Rheological study of Hall current and slip boundary conditions on fluid–nanoparticle phases in a convergent channel

Purpose: the purpose of this theoretical study was to analyze the heat transfer in the fluid–particle suspension model under the effects of a porous medium, magnetic field, Hall effects, and slip boundary conditions in a convergent channel with the addition of electrokinetic phenomena. The Darcy–Brinkman (non-Darcy porous medium) model was used to assess the effects of the porous medium. Methodology: the rheological equations of both models were transformed into a dimensionless form to obtain the exact solutions of the fluid and particle phase velocities, pressure gradient, volumetric flow rate, stream function, temperature distribution, and heat-transfer rate. To obtain an exact solution to the models, the physical aspects of the parameters are discussed, analyzed, and reported through graphs, contour plots, and in tabular form. Findings: mixing in hafnium particles in a viscous fluid provide 1.2% more cooling compared to with a regular fluid. A reduction of the streamlines was observed with the contribution of the slip condition. The utilization of the Darcy parameters upgraded both the fluid flow and temperature profiles, while the heat-transfer rate decreased by up to 3.3% and 1.7% with the addition of a magnetic field and porous medium, respectively. Originality: the current study is an original work of the authors and has not been submitted nor published elsewhere.


Introduction
Colloidal suspensions of nanoparticles generated by carbides, metal oxides, etc. appearing in a regular uid (such as water, glycol, oil, and ethylene) can form a nanouid.Much research is devoted to the search for different types of nanoparticles that have good mechanical properties (i.e., thermal conductivity, a certain size of the particles, low particle momentum, and high mobility) to promote the heat-transfer analysis of the systems.Due to the distinguished properties of nanoparticles to increase heat transfer, nanouids are used in various scientic processes (such as electronics, nuclear reactors, and biomedicine).The nanoparticles are also effectively used in many chemical processes and modern biotechnology (such as, articial heart surgery, cancer therapy, drug delivery, sensor technology, disease diagnosis, and brain tumor therapy, due to their extensive thermophysical properties.Various researchers have performed work on different types of nanoparticles and base uids in many diverse shapes of geometries to highlight the applications of nanouids.For instance, Ellahi et al. 1 reported the applications of nanoparticles in a cooling process.They considered the spherical shape of aluminum nanoparticles in kerosene oil (as a base uid) with a maximum volume fraction of 4% to promote their applications in the cooling process.Bhatti et al. 2 used a suspension of cobalt oxide and graphene nanoparticles in a carrier uid to examine the applications of nanouids in solar energy.They employed a successive linearization method to obtain numerical solutions to the problem.Their ndings revealed that both nanoparticles could upgrade the heat-transfer rate, while skin friction displayed the opposite behavior.Hussain et al. 3 suspended gold nanoparticles in a couple of stress uids to discover their applications in gland and tumor remedies.They used a semi-analytical technique to obtain computational results for their model.Their results revealed that gold nanoparticles are the best option to kill the affected cells of tumors or glands due to their larger atomic number.Xu et al. 4 showed the brilliant advantages of ultrasmall-sized nanoparticles in biomedical engineering (tumor therapy) and showed that nanoparticles sized less than 7 nm were more effective in kidney and tumor treatment compared to larger-sized nanoparticles (100-200 nm).Aljohani et al. 5 chose a fractional derivative approach to show the applications of different types of nanoparticles in solar collectors to store solar energy.Some important results of nanouids in different congurations were also reported by Dharmaiah et al. [6][7][8] and Vedavathi et al. 9 Fluid and heat transfer in a porous medium are attracting increasing interest from researchers due to their wider applications in geothermal systems, food industries, the insulation of buildings, the design of nuclear reactors, the manufacturing of thermal isolators, oil production, solar power reactors, hot rolling, drying technologies, the control of pollutant spread in groundwater, and in compact heat exchangers, [10][11][12][13] etc. Various models have been proposed by different authors to simulate the porous medium effects, such as Darcian and non-Darcian models, and non-equilibrium models.In 1856, Darcy proposed the porous medium model for assessing uid ow through a porous medium experimentally by considering the linear relationship between the drop in pressure and the ow rate.The extensive body of literature suggests the importance of the porous medium in heat and ow analysis.For instance, Al Hajri et al. 14 analyzed the importance of the porous medium in the heat-transfer analysis of a Maxwell uid through a square conduit.The applications of a porous medium with heat transfer along a stretched cylinder were provided by Reddy et al. 15 Asghar et al. 16 used the Sisko uid model to discuss the heat transfer with the porous medium in a curved channel by using the implicit nite difference method.Ramesh 17,18 chose a non-uniform tilled channel model to discuss peristaltic ow under the inuence of a porous medium and presented an exact solution to the problem.
Various engineering applications of heat transfer with porous media have been observed where cooling or heating is a very important factor, such as combustion systems, the cooling of turbine blades, the cooling of electronic devices, chemical reactors, storage in thermal-transport systems, and composite fabrication.The mixtures of low or high thermal uids that appear in such applications can affect the output of these devices.In this situation, this issue can be resolved, and the performance of these devices can be increased (i.e., the heat transfer) by utilizing a porous medium with nanouids.Zhao et al. 22 conducted a study on the utilization of Mg gas inltration to produce MgB2 pellets, incorporating micro-sized B and nanosized powders.Zhang et al. 23 investigated the energy absorption of water jet penetration on the 2A12 aluminum alloy.Chen et al. 24 focused on the signicance of the rst hiddencharm pentaquark in relation to its strangeness.Also, the applications of nanouids with a porous medium in heattransfer analysis were reported by Nabway et al., 19 Kasaeian et al., 20 Mahdi et al., 21 Hussain et al., 25,26 Ge-JiLe et al. 27 Cai et al., 28 Chen, 29 and Du et al. 30 in recent works on the applications of nanouids in different applied elds.
The inuences of the slip boundary conditions on the heattransfer analysis of a uid-particle suspension of a magnetohydrodynamic (MHD) electro-osmotic ow of a rheological uid through a convergent shape geometry with a porous medium and the Hall effect have not been considered yet.Yet, we were motivated investigate these because of the potential applications of the slip boundary conditions in electro-osmotic ow to aid the design of reliable microuidic devices and to ensure the effective operation of such devices.Here, the problem of slip boundary conditions can have a signicant role in the study of heat and ow analysis to characterize the behavior of micro-and nano-uids through a microchannel.Currently, "no-slip" conditions are commonly used in the problem of microuidic ows, but a key limitation of the no-slip condition is that it may fail to solve micro-, nanoscale uid problems (depending on the roughness interface and the uid-solid interface interaction). 31Due to this reason, it is important to use the slip boundary conditions during the study of the ow of uids in a microchannel.It was Navier who rst presented the slip boundary condition, in which he reported a linear relationship between the wall shear rate and the velocity of the slip.Aer his development, different researchers presented different types of slip boundary conditions, 32 but the Navier slip boundary conditions are most commonly used due to their easiness and reliability.This article reports the solution of a uid-particle suspension of an MHD electro-osmotic ow with heat transfer analysis of a rheological uid under the consideration of the slip boundary conditions, porous medium, and Hall effects through a convergent geometry.In prior research, the crystallographic orientation, precipitation, mechanical characteristics, and phase transformation of a Ni-rich NiTi alloy were investigated by Wang et al. 47 The authors examined the effects of the deposition current for a dual-wire arc on the alloy's properties.In recent studies, Zhao 48 and Zhang et al. 49 also explored the applications of co-precipitated Ni/Mn shell-coated nano Cu-rich core structures and analyzed the microstructural properties of alkali-activated composite nanomaterials.Additionally, Lu et al. 50and Kong et al. 51 emphasized the importance of thermo-electric thermal uid ows and conducted microspectroscopy analysis under high pressure.

Mathematical analysis
Consider the time-independent ow of a rheological uid suspended by the addition of 40% hafnium solid spherical particles in as convergent channel as shown in Fig. 1.Here, V vf = [u vf (x,y),v vf (x,y),0] and V vp = [u vp (x,y),v vp (x,y),0] are the velocity vectors of the uid and particulates.
The following assumptions are taken to simplify the given ow problem.
1.An external electric eld E x is applied along the x-axis direction.
3. The ow is considered in the Cartesian coordinate systems in which x and y are chosen in the axial and normal-to-ow directions, respectively.
4. The uid is electrically conducted while the channel is non-electrically conducted.
5. A constant magnetic eld B 0 is applied normal to the ow direction.
6.The magnetic Reynolds number is very small, so the induced magnetic eld is negligible.
7. The Hall current is considered while the ion slip effects and buoyancy force are neglected.
8. The ow and particle interact as a continuum.9.The uid and particle velocities are irrotational.10.The Joule heating and thermal radiative heat ux effects are omitted.
11.The ow is symmetric along the center line of the channel y = 0 while y = h(x) and y = −h(x) are the upper and lower boundaries of the divergent channel, respectively.
12. The hafnium particles are considered to be in a spherical shape with equal size and uniformly distributed in the uid.
13.The uid and hafnium particles are coupled in terms of the drag force and heat transfer between them.
14.The top and lower walls of the channel maintain the temperatures T 0 and T 1 , respectively.

Physical model of the uid phase
The rheological equations for the uid phase in the vector form are dened as 33,34 vr The Darcy resistance R = (R X ,R Y ,0), the electric current J = (J X ,J Y ,0), and the Cauchy stress tensor T ij are expressed in the following form: Here D f (=6pmr) is called the Stoke drag coefficient, r is the radius of the hafnium particle, 4(0 < 4 < 1) is known as the porosity of porous medium, k*(k* > 0) is the permeability of the porous medium, s e is the electrical conductivity, u e is the cyclotron frequency of electrons, s e is the electron collision time, A is known as the kinematical tensor, E is the electric eld, and the total magnetic eld is B = B 0 + b, in which B 0 is the applied magnetic eld and b represents the induced magnetic eld, which is assumed to be negligible due to taking a low magnetic Reynolds number.Here, we chose E = (0,0,0) = 0, due to the absence of applied polarization voltage.So the total uniform magnetic eld with magnetic ux density is B = B 0 = (0,0,B 0 ).From the above assumption, eqn (5) can take the following form: 35-38 From the above eqn ( 7), we have where m 1 = u e s e z O( 1) is called the Hall current parameter.Solving eqn ( 8) and ( 9) simultaneously, we get In eqn (10) and (11), J x and J x are the components of the current vector J in the x-and y-directions, respectively while m is the Hall current parameter.

Physical model of the particle phase
The rheological equations for the particle phase in vector form are dened as 39,40 The vector form of the heat equation with viscous dissipation effects is dened as 41

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The components form of the uid and particle phases ow equations are dened as where the viscous dissipation term from the present ow problem is expressed as The boundary conditions are Introducing the dimensionless variable to convert the abovegoverning equations into dimensionless form In view of the above equation, the dimensionless uidparticle phase problem is expressed as (the bar has been removed) The boundary conditions are where In eqn (24), is known as the porous medium parameter, D f is the drag force, and Br is called the Brinkman number.

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The exact solution, i.e., the exact expressions for the uid, particle, and temperature, is expressed as The expression of the total volumetric ow is obtained from the following mathematical expression as Solving the above equation for dp/dx gives The expression of the stream function is calculated from the following mathematical relation u fv ¼ vj vy and given as

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The expression of temperature distribution can be obtained from eqn (17) with boundary conditions ( 21) As an important physical quantity, the heat-transfer rate is calculated by using the relation H tr ¼ À vT f;p vy y¼h and allows obtaining the following analytical expression: The constants G 1 ,G 2 ,G 3 ,G 4 ,G 5 , and R1,R2,.,R22are dened in the appendix.

Particular cases
1.It is interesting to note that for a m / 0, the problem is reduced to the single phase of an MHD electro-osmotic Newtonian uid ow with heat transfer analysis in a convergent channel under the effects of slip boundary conditions, a porous medium, and Hall current without hafnium particles.
2. When m 1 / 0, the problem is reduced to the multiphase electro-osmotic ow of a Newtonian uid with heat-transfer analysis in a convergent channel under the effects of slip boundary conditions, a porous medium, and a magnetic eld with a 40% suspension of Hafnium particles.
3. The nite value of D p / N corresponds to heat-transfer analysis of a biphase liquid ow of an MHD Newtonian uid under the action of the Hall current and slip boundary conditions in a convergent channel with a 40% suspension of hafnium particles.
4. For m 0 / 0, the problem is transformed into a simple multiphase pressure-driven ow of an MHD Newtonian uid with heat-transfer analysis under the impact of the Hall current and slip boundary conditions in a convergent channel with a 40% suspension of hafnium particles.
5. When Ha / 0, the problem is reduced to the thermal transport of an MHD Newtonian uid with a porous medium in a uid-particle suspension through a convergent channel under the inuence of the slip boundary conditions and Hall current.
6.For v s = t p = 0, the problem is reduced to thermal analysis of an MHD Newtonian uid in a uid-particle suspension in convergent channels without wall properties.
7. When D p / N and m 1 = 0, the results of Hussain et al. 33 and Ellahi et al. 34 can be recovered, i.e., the ow of a Newtonian nanouid through a convergent channel without a heat-transfer mechanism.

Results and discussion
In this section, we analyze the graphical behavior of the important physical quantities, namely the uid and particle velocities, stream function, temperature distribution, and heattransfer rate under the involved parameters of the study, namely the Debye length parameter , Hall parameter (m 1 = u e s e ), velocity slip parameter , Darcy parameter , particle suspension parameter (a m ), and thermal slip parameter Nanoscale Advances Paper solution to the considered problem is obtained with the help of mathematical soware and presented as closed-form expressions of the uid and particle velocity, stream function, total volumetric ow rate, pressure gradient, temperature distribution, and local heat-transfer rate.The range of the involved parameters for computation results is listed in Table 1.The author present seven gures.Fig. 2 shows the variation of the uid and particle velocities, while Fig. 3 reports the temperature distribution, Fig. 4-7 show the stream function behavior, and the graphs of the pressure rise are displayed in Fig. 8.The variation of heat-transfer rate is presented in Table 2.The effects of the magnetic eld parameter (M p ) on both the uid and particle velocities are displayed in Fig. 2(a).The dashed and solid lines are used in the graphs to differentiate the particle and uid velocity distribution, respectively.This gure shows that increasing the magnetic eld parameter led to decreases in both velocities in the divergent channel, and this behavior was due to resistive force i.e., Lorentz force (J × B).Basically, the greater the values of the magnetic eld parameter, the larger the magnetic eld produced relative to the viscosity of the uid, which enhances the Lorentz force acting on the uid particles and reduces the motion of the uid particles.The inuence of the Hall parameter on the velocity distribution is highlighted in Fig. 2(b).Here we observed that the Hall parameter and magnetic eld had a direct and inverse relation on the velocity proles, respectively, i.e., the magnetic eld parameter disturbed the uid ow, which caused a reduction in the uid velocity, while the Hall parameter supported the motion of the uid particle, and the velocities distribution increased against it.Since m 1 = u e s e , i.e., the cyclotron frequency of electrons and the electron collision time have a direct relationship with the Hall parameter, which means that when the Hall parameter is increased, then the cyclothron frequency of the electrons in the uid particles increases, which causes an increase in both velocities' distribution.This can also be expressed by another mathematical relation s e B 0 1 þ m 1 2 , i.e., the greater values of the Hall parameter diminish the electrical conductivity, which causes a reduction in the damping force, and as a result the velocity of both phases increases against the Hall parameter.Fig. 2(c) shows the increasing behavior of the velocity distribution against the Darcy parameter (D p ). Physically, this means that the Darcy parameter reduces the drag force in the uid particles, which causes an enhancement in the velocity prole via the Darcy parameter.We can also express that, for increasing values of the Darcy parameter, a more permeable porous medium will exist, which will provide less resistance to the uid particles as a result of the velocities proles enhancement.The effects of the Debye length parameter (m 0 ) on the velocities are examined in Fig. 2(d), in which the increasing trend of the velocities distribution is captured against this parameter.Since the Debye length parameter , i.e., the Debye length parameter is dened as "the height of the channel (a) over the Debye thickness (l d )", when the height of the channel increases, then the velocity distribution will increase, and as a result an electrical double layer arises.The effects of the coefficient of the volume fraction on velocity proles are shown in Fig. 2(e), in which the velocity proles decrease when increasing the coefficient of the volume fraction.From the computational results, it was observed that the velocity of the clean/simple uid was greater than the velocity of the multiphase uid (i.e., the uid with the suspension of hafnium particles).The physical reason for this is that when the solid tiny particles of hafnium are mixed in simply, then these particles generate internal friction in the resultant uid (multiphase uid), which retards the uid ow.Fig. 2(f) was constructed to analyze the impact of the velocity slip parameter on the uid-particle phase velocities.From this gure, it can be noted that the velocity of both phases showed a mixed behavior via the velocity slip parameter, i.e., the velocity proles increased and decreased in the interval x ˛[0,0.65]U[1.35,2.0]and x ˛[0.65,1.35],respectively.The impact of the magnetic eld parameter on the temperature distribution is illustrated in Fig. 3(a), which shows the inverse relationship between the temperature prole and the magnetic eld parameter.It could be observed that the hafnium particles lost their temperature when the magnetic force was applied, and we can also say that the reduction in temperature distribution happened due to departing effects.reason is that when hafnium particles are suspended in the uid, they absorb the heat, and as a result, the temperature decreases.Thus, we can say that the viscous uid with the suspension of hafnium particles plays a signicant role in the cooling process compared to in a regular viscous uid.Fig. 3(g) and (h) characterize the temperature behavior against the Brinkman number and thermal slip parameter . These gures show increasing trends via The physical reason for the increasing temperature against the Brinkman number is that when the Brinkman number increases, the viscous dissipation produces the condition of the heat-transfer mode, which enhances the temperature prole.The impacts of the magnetic parameter, Darcy parameter, Hall parameter, and velocity slip parameter on the stream function are captured in Fig. 4-7, respectively.From these gures, a signicant change can be observed in the stream function graphs.The magnitude of the stream is reduced with the contribution of the magnetic eld, which happened due to the Lorentz force produced against the applied magnetic eld (see Fig. 4).On the other hand, the Darcy parameter controls the magnitude of the stream function, i.e., the stream function obtains a greater magnitude with consideration of a porous medium as compared to a clear medium (see Fig. 5).A similar trend for the stream function was recorded against the Hall current parameter and velocity slip parameter (see Fig. 6 and 7).The variation of the heat-transfer rate under the effects of the parameters of the study is shown in Table 2. From this table, it can be noted that there was a 1.2% reduction in the heattransfer rate in the case of the suspension with hafnium particles, suggesting that the hafnium particles absorb heat when they are suspended in viscous uid.Thus, the hafnium particles provided 1.2% greater cooling when they were mixed in the

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viscous uid compared to a regular uid.The percentage reduction in heat transfer against the velocity parameter was 8.7%, which means that the velocity slip parameter controlled the heat transfer of the system i.e., the slip parameter provided 8.7% more cooling compared to the case without slip conditions.The heat-transfer rate also decreased by up to 3.3% and 1.7% with the addition of a magnetic eld and porous medium, respectively.These results suggest that these parameters also favor the cooling process of the system.The thermal slip parameter, Debye length parameter, Hall parameter, and Brinkman number thus support the heat-transfer rate, i.e., these parameters increase the heat transfer rate when increasing these parameters, and the percentages heat transfers were 2.4%, 1.3%, 2.6%, and 53.8%, respectively, see Table 2. Thus, the thermal slip parameter, Debye length parameter, Hall parameter, and Brinkman number favor the heating system.

Solution validation
A comparative analysis was performed with the existing results available in the literature to verify the current calculated results.
For this, we compared our results with the computational results of multiphase ow for a Newtonian uid with a suspension of hafnium nanoparticles through a convergent channel as presented by Ellahi et al. 34 Ellahi et al. 34 developed the mathematical model of the two-phase ow of a Newtonian uid through the diverse shape of the channel under the action of electro-osmotic and magnetic forces, and velocity slip conditions, and presented the exact solution through MATHEMATICA 12.0.The present results are in full agreement with the results of Ellahi et al. 34 for the limiting case m 1 / 0 and D p / N. The variations of the uid and particle phase velocities are illustrated in Fig. 9 under the effects of the velocity slip parameter.It could be observed that the impact of the velocity slip parameter in the current study (the red solid graph) was analogous to the results in the existing literature (black solid graph).

Concluding remarks
This study was conducted to analyze the heat-transfer rate of an MHD rheological uid with electro-osmotic ow with a uid and particle model through a convergent channel under the consideration of a porous medium, slip conditions, and Hall effects.The rheological equations were simplied and solved in the dimensionless form in MATHEMATICA soware and exact solutions for the uid-particle velocities, stream function, volumetric ow rate, pressure gradient, temperature distribution, and heat transfer rate were presented.Then to analyze the physical behavior of the important physical quantities, graphs and tables were constructed.The important ndings are listed below: The Hall current parameters increased the velocity and temperature of the uid-particle phases.
The velocity slip parameter diminished the temperature distribution while the thermal slip parameter enhanced the temperature.
Reduction of the streamlines was observed with the contribution of the slip condition.
The Darcy parameter upgraded both the uid and temperature proles.
The magnitude of the stream function increased against the Debye length parameter.
The viscous uid with a suspension of hafnium particles played a signicant role in the cooling process compared to the regular viscous uid.
The heat-transfer rate decreased by up to 3.3% and 1.7% with the addition of a magnetic eld and porous medium, respectively.
The thermal slip parameter, Debye length parameter, Hall parameter, and Brinkman number support the heat-transfer rate, i.e., these parameters increased the heat-transfer rate by increasing these parameters, and the percentage increases in terms of heat transfer were 2.4%, 1.3%, 2.6%, and 53.8%, respectively.
The hafnium particles provided for 1.2% more cooling when they were mixed in the viscous uid compared to the regular uid.
This study can be extended by utilizing the non-Newtonian uid models with heat-transfer analysis and slip boundary conditions.

Appendix
Nomenclature The stream function (m 2 s −1 ) 3 The permittivity of the uid F m −1 m 1 The Hall parameter l The wavelength (m) u hs The electro-osmotic velocity (m s −1 ) a The wave amplitude (m) t p The thermal slip parameter x The coordinate axis (m) h The wall of the channel Q f The volumetric ow rate of uid velocity b The amplitude ratio u vp The component of uid velocity (m s −1 ) T f,p The temperature of uid and particle phase (K) n 0 The bulk ionic concentration K The thermal conductivity (W mK −1 ) T 1 The temperature of the upper wall (K) u e The cyclotron frequency of electrons B 0 The strength of the magnetic eld (Wb m −2 ) B The contribution of viscous dissipation term p y The yield stress of the uid T ij The (i,j) th components of the stress tensor V vf The velocity vector of the uid phase (m s −1 ) g The gravitational force (m s −2 ) t The time (s) s e The electron collision time h b The Boltzmann constant E y The component of the electric eld in y direction (V m −1 ) p The pressure (N m −2 )

Subscripts f
The uid phase p The particle phase v f The uid velocity v p The particle velocity

Fig. 1
Fig. 1 Physical sketch of the flow problem.

Fig. 3 (
b) depicts the inuence of the Hall parameter (m 1 = u e s e ) on the temperature prole, and we could observe that this parameter enhanced the temperature distribution, and the maximum temperature was noted to be in the center of the channel.The variations of temperature against the Darcy parameter Fig.3(c) and (d), respectively.Both gures indicate the direction variation of the temperature proles with these parameters.Fig. 3(e) and (f) exhibit the impact of the coefficient of the volume fraction (a m ) and velocity slip parameter v s ¼ l a on the temperature of the uids and particles, respectively.Both gures show the decreasing behavior of the temperature distribution via the coefficient of the volume fraction and velocity slip parameter.The physical

Fig. 8
illustrates an analysis of the characteristics of the pumping phenomena.It could be observed that the pressure rise decreased in the region 0 < Q & 0 < DP (i.e., the retrograde pumping region) and increased in the region Q > 0 & DP > 0 (i.e., the co-pumping region) when increasing the coefficient of the volume fraction and the Darcy parameter (see Fig.8(a)).The pressure rise also increased via the magnetic eld parameter in the retrograde pumping region (Q > 0 & DP > 0) and decreased via the magnetic eld parameter in the co-pumping region (0 < Q & 0 < DP), see Fig.8(b).Fig.8(c) examines the behavior of the pressure rise against the velocity slip parameter for m 0 = 0 and m 0 s 0 and it could be noted that the pressure rise showed a similar trend against the velocity slip parameter and Debye length parameter, as can be seen in Fig.8(a).Considering the characteristics of the pressure rising versus the coefficient of the volume fraction for the two different cases, i.e., when the magnetic eld and slip conditions are ignored, and when the magnetic eld and slip conditions are included, we see that the pressure rise decreased in the region 0 < Q & 0 < DP (i.e., the retrograde pumping region) and increased in the region Q > 0 & DP > 0 (i.e. the co-pumping region) when increasing the coefficient of the volume fraction (see Fig.8(d)).
tr The heat transfer rate b The width of the channel (m) a m The coefficient of particle fraction (kg m 3 ) d The wave number D f The drag force m 0 The Debye length parameter (m) v s The velocity slip parameter Br Brinkman number y The coordinate axis (m) Q Total volumetric ow rate Q p The volumetric ow rate of particle velocity m s The viscosity kg m −1 s −1 v vp The component of particle velocity (m s−1) M p The magnetic eld parameter F The electro-osmotic potential function (V) T 0 The temperature of the lower wall (K) l The length of the channel (m) r f The density of the uid (kg m −3 ) s e The electrical conductivity (s m −1 ) V The dell operator D p The Darcy parameter 4 The porosity of the porous medium k* The permeability of the porous medium (m 2 ) V vp The velocity vector of the particle phase (m s −1 ) d dt The material time derivative T a The absolute temperature (K) E x The component of the electric eld in x direction (V m −1 ) k 1 The thermal slip constant c

Table 1
Range of the physical parameters used in the current computational analysis © 2023 The Author(s).Published by the Royal Society of Chemistry Nanoscale Adv., 2023, 5, 6473-6488 | 6479 Paper Nanoscale Advances