Hydrothermal analysis on MHD squeezing nanofluid flow in parallel plates by analytical method
 Kh. Hosseinzadeh^{1},
 M. Alizadeh^{1} and
 D. D. Ganji^{1}Email author
https://doi.org/10.1186/s4071201800897
© The Author(s). 2018
Received: 15 September 2017
Accepted: 19 January 2018
Published: 26 February 2018
Abstract
Background
In this paper, the heat and mass transfer of MHD nanofluid squeezing flow between two parallel plates are investigated. In squeezing flows, a material is compressed between two parallel plates and then squeezed out radially. The significance of this study is the hydrothermal investigation of MHD nanofluid during squeezing flow. The affecting parameters on the flow and heat transfer are Brownian motion, Thermophoresis parameter, Squeezing parameter and the magnetic field.
Methods
By applying the proper similarity parameters, the governing equations of the problem are converted to nondimensional forms and are solved analytically using the Homotopy Perturbation Method (HPM) and the Collocation Method (CM). Moreover, the analytical solution is compared with numerical Finite Element Method (FEM) and a good agreement is obtained.
Results
The results indicated that increasing the Brownian motion parameter causes an increase in the temperature profile, while an inverse treatment is observed for the concentration profile. Also, it was found that enhancing the thermophoresis parameter results in decreasing the temperature profile and augmenting the concentration profile.
Conclusions
Effects of active parameters have been considered for the flow, heat and mass transfer. The results indicated that temperature boundary layer thickness will increases by augmentation of Brownian motion parameter and Thermophoresis parameter, while it decreases by raising the other active parameters.
Keywords
Background
Investigation of heat and mass transfer of viscous flow between two parallel plates is one of the most important and wellknown academic research topics because of the wide range of its scientific and engineering approaches, such as polymer processing, systems of lubricating, cooling towers, and food processing.
Utilizing nanoscale particles in the base fluid seems to be a creative technique to improve the heat transfer rate. As known, nanoparticlecontaining fluids are called nanofluids. Choi 1995 was the first one ever to name the nanoparticlecontaining fluids as nanofluids.
Sheikholeslami and Ganji (2013a) studied analytically the heat transfer of a nanofluid flow compressed between parallel plates using Homotopy Perturbation Method (HPM). They indicated that the Nusselt number is directly related to nanoparticle volume fraction along with Squeeze number and Eckert number for two separated plates, while there is an inverse relationship between the Nusselt number and Squeeze number when two plates are squeezed.
Sheikholeslami and Bhatti (2017a) studied the heat transfer enhancement of nanofluid flow by using EHD. Results indicated that the effect of Coulomb force is more considerable in lower values of Reynolds number. The shape effects of nanoparticles on the natural convection nanofluid flow in a porous semiannulus were studied by Sheikholeslami and Bhatti (2017b). Results illustrated that the maximum rate of heat transfer is obtained at platelet shape. Bhatti and Rashidi (2016) investigated the influences of thermosdiffusion and thermal radiation on nanofluid flow over a stretching sheet. The authors showed that the temperature profile is an increasing function of thermal radiation and thermophoresis parameters. Ghadikolaei et al. (2017) reviewed the micropolar nanofluid flow over a porous stretching sheet. It was found that raising the radiation parameter enhances the boundary layer thickness. Ghadikolaei et al. (2017) analyzed the stagnationpoint flow of hybrid nanofluid over a stretching sheet. They concluded that using hybrid nanofluid instead of conventional nanofluid results in higher Nusselt numbers. Dogonchi et al. (2017) investigated the influence of thermal radiation on MHD nanofluid flow in a porous channel. Results illustrated that there is a direct relationship between the Nusselt number and nanofluid volume fraction. Recently, many authors analyzed the effect of nanoparticles in theirs studies (Abbas et al. 2017; Bhatti et al. 2017b; Saedi Ardahaie et al. 2018; Abbas et al. 2017).
The results of the timedependent chemical reaction on the viscous fluid flow over an unsteady stretching sheet were considered by AbdEl Aziz (2010). Furthermore, for a certain viscous fluid between parallel disks, the magneto hydrodynamic squeezed flow was investigated by Domairry and Aziz (2009). Also, for the fluid flow between parallel plates, the Homotopy Analysis Method (HAM (Domairry and Ziabakhsh 2009a; Domairry and Ziabakhsh 2009b; Ziabakhsh et al. 2009)) was utilized by Mustafa et al. (2012).
The majority of the problems in the field of engineering especially heat transfer equations contain nonlinear equations. In this case, some of these nonlinear equations can be solved by numerical approaches, while some others are solvable using various analytical methods such as Perturbation method (PM) (Bhatti and Lu 2017), Collocation method (CM) (Rahimi et al. 2017; Atouei et al. 2015), Homotopy perturbation method (HPM), and Variational iteration method (VIM). Hence, the elimination of small parameter has been the critical issue for the scientists nowadays which has led to introduction of various ways to solve these special problems. One of these ways is the implementation of the semiexact method called “HPM” which does not need any small parameters. The Homotopy perturbation method was suggested and modified by He (2004). This method results in a rapid convergence of the solution series in most cases. Including both the efficiency and the accuracy in solving a large number of nonlinear problems, the HPM proved itself capable in dealing with such problems. Ijaz et al. (2018) applied the HPM to investigate the effect of liquidsolid particles interaction in a wavy channel. Dogonchi et al. (2015) analyzed the sedimentation of nonspherical particles in Newtonain media using DTMPade approximation. It was found that enhancing the sphericity of particles results in augmenting the velocity profile.
Mosayebidorcheh et al. (2016) studied the analysis of turbulent MHD Couette nanofluid flow and heat transfer using hybrid DTM–FDM. Sheikholeslami et al. (2011) investigated the rotation of MHD viscous flow along with the heat transfer between stretching and porous surfaces using HPM. Results showed that an increase in the rotation parameter along with the increase in the blowing velocity parameter and Prandtl number would cause an increase in the Nusselt number. The profiles of the variables such as the velocity, temperature, and the concentration of the nanofluids affected by the magnetic field are investigated by Uddin et al. (2014). They comprehended that the presence of magnetic field would cause a decrease in the velocity field and an increase in temperature and concentration profiles. Also, it was found that the convective heating parameter leads to augment the velocity, temperature, and concentration profiles. Uddin et al. (2014) also studied nonNewtonian nanofluid slip flow over a permeable stretching sheet. Their result indicated that the skin friction factor plays a key role in the characteristics of nanofluid flow. In addition, the chemical reaction of nanofluid in free convection in the presence of magnetic field was investigated by Uddin et al. (2015). During the study of Jing et al. (2015), they discovered that the presence of nanoparticles within the fluid can extremely increase the effective thermal conductivity of the fluid, and as a result, the heat transfer characteristics will be improved. Sheikholeslami and Ganji (2013b) examined the nanofluid flow squeezed between parallel plates utilizing Homotopy perturbation method (HPM). They reported that the Nusselt number has direct relationship with nanoparticle volume fraction, the Squeeze number and the Eckert number in the case of separated plates, while its relationship with the Squeeze number in the case of squeezed plates is vice versa. Paying attention to the nanoparticle migration, the mixed convection of alumina–water nanofluid inside a concentric annulus was investigated by Malvandi and Ganji (2016). Sulochana et al. (2016) examined the effect of transpiration on the magnetohydrodynamic stagnationpoint flow of a Carreau nanofluid toward a stretching/shrinking sheet in the presence of thermophoresis and Brownian motion, numerically. They discovered that by raising the thermophoresis parameter, both the heat and mass transfer rates will be increased, whereas the Weissenberg number enlarges the momentum boundary layer thickness along with the heat and mass transfer rate. Sheikholeslami et al. (2016) studied the effect of Lorentz forces on forcedconvection nanofluid flow over a stretched surface. Their results indicated that the skin friction coefficient increases by amplifying the magnetic field, while it decreases by enhancing the velocity ratio parameter. Sudarsana Reddy and Chamkha (2016) analyzed the influence of size, shape, and type of nanoparticles along with the type and temperature of the base fluid on the natural convection MHD nanofluid flow. Their results revealed that decreasing the size of the nanoparticles leads to a significant natural convection heat transfer rate. Moreover, types of nanoparticles and the base fluid also impressed the natural convection heat transfer. Mishra and Bhatti (2017) investigated the MHD stagnationpoint flow over a shrinking sheet, numerically. The authors compared the accuracy of their solution with previous studies and found that a good agreement was obtained. Newly, the study of MHD flow in different geometries has attracted many attentions (Bhatti et al. 2017a; Ghadikolaei et al. 2017; Hatami et al. 2014; Bhatti et al. 2018).
The main goal of the present study is to investigate the effect of Brownian motion and thermophoresis phenomenon on the squeezing nanofluid flow and heat transfer between two parallel flat plates in the presence of variable magnetic field. Both the flow and heat transfer characteristics have been examined under the effects of Squeeze number, suction parameter, Hartmann number, Brownian motion parameter, Thermophoretic parameter, and Lewis number.
Problem description and governing equations
HPM and CM solution when A = 1, M = 1, S = 1, Le = 1, Nb = 1, Nt = 1, Pr = 6.2.
η  f(η)  θ(η)  ϕ(η)  

HPM  CM  FEM  %Error  HPM  CM  FEM  %Error  HPM  CM  FEM  %Error  
0  1  1  1  0  1  1  1  0  1  1  1  0 
0.1  0.9828509855  0.9824627697  0.9824361313  0.0000266384  0.9346923870  0.8308268138  0.830835128  0.0000083142  0.7793705558  0.9000934056  0.9112683052  −0.0111748996 
0.2  0.9392513782  0.9375660234  0.9375269886  0.00003900384  0.8640178590  0.6904013197  0.6910783899  − 0.0006770702  0.6012650162  0.7989509826  0.8108534964  − 0.0119025138 
0.3  0.8788058754  0.8752398967  0.8752157845  0.0000241122  0.7884934362  0.5714471980  0.5725408493  − 0.0010936513  0.4559056726  0.6970283829  0.7064057452  − 0.0093773623 
0.4  0.8090072079  0.8036034623  0.8035984828  0.0000049795  0.7076406847  0.4683468174  0.4695239098  − 0.0011770924  0.3359299240  0.5947812584  0.6016217396  − 0.0068404812 
0.5  0.7359775756  0.7293903870  0.7293979521  0.0000075651  0.6202061754  0.3767871883  0.37785019  − 0.0010630017  0.2360877164  0.4926652610  0.4980783825  − 0.0054131215 
0.6  0.6650106060  0.6583174081  0.658336159  − 0.0000187509  0.5243799637  0.2934637222  0.2943510479  − 0.0008873257  0.1530318820  0.3911360429  0.3962207368  − 0.0050846939 
0.7  0.6009852669  0.5953956265  0.5954289745  − 0.000033348  0.4179140248  0.2158417967  0.2165647056  − 0.0007229089  0.08537940058  0.2906492557  0.2959127913  − 0.0052635356 
0.8  0.5487041816  0.5451846184  0.5452260815  − 0.0000414609  0.2979942644  0.1419761291  0.1425481689  − 0.0005720398  0.03409554226  0.1916605515  0.196753453  − 0.0050929015 
0.9  0.5131945405  0.5119893652  0.5120144707  − 0.0000251055  0.1607055638  0.07038795247  0.0707603109  − 0.00037235843  0.003119131100  0.09462558233  0.0982635139  − 0.00363793157 
1  0.4999999999  0.5000000008  0.5  0.0000000008  0  0  0  0  0  0  0  0 
The “×” marks show that magnetic field is perpendicular to the illustrated plane. “+” and “−” indicate the positive and negative charges, respectively. T_{ w } and C_{ w } represent the temperature and concentration of nanoparticles at the bottom disk, while the temperature and concentration of nanoparticles at the upper disk are denoted by T_{ H } and C_{ H }. The upper disk at Z = h(t) can move toward or away from the motionless bottom disk with the velocity of \( \raisebox{1ex}{$ dh$}\!\left/ \!\raisebox{1ex}{$ dt$}\right. \) .
For a > 0 and a < 0, two plates are squeezed and separated, respectively. The viscous dissipation effect along with the generated heat remained intact due to the friction caused by shear forces in the flow. It should be noted that when the fluid is largely viscous or is flowing at a high speed, the dissipation effect is quite important. Knowing that the nanofluid is a twocomponent mixture, the following assumption has been considered:
Incompressible; nochemical reaction; with negligibleness of viscous dissipation and radiative heat transfer; nanosolidparticles and the base fluid are in thermal equilibrium and without any slip between them. The equations which govern the flow, heat, and mass transfer in viscous fluid are as follows (Hashmi et al. 2012; Turkyilmazoglu 2017):
where S is the Squeeze number, A is the suction/blowing parameter, M is the Hartmann number, Nb is the Brownian motion parameter, Nt is the Thermophoretic parameter, Pr is the Prandtl number, and Le is the Lewis number and are defined as follows:
The continuity equation is identically satisfied. It should be noted that A > 0 indicates the suction of fluid from the lower disk, while A < 0 represents the injection flow.
Methods
Collocation method (CM)
Now, we can substitute \( \tilde{u} \) from Eq. (12) into the Eq. (11), generally p (x) is not the result of the operations. Therefore an error or residual will exist:
The basic principle of the Collocation method is to lead an error or the residual to zero in some average sense over the domain as follows:
Application of CM
Eventually, by substitutions values of Eq. (17) into residual function R (c1, c2, c3, η), a set of three long equations with three unknown coefficients are obtained. After solving these unknown parameters (c1, c2, c3), the temperature distribution equation, Eq. (16), will be determined.
To find the solution, the parameters can be considered as Pr = 6.2, A = 1, Nt = 1, and M = S = Le = Nb = 1, and based on Eq. (10) (Sabbaghi et al. 2011), the f(η), θ(η) and ϕ(η) formulation are obtained as follows:
Homotopy Perturbation Method (HPM)
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 domainΩ.
Application of HPM
Numerical Finite Element Method (FEM)
Some several methods can be useful to find a solution of fluid flow and heat transfer problems such as the Finite Difference Method, the Finite Volume method (FVM), and the Finite Element Method (FEM). The control volume Finite Element Method (CVFEM) contains interesting features from both the FVM and FEM. The CVFEM benefits from the flexibility of the FEMs to discretize complex geometry with conservative formulation of the FVMs, in which the variables can be easily interpreted physically in terms of fluxes, forces, and sources (Kandelousi and Ganji n.d.).
FlexPDE is a scripted Finite Element model builder and numerical solver. This software performs the essential operations to turn a description of a partial differential equations system into a Finite Element model and finally solve the system, and present graphical and tabular output of the results (Table 1).
Results and discussion
Conclusion
The present study examines the effect of Brownian motion and Thermophoresis phenomenon on the heat and mass transfer of MHD nanofluid flow between parallel plates. To examine this problem, a number of methods such as the Collocation Method (CM), the Homotopy Perturbation Method (HPM), and the Finite Element Method (FEM) were applied. The results indicated that the outcomes of Collocation Method have the best agreement with the numerical solutions. The crucial effect of Brownian motion and thermophoresis parameter has been included in the model of nanofluid. Effects of active parameters have been considered for the flow, heat, and mass transfer. The results indicated that temperature boundary layer thickness will increase by augmentation of Brownian motion parameter and Thermophoresis parameter, while it decreases by raising the other active parameters. Also, it can be concluded that the thickness of concentration boundary layer declines by enhancing the Brownian motion parameter, while an inverse trend is observed by augmenting the Thermophoresis parameter.
Abbreviations
 T _{ w } :

Nanoparticle temperature (K)
 C _{ w } :

Nanoparticles concentration (%wt)
 A :

Suction/blowing parameter
 B :

Magnetic field (T)
 FEM:

Finite Element Method
 H :

Height between the plate (m)
 HPM:

Homotopy Perturbation Method
 k :

Thermal conductivity (w/m. k)
 Le:

Lewis number
 M :

Hartmann number
 Nb:

Brownian motion parameter
 Nt:

Thermophoretic parameter
 p :

Pressure (Pa)
 P :

Pressure term
 Pr:

Prandtl number
 S :

Squeeze number
 t :

Time (s)
 Z :

Vertical direction
Greek symbols
 α :

Thermal diffusivity
 σ:

Stefan–Boltzmann constant (w/m^{2}k^{4})
 ρ :

Density (kg/m^{3})
 θ :

Dimensionless temperature
 ϕ :

Nanoparticle volume fraction
 η :

Dimensionless variable
Subscripts
 f :

Base fluid
 nf :

Nanofluid
 CM:

Collocation Method
Declarations
Acknowledgements
Not applicable
Funding
This research doesn’t have any funding.
Availability of data and materials
Not applicable
Authors’ contributions
Kh. Hosseinzadeh has involved in conception and design of the problem and analysis and interpretation of data. M. Alizadeh made substantial contributions to the study specifically in the conception, design and running most of the simulation analysis. D.D. Ganji contributed in the development of the concept and understood the theory behind the concept. All authors contributed to the problem formulation, drafted the manuscript, and read and approved the final manuscript.
Authors’ information
Kh. Hosseinzadeh is PhD student in mechanical engineering at Babol Noshirvani University of Technology.
M. Alizadeh is PhD student in mechanical engineering at Babol Noshirvani University of Technology.
D.D.Ganji is full professor in mechanical engineering faculty at Babol Noshirvani University of Technology.
Competing interests
The authors declare that they have no competing interests.
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