Thermo-Magneto-Solutal Squeezing Flow of Nanofluid between Two Parallel Disks Embedded in a Porous Medium: Effects of Nanoparticle Geometry, Slip and Temperature Jump Conditions

,e various applications of squeezing flow between two parallel surfaces such as those that are evident in manufacturing industries, polymer processing, compression, power transmission, lubricating system, food processing, and cooling amongst others call for further study on the effects of various parameters on the flow phenomena. In the present study, effects of nanoparticle geometry, slip, and temperature jump conditions on thermo-magneto-solutal squeezing flow of nanofluid between two parallel disks embedded in a porous medium are investigated, analyzed, and discussed. Similarity variables are used to transform the developed governing systems of nonlinear partial differential equations to systems of nonlinear ordinary differential equations. Homotopy perturbation method is used to solve the systems of the nonlinear ordinary differential equations. In order to verify the accuracy of the developed analytical solutions, the results of the homotopy perturbation method are compared with the results of the numerical method using the shooting method coupled with the fourth-order Runge–Kutta, and good agreements are established. ,rough the approximate analytical solutions, parametric studies are carried out to investigate the effects of nanoparticle size and shape, Brownian motion parameter, nanoparticle parameter, thermophoresis parameter, Hartmann number, Lewis number and pressure gradient parameters, slip, and temperature jump boundary conditions on thermo-solutal and hydromagnetic behavior of the nanofluid. ,is study will enhance and advance the understanding of nanofluidics such as energy conservation, friction reduction, and micromixing of biological samples.


Introduction
e study of squeezing flow of fluid between two parallel surfaces has received considerable and appreciable attentions in the last few decades due to its various industrial and biological applications. ese applications are evident in manufacturing industries, polymer processing, compression, power transmission, lubricating system, food processing, and cooling amongst others. In the past efforts to analyze fluid flow behavior between two parallel surfaces under the influences of various flow, fluid, and external properties, Mustafa et al. [1] presented an analysis of heat and mass transfer between parallel plates undergoing unsteady squeezing fluid flow while in the previous year, Hayat et al. [2] studied the squeezing flow behavior of a second grade fluid between parallel disk under the influence of presence of magnetic field. In an earlier work, Domairry and Aziz [3] applied the homotopy perturbation method to investigate the effects of suction and injection on magnetohydrodynamic squeezing flow of fluid between parallel disks. Also, Siddiqui et al. [4] investigated squeezing flow of viscous fluid between two parallel plates under the unsteady flow condition. In the same year, Rashidi et al. [5] examined the unsteady squeezing flow between parallel plates and presented different approximate analytical solutions. ree years later, Khan and Aziz [6] submitted a study on the flow of nanofluid between parallel plates due to natural convection. In a further work in the same year, the same authors [7] analyzed a double-di usive natural convective boundary-layer uid ow through porous media saturated with nano uid while in an earlier work, Kuznetsov and Nield [8] studied the ow of nano uid between two parallel plates considering the e ect of the natural convective boundary layer. Hashimi et al. [9] presented stimulating exploration of analytical solutions for the study of squeezing ow of nano uid. Most of the above reviews have been limited to the analysis of squeezing ow under no slip and no temperature jump boundary conditions. However, when ow system characteristic size is small or at a low ow pressure, the assumption of no slip boundary condition becomes insu cient and inadequate in predicting the ow behavior of the uid under such scenario. erefore, additional boundary conditions are required to adequately predict the low ow pressure or low system characteristics size. Moreover, it has been established that in many cases of uid and ow problems, such as polymeric liquids, thin lm problems, nano uids, rare ed uid problems, uids containing concentrated suspensions, and ow on multiple interfaces, slip condition prevails at the boundary of the ow [10][11][12][13][14][15][16][17][18][19][20][21]. Such slip boundary condition was rst initiated by Navier [22] upon which other researchers have built their analysis [10,11]. erefore, in recent years, the e ects of slip e ect on uid ow have been considered by many researchers [12][13][14][15][16][17][18][19][20][21] due to its signi cance to most practical uid ow situations. Furthermore, the e ect of stretching sheet wall problem adopting natural convective boundary conditions was investigated by Yao et al. [23], Kandasamy et al. [24], and Makinde and Aziz [25]. More works on parametric studies of the ow process can be found in [26][27][28][29][30][31][32][33][34]. Additionally, analyses of magnetohydrodynamic ow in the porous medium have been presented in various works [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] while various studies of magneto-nano uid ow have also been presented in the past works [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68]. e development of analytical solutions through some approximate analytical methods such as the Adomian decomposition method, homotopy analysis method, variation iteration method, and di erential transformation method for the ow process has been semianalytic in procedures and applications. In the earlier e orts to develop total analytic methods for solving nonlinear equations, di erent perturbation methods such as regular, singular, and homotopy perturbation methods have been applied. However, among these methods, the traditional perturbation methods such as methods of regular and singular perturbations require the inclusion of a small parameter in the governing di erential equations for the approximate analytical solution(s) to be realized for the governing differential equations. In the class of the newly developed approximate analytical methods for solving nonlinear equations, the homotopy perturbation method has been considered to be relatively simple with fewer requirements for mathematical rigour or skill. e homotopy perturbation method (HPM) gives highly accurate analytical solutions without the need for small perturbation parameters as required in traditional perturbations [69][70][71][72][73]. e results of HPM are completely reliable and physically realistic. Unlike the other approximate analytical methods, the HPM does not involve the search for a particular value that will satisfy the end boundary condition. Furthermore, to the best of our knowledge, there is no study where the thermal-di usion and di usion-thermal of magnetohydrodynamic squeezing unsteady ow of nano uid between two parallel disks embedded in a porous medium under the in uences of di erent nanoparticle geometries, slip and temperature jump conditions are analyzed. erefore, in this work, the homotopy perturbation method is used to study the e ects of nanoparticle geometry, slip, and temperature jump conditions on thermo-magneto-solutal squeezing ow of nano uid between two parallel disks embedded in a porous medium. e obtained analytical solutions are used to investigate the e ects of the Brownian motion parameter, thermophoresis parameter, Hartmann number, Lewis number and pressure gradient parameters, slip, and temperature jump boundary conditions on uid behavior of the nano uid.

Model Development and Analytical Solutions
Consider an axisymmetrical ow of nano uid through two parallel disks as shown in Figure 1. e upper disk is moving towards the stationary lower disks under a uniform magnetic eld strength applied perpendicular to disks as depicted in Figure 1. e uid conducts electrical energy as it ows unsteadily under the in uence of magnetic force eld. It is assumed that the uid structure is everywhere in thermodynamic equilibrium and the plate is maintained at constant temperature. e details of the governing equation and nondimensional parameters have been described by Hashimi et al. [9] and Das et al. [20] which can be introduced under the stated assumptions in the previous studies as zu zr where e appropriate initial and boundary conditions given as Using the following dimensionless quantities and similarity transformations, e dimensionless equations are given as And the dimensionless boundary conditions are given as where m in the above Hamilton Crosser's model in (4) is the shape factor, and its numerical values for different shapes are given in Table 1. e physical and thermal properties of the basefluid and nanoparticles are given in Tables 2 and 3, respectively.
It should be noted that the shape factor m � 3/ψ, where ψ is the sphericity (the ratio of the surface area of the sphere and the surface area of the real particles with equal volumes). Sphericity of sphere, platelet, cylinder, laminar, and brick are 1.000, 0.526, 0.625, 0.185, and 0.811, respectively. Hamilton-Crosser's model becomes Maxwell-Garnett's model, when the shape factor of the nanoparticle is m � 3. SWCNTs represent single-walled carbon nanotubes.

Method of Solution by the Homotopy
Perturbation Method e comparative advantages and the provision of acceptable analytical results with convenient convergence and stability [69][70][71][72][73] coupled with total analytic procedures of the homotopy perturbation method compel us to consider the method for solving the system of nonlinear differential equations in (5)-(7).

e Basic Idea of the Homotopy Perturbation Method.
In order to establish the basic idea behind the homotopy perturbation method, consider a system of nonlinear differential equations given as With the boundary conditions where A is a general differential operator, B is a boundary operator, f(r) is a known analytical function, and Γ is the boundary of the domain Ω. e operator A can be divided into two parts, L and N, where L is a linear operator and N is a nonlinear operator. Equation (10) can be therefore rewritten as follows: By the homotopy technique, a homotopy U(r, p) : Ω × [0, 1] → R can be constructed, which satisfies or In the above (13) and (14), p ∈ [0, 1] is an embedding parameter and u 0 is an initial approximation of equation of (10), which satisfies the boundary conditions. Also, from (13) and (14), one has or e changing process of p from zero to unity is just that of U(r, p) from u 0 (r) to u(r). is is referred to homotopy in topology. Using the embedding parameter p as a small parameter, the solutions of (13) and (14) can be assumed to be written as a power series in p as given in (17): It should be pointed out that of all the values of p are between 0 and 1, and p � 1 produces the best result. erefore, setting p � 1, results in the approximation solution of (9): e basic idea expressed above is a combination of the homotopy and perturbation method. Hence, the method is called the homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods (regular and singular perturbation methods). On the other hand, this technique can have full advantages of the traditional perturbation techniques. e series (18) is convergent for most cases.
(35) e boundary conditions for (33)-(35) are On solving (25), applying the boundary condition (28) gives Also, on solving (29), applying the boundary condition (32) yields And the solution of (33), by applying the boundary condition (36) is On solving (26) and applying the boundary condition (28), one arrives at 6 Modelling and Simulation in Engineering On solving (30), applying the boundary condition in (32) gives It can easily be shown that the solution of (34) after applying the boundary condition of (36) yields

Modelling and Simulation in Engineering
In the same way, f 2 (η), θ 2 (η), andϕ 2 (η) are (27), (31), and (35) are solved using the boundary conditions in (28), (32), and (36), respectively. Although the resulting solutions and the other subsequent solutions are too long to be shown in this paper, they are included in the simulated results shown graphically in the results and discussion section.

Modelling and Simulation in Engineering
Similarly, substituting (39) and (42) into the power series (24) yields

Results and Discussion
In order to verify the accuracy of the homotopy perturbation method, the developed nonlinear equations are also solved using the shooting method coupled with the Runge-Kutta method. Table 4 shows the comparison of the results of the numerical method (NM) and the homotopy perturbation method. From the table, it could be inferred that the results of the present work agree with the results of the numerical method using the shooting method with the Runge-Kutta method.
In order to further establish the accuracy of the solution of the HPM, the results of the present study (in the absence of the slip parameter, i.e., no-slip condition) are further compared with the results of the numerical method using the shooting method with the six-order Runge-Kutta method. e values of the local Nusselt number Nu L and local Sherwood number Sh L have been calculated for various values of Nb and Nt. An excellent agreement found between the two set of results is as shown in Table 5. erefore, the use of the homotopy perturbation method for the analysis of the double-diffusive models is justified.
In order to further establish the accuracy of the solution of HPM, the results of the present study (in the absence of the slip parameter) are further compared with the results of the numerical method using the shooting method with the Runge-Kutta method as presented in the literature [20]. e values of the local Nusselt number Nu L and local Sherwood number Sh L have been calculated for various values of Nb and Nt. An excellent agreement is found between the two set of results as shown in Table 5. Also, the results that agree with the previous studies using the homotopy analysis method have been presented in the literature [9]. erefore, the use of the homotopy perturbation method for the analysis of the thermal-diffusion and diffusion-thermal of magnetohydrodynamic squeezing unsteady flow of nanofluid between two parallel disks embedded in a porous medium under the influences of slip and temperature jump conditions is justified. Figure 2 shows the effects of nanoparticle volume fraction on the nanofluid dynamic viscosity ratio. It is inferred from the results that nanofluid dynamic viscosity ratio decreases as the nanoparticle volume fraction increases. e variation of nanoparticle volume fraction with thermal conductivities ratio of copper (II) oxide-water nanofluid with different particle shapes is shown in Figure 3. It is depicted in Figure 3 that the thermal conductivity of nanofluid varies linearly and increases with increase in nanoparticle volume fraction. Irrespective of the nanoparticle volume fraction, it is established that the nanofluid with spherical shape nanoparticles has the lowest thermal conductivity ratio while the nanofluid with lamina shape nanoparticles has the highest thermal conductivity ratio.
is also shows that the thermal conductivity ratio of the nanofluid is directly proportional to the nanoparticles shape factor. e suspensions of nanoparticles with a high shape factor or low sphericity in a basefluid gives higher thermal conductivity ratio to the fluid as a nanofluid than when nanoparticles with a low shape factor or high sphericity is suspended in the same fluid. e influence of nanoparticle volume fraction on the velocity of the nanofluid during the squeezing flow is depicted in Figure 4. It is established from the results that as the nanoparticle volume fraction increases, the velocity of the nanofluid increases between 0 ≤ η ≤ 0.42 (not accurately determined) because the nanoparticle volume fraction increases the skin friction coefficient of the nanofluid. However, the trend of this effect is reversed when η > 0.42.
Effect of the slip parameter on the velocity profile of the flow process is illustrated in Figure 5. It is shown in the figure that the radial velocity component increases with an increase in the slip parameter near the lower disk, that is, η > 0 and η > 0.5 (not accurately determined). A reverse case is recorded as the flow approaches the upper disk, that is, η > 0.5 and η < 1 (not accurately determined). e trend in the graph and the behavior of the fluid as depicted in the figure can be physically explained that as the slip parameter increases, there is a corresponding decrease in shear stress which consequently increases the radial velocity component near the stationary lower disk while a reverse trend occurs as the nanofluid moves and flow approaches the upper disk. Figure 6 shows the effect of the increasing Hartmann parameter (M) on the velocity profile of the nanofluid flow between the parallel disk. It is observed as presented in the figure that, at increasing values of M, the velocity decreases slightly near the lower disk and as the upper disk is approached the velocity increases slightly due to the increase in the boundary layer thickness caused by the Lorentz or magnetic force eld; that is, for the electrically conducting uid in the presence of magnetic eld, there is a Lorentz force which slows down the motion of the uid in the boundary layer region. Moreover, during the squeezing ow (when the disks are moving towards each other), the situation together with the Lorentz force creates adverse pressure gradient in the ow. Whenever such forces act over a long time, there might be a point separation and back ow occurs. However, when the disks are moving apart, the reason for the behavior is quite di erent as there exist a vacant space and the nano uid in that region goes with high velocity so as not to violate the mass conservation.
at is, since the mass ow rate is kept conservative, decrease in the uid velocity near the wall region will be compensated by increasing the uid velocity near the central region.     Table 3: Physical and thermal properties of nanoparticles [74,75].  Table 2: Physical and thermal properties of the base uid [74,75]. As squeeze parameter (S) increases which is demonstrated in Figure 7, the radial velocity component increases. e e ect is maximum at the lower disk and minimum at the upper disk. Figure 8 depicts the e ect of increasing the pressure term (C) on the velocity of the uid ow; it is shown that with, increasing C, a very slight increase in the velocity component is observed.
e e ect of the temperature jump parameter (c) on the temperature pro le is shown in Figure 9. It is observed that as the temperature jump parameter increases, temperature distribution increases towards the lower disk where it decreases towards the upper disk. In the absence of slip, that is, c 0, it is observed that temperature distribution equals to unity at the lower plate and zero at the upper plate. In uence of the thermophoresis parameter (Nt) is demonstrated in Figure 10 which depicts that, with increasing values of Nt, the temperature distribution increases and is maximum at the lower disk but falls rapidly towards the upper disk. e e ect of the squeeze parameter (S) on temperature distribution is displayed in Figure 11. e gure illustrates that, at increasing values S, the temperature distribution at the lower disk reduces while the   temperature distribution at the upper disk increases which can be physically explained as increase in S that leads to a corresponding decrease in kinematic viscosity and vice versa. It is obvious from Figure 12 that the increasing pressure term (C) has no signi cant e ect on temperature distribution. Figure 13 displays the in uence of nanoparticle volume fraction on the temperature pro le of the nano uid during the squeezing ow. e gure shows that the temperature increases as the nanoparticle volume fraction increases in the nano uid. e reason behind this behavior is that when the two disks move towards each other (squeezing ow) and the nanoparticle volume fraction increases, there is more collision between the nanoparticles and the particles with the boundary surface of the disk. Consequently, the resulting friction gives rise to increases in temperature within the nano uid near the boundary region. It should be pointed out that an opposite to the trend witnessed during the squeezing ow is established when the disks are moving apart. e e ect of the nanoparticle shape factor on the temperature pro le of the nano uid during the squeezing ow is shown in Figure 14.
e di erent shaped nanoparticles considered are sphere, brick, cylinder, and lamina. It is shown from the gure that the temperature increases as the nanoparticle shape factor increases. is is because thermal conductivity of the nano uid increases with increase in the nanoparticle shape factor. Regardless of the nanoparticle volume fraction, it is established that the nano uid with spherical shape nanoparticles has the lowest thermal conductivity ratio while the nano uid with lamina shape nanoparticles has the highest thermal conductivity ratio. e suspensions of nanoparticles with a high shape factor or low sphericity in a base uid gives higher thermal conductivity ratio to the uid as a nanouid than when nanoparticles with a low shape factor or high sphericity is suspended in the same uid. erefore, as the nanoparticle volume fraction increases due to the increased shape factor, there are increased interactions and collisions between the nanoparticles and the particles with the boundary surface of the disk, and as a consequence, more friction is created that gives rise to increase in temperature within the nano uid near the boundary region.
In uence of the Lewis number (Le) on the concentration pro le of the nano uid ow process is depicted in Figure 15. It is illustrated that increasing Le evokes a corresponding increase in concentration distribution at the region closer to the lower disk, though it decreases rapidly as it moves towards the upper disk. ermophoresis parameter (Nt) e ect on the concentration pro le of the ow of nano uid is observed in Figure 16, at increasing values of Nt the concentration pro le increases signi cantly but towards the upper disk there is a steady reduction. Figure 17 depicts the e ect of the Brownian motion parameter (Nb) on the concentration pro le of the nano uid ow. As observed and presented in the gure, it is established that increasing value of Nb concentration pro le decreases signi cantly, which falls rapidly as it approaches the upper disk at suction. Also, when the squeeze parameter (S) is increased, the e ect on the concentration pro le of the nano uid ow process is shown in Figure 18. From the gure, it is shown that the concentration distribution increases signi cantly near the wall close to the lower disk but falls rapidly as the upper disk is approached during suction. Figures 19 and 20 display the in uence of nanoparticle volume fraction and nanoparticle shape factor on the concentration pro le of the nano uid, respectively. It can be seen in the gure that a maximum temperature occurs between 0.2 < η < 0.3 for all the nanoparticle volume fraction and nanoparticle shape factor considered. e gure shows that the concentration of the nano uid increases as the nanoparticle volume fraction and nanoparticle shape factor increase. e increase of nanoparticle volume fraction or the use of nanoparticle with a high shape factor in a base uid increases the surface area for chemical reaction. erefore, as the nanoparticle volume fraction or nanoparticle shape factor increases, there are increased chemical interactions between the nanoparticles and the particles with the boundary surface of the disk, and as a consequence, more species are formed which increase the concentration of the species within the nano uid.

Conclusion
In this study, e ects of nanoparticle geometry, slip, and temperature jump conditions on thermo-magneto-solutal squeezing ow of nano uid between two parallel disks embedded in a porous medium have been investigated analytically using the homotopy perturbation method. Also, the in uences of various ow parameters such as thermophoresis, the Brownian motion, and the Lewis number and pressure gradient on ow, heat, and mass transfer of the process were investigated. It is established in the study that the addition of nanoparticles to the base uid enhances its thermal conductivity. Based on the study, the following remarks were made: (i) Irrespective of the nanoparticle volume fraction, the thermal conductivity ratio of the nano uid is directly proportional to the nanoparticles shape factor. is fact presents overall e ects on the velocity, temperature, and concentration of the nano uid during the ow process.
(ii) e radial velocity component increases with an increase in the slip parameter near the lower disk.
A reverse case is recorded as the ow approaches the upper disk. (iii) Increasing the values of the magnetic eld parameter, the velocity decreases slightly near the lower disk and increases towards the upper disk. (iv) As the squeeze number increases, the radial velocity component increases and the temperature distribution at the lower disk decreases, while the temperature at the upper disk increases and the concentration distribution increases signi cantly near the wall close to the lower disk but falls rapidly as the upper disk is approached during suction. (v) Increasing the pressure gradient term, a very slight increase in the velocity component was observed while the pressure gradient term has no signi cant e ect on temperature distribution. (vi) e temperature and concentration of the nanouid increase as the nanoparticle volume fraction and nanoparticle shape factor of the nano uid increase. (vii) As the temperature jump parameter increases, temperature distribution increases towards the lower disk where it decreases towards the upper disk. (viii) Increasing values of the thermophoresis parameter, the temperature distribution increases and is maximum at the lower disk but falls rapidly towards the upper disk, while the concentration pro le increases signi cantly, but towards the upper disk there is a steady reduction. (ix) Also, increasing the Lewis number evokes a corresponding increase in concentration distribution at the region closer to the lower disk. (x) Increasing the value of the Brownian number, the concentration profile decreases significantly, which falls rapidly as it approaches the upper disk at suction.
Important significance of the study includes energy conservation, friction reduction, and micromixing of biological samples.