile of $N(\eta )$ and $M(\eta )$ for different values of $B_{t}$ and fixed values of $\mathit{Sc}=1.5$, $\delta =4.0$, $\alpha _{1}=1.2$, $\alpha _{2}=1.8$, $B = 1.7$, $S = -1.1$, $K_{1}=1.8$, $K_{2}=2.6$, $\mathit{Pr} =1.6$, $\mathit{Ec}=0.5$

The effect of Schmidt number $S_{c}$ on $H(\eta )$ and $K(\eta )$ is depicted in Fig. 6. The Schmidt number is the ratio of the momentum to the mass diffusivities. An increase in $S_{c}$ means increase in momentum or decrease in the mass diffusion. It is noticed that decreasing the mass diffusion is decreasing the anion distribution from the lower to the upper plate. Also it can be seen that near the lower plate, the anion distribution increases with increase in the Schmidt number due to smaller mass diffusion with greater $S_{c}$. The influence of $P_{r}$ and $E_{c}$ on temperature distribution $\theta (\eta )$ is depicted in Fig. 7. The Prandtl number is the ratio of momentum and thermal diffusivities. It is noticed that increase in $P_{r}$ causes decrease in temperature because of decrease in thermal diffusion. The temperature is increasing after the central region because of heat diffusion from the upper plate at $\eta = 1$. Similarly an increase in $E_{c}$ means increase in the kinetic energy of fluid molecules which is maximum near the upper plate due to their motion.

Figure 6

Profile of $H(\eta )$ and $K(\eta )$ for different values of $S_{c}$ and fixed values of $Bt=0.3$, $\delta =4.0$, $\alpha _{1}=1.2$, $\alpha _{2}=1.8$, $B = 1.7$, $S = -1.1$, $K_{1}=1.8$, $K_{2}=2.6$, $\mathit{Pr} =1.6$, $\mathit{Ec}=0.5$

Figure 7

Profile of $\theta (\eta )$ for different values of $P_{r}$, $E_{c}$ and fixed values of $\mathit{Sc} = 1.5$, $Bt=0.3$, $\delta =4.0$, $\alpha _{1}=1.2$, $\alpha _{2}=1.8$, $B = 1.7$, $S = -1.1$, $K_{1}=1.8$, $K_{2}=2.6$

Figures 8–12 are made to study the entropy generation and the Bejan number of squeezing flow for the effect of different involved parameters. The effect of squeezing parameter is depicted in Fig. 9. It is investigated that both entropy and Bejan number are maximum near the two plates because of the maximum disorderness due to plate movements and have minimum value in the center at $\eta = 0.5$. The effects of Prandtl number and Eckert number on $\mathit{Ns}(\eta )$ and $\mathit{Be}(\eta )$ are almost same and are shown in Figs. 9–10. An increase in the Prandtl number means an increase in the momentum diffusivity that reaches its maximum value near the two movable plates. The pace of this diffusion is in the descending order from one layer to the other, which clearly shows that the rate of the entropy generation and the Bejan number are maximum near the adjacent plate due to maximum disorderness of molecules and are minimum in the middle of the liquid domain. An increase in Ω increases the rate of the entropy generation due to the thermal propagation (due to the friction of liquid and plates) of the moving plates. This thermal diffusion is higher near the plates, and so the entropy generation rate decreases gradually from one layer to another, and thus constitutes the minimum in the center of the liquid domain as shown in Fig. 11. Results of physical interest for the skin friction and the Nusselt number for different values of S, $B_{t}$, $P _{r}$, and $E_{c}$ are plotted in Figs. 12–13. Here it is shown that the skin friction decreases with an increase in S, while the heat transfer increases with an increase in S. The reason is that heat generation increases with increase in S due to collision of molecules. Also this phenomenon increases the fluid velocity due to which the skin friction decreases. No change is noticed for the Prandtl number on skin.

Figure 8

Profile of $N_{s}(\eta )$ and $B_{e}(\eta )$ for different values of S and fixed values of $\mathit{Sc}=0.5$, $\delta =2.0$, $\alpha _{1}=1.2$, $\alpha _{2}=0.4$, $Bt = 0.3$, $B = 1.7$, $K_{1}=1.8$, $K_{2}=2.6$, $\mathit{Pr} =1.6$, $\mathit{Ec}=0.5$, $\varOmega =1.2$, $\lambda =0.3$

Figure 9

Profile of $N_{s}(\eta )$ and $B_{e}(\eta )$ for different values of $P_{r}$, $E_{c}$ and fixed values of $\mathit{Sc} = 1.5$, $Bt=0.3$, $\delta =4.0$, $\alpha _{1}=1.2$, $\alpha _{2}=1.8$, $B = 1.7$, $S = -1.1$, $K_{1}=1.8$, $K_{2}=2.6$, $\varOmega =1.2$, $\lambda =0.3$, $\mathit{Ec}=0.5$

Figure 10

Profile of $N_{s}(\eta )$ and $B_{e}(\eta )$ for different values of $E_{c}$ and fixed values of $\mathit{Sc} = 1.5$, $Bt=0.3$, $\delta =4.0$, $\alpha _{1}=1.2$, $\alpha _{2}=1.8$, $B = 1.7$, $S = -1.1$, $K_{1}=1.8$, $K_{2}=2.6$, $\varOmega =1.2$, $\lambda =0.3$, $\mathit{Pr} =1.6$

Figure 11

Profile of $N_{s}(\eta )$ and $B_{e}(\eta )$ for different values of Ω and fixed values of $\mathit{Sc} = 1.5$, $Bt=0.3$, $\delta =4.0$, $\alpha _{1}=1.2$, $\alpha _{2}=1.8$, $B = 1.7$, $S = -1.1$, $K_{1}=1.8$, $K_{2}=2.6$, $\mathit{Ec}=0.5$, $\lambda =0.3$, $\mathit{Pr} =1.6$

Figure 12

(a) The effect of skin friction for different values of S. (b) The effect of skin friction for different values of Bt. With fixed values are $\mathit{Sc} = 1.5$, $Bt=0.3$, $\delta =1.0$, $\alpha _{1}=1.2$, $\alpha _{2}=1.8$, $B = 1.7$, $S = -1.1$, $K_{1}=1.8$, $K_{2}=2.6$, $\varOmega =1.2$, $\lambda =0.3$, $\mathit{Ec}=0.3$, $\mathit{Pr} = 0.6$

Figure 13

(a) The effect of Nusselt number for different values of Pr. (b) The effect of Nusselt number for different values of Ec. With fixed values are $\mathit{Sc} = 1.5$, $Bt=0.3$, $\delta =1.0$, $\alpha _{1}=1.2$, $\alpha _{2}=1.8$, $B = 1.7$, $S = -1.1$, $K_{1}=1.8$, $K_{2}=2.6$, $\varOmega =1.2$, $\lambda =0.3$, $\mathit{Ec}=0.3$, $\mathit{Pr} = 0.6$

7 Concluding remarks

The mathematical formulation for the constitutive expressions of unsteady Newtonian fluid is employed to model the flow between the rectangular space of porous and squeezing plates in the form of Eqs. (10)–(16) subject to the boundary conditions given in Eq. (17). Parametric continuation method is used to determine the solution of velocity components $f(\eta )$, $f'(\eta )$, magnetic field components $M(\eta )$, $N(\eta )$, electric field components $H(\eta )$, $K(\eta )$, electric potential $m(\eta )$, and temperature distribution $\theta (\eta )$. Parametric analysis is carried out for the involved dimensionless parameters.

Main upshots of this paper are presented below:

• It is concluded that an increase in the magnitude of squeeze Reynolds number S leads to increase in fluid’s density, which gives strength to the magnetic field from the lower plate to the upper plate.

• The Batchelor number decreases the strength of the horizontal component of magnetic field; this is due to the fact that magnetic field effect is weaker near the lower plate which is getting stronger as fluid moves towards the upper plate.

• Decreasing the mass diffusion decreases the anion distribution from the lower to the upper plate.

• Entropy generation and the Bejan number are maximum near the two plates because of the maximum disorderness due to plate movements and have minimum value in the fluid’s center.

• Furthermore, the variations of entropy generation with an increasing value of the Prandtl number. It is analyzed that for a fixed value of domain, entropy generation amplifies with increase in $P_{r}$. Heat is a disorganized form of energy, and increasing the Eckert number increases viscous heating, therefore entropy production increases near the plates.