The purpose of this paper is to examine the combined effects of thermal radiation and magnetic field of molybdenum disulfide nanofluid in a channel with changing walls. Water is considered as a Newtonian fluid and treated as a base fluid and MoS2 as nanoparticles with different shapes (spherical, cylindrical and laminar). The main structures of partial differential equations are taken in the form of continuity, momentum and energy equations.
The governing partial differential equations are converted into a set of nonlinear ordinary differential equations by applying a suitable similarity transformation and then solved numerically via a three-stage Lobatto III-A formula.
All obtained unknown functions are discussed in detail after plotting the numerical results against different arising physical parameters. The validations of numerical results have been taken into account with other works reported in literature and are found to be in an excellent agreement. The study reveals that the Nusselt number increases by increasing the solid volume fraction for different shapes of nanoparticles, and an increase in the values of wall expansion ratio α increases the velocity profile f′(η) from lower wall to the center of the channel and decreases afterwards.
In this paper, a numerical method was utilized to investigate the influence of molybdenum disulfide (MoS2) nanoparticles shapes on MHD flow of nanofluid in a channel. The validity of the literature review cited above ensures that the current study has never been reported before and it is quite new; therefore, in case of validity of the results, a three-stage Lobattoo III-A formula is implemented in Matlab 15 by built in routine “bvp4c,” and it is found to be in an excellent agreement with the literature published before.
Raza, J., Mebarek-Oudina, F. and Chamkha, A.J. (2019), "Magnetohydrodynamic flow of molybdenum disulfide nanofluid in a channel with shape effects", Multidiscipline Modeling in Materials and Structures, Vol. 15 No. 4, pp. 737-757. https://doi.org/10.1108/MMMS-07-2018-0133
Emerald Publishing Limited
Copyright © 2019, Emerald Publishing Limited
external uniform magnetic field
thermal conductivity of the solid fraction (W/m.K)
thermal conductivity of the nanofluid (W/m.K)
density of the solid fraction (Kg/m3)
specific heat of nanofluid
fluid temperature (K or °C)
thermal conductivity of the fluid (W/m.K)
shape factor through H-C Model
specific heat at constant pressure (J/(kg K)
- (u, v)
velocity component in Cartesian coordinate
- Greek symbols
wall expansion ratio
scaled boundary layer coordinate
effective electrical conductivity of nanofluid
thermal conductivity of the nanofluid (W/m.K)
mean absorption coefficient of the nanofluid
nanoparticle volume fraction parameter
effective dynamic viscosity of nanofluid
- Dimensionless numbers
- R = avw/υ
cross-flow Reynolds number
- ρnf = ρf(1−φ) + ρs
density of the nanofluid
- μnf = μf/(1 − φ)2.5
dynamic viscosity of the nanofluid (Pa.s)
ratio of effective electrical conductivity of nanofluid to the base fluid
The present world is encompassed by various uses of nanofluids in the field of engineering and sciences. The conceptualization of these fluids is in much practice in the field of biomedical and food security. Additionally, these fluids are utilized for the cooling of atomic reactors in aerodynamic process and in next-generation solar film collector. Choi (Choi and Eastman, 1995) was the first one to present the hypothesis of nanofluids. At the point when a convectional base fluid is joined with nanometer-estimated strong particles, e.g., copper, aluminum, silver, gold, and silicon, nanofluids are formed. The reasons for utilizing nanoparticles in the base fluid are to improve thermal conductivity of base fluids. According to Choi (Choi and Eastman, 1995), by including a little measure of nanoparticles to convectional heat bearer’s fluids, the thermal conductivity can be expanded approximately up to two times. Choi (Choi and Eastman, 1995) broke down the thermal conductivity of nanofluids wherein nano-estimated copper particles were included and the result demonstrated heat exchange upgrade as an element of thermal conductivity. The scientific displaying of nanofluids is coordinated as a single-phase model and a two-phase model. Buongiorno (2006) examined the mathematical modeling of two-phase fluid models. He revealed that Brownian motion and thermophoresis have an indispensable commitment toward upgrade of thermal conductivity of fluids. Khan and Pop (2010) investigated the impacts of the thermophoresis number and Brownian motion number on an enduring boundary layer flow over an extending surface. Sheikholeslami et al. (2012) examined the laminar boundary layer flow in a semi-permeable channel in which transverse magnetic field was connected to discover the impact of Harman number on the flow. The authors built up the table of error percentage between three techniques, namely HAM, OHAM and VIM. In 2012, Ibrahi et al. (2013) examined the impact of MHD and heat exchange on nanofluid over stretching sheet. The flow is seen close to the stagnation point. A work has been accomplished by Sheikholeslami (2018b) on heat transfer in a forced convection. The flow is considered in a lid-driven porous cavity, encased with Al2O3-water nanofluid under the influence of magnetic field. Lattice–Boltzmann method was utilized to take care of the issue.
The study of Sheikholeslami, Shehzad, Li and Shafee (2018) modeled the alumina nanofluid magnetohydrodynamic flow through a permeable enclosure using CVFEM. The results show that the Lorentz forces reinforce the conduction mechanism. The Brownian motion and shape factor influence the viscosity and thermal conductivity of alumina. Of late, studies associating with heat exchange and nanofluid flows are as follows: Sheikholeslami, Jafaryar and Li (2018), Sheikholeslami, Li and Shafee (2018), Sheikholeslami, Darzi and Sadoughi (2018), Hussein and Mustafa (2018), Sheikholeslami (2018a), Sheikholeslami, Shehzad, Abbasi and Li (2018), Sheikholeslami, Darzi and Li (2018), Mebarek-Oudina and Makinde (2018), Reza et al. (2018), Mebarek-Oudina (2018) and Sheikholeslami (2017). Raza, Rohni, Omar and Awais (2016) had done a bit of work on nanofluids in a rotating channel wherein MHD three-dimensional flow was investigated. The problem was solved by numerical strategy. Investigations of the fluid flow problems in a channel with permeable expanding or contracting walls have attracted most of the researchers due to the vast applications in the fields of chemical, biomedical, engineering and science. The blood flow in vessels, artificial kidneys and the air flow in respiratory system are most prominent examples. The principal endeavor focused on the viscous flow inside a porous pipe of contracting cross section, as examined by Uchida and Aoki (1977). Later, this issue was enlightened by Bujurke et al. (1998), both numerically and approximately. Goto and Uchida (1990) proposed the model of the laminar incompressible flow in a semi-infinite permeable pipe to recreate the laminar stream field in tube-shaped strong rocket engines. Furthermore, Majdalani et al. (2003), Dauenhauer and Majdalani (2003) and Majdalani and Zhou (2003) explored the issue of laminar flow in a channel with permeable extending the channel numerically and asymptotically. Afterwards, logical arrangements were acquired by Rahimi et al. (2016) for the instance of extending and contracting permeable channel walls. Reddy et al. (2013) utilized perturbation technique to examine the impacts of heat and mass transfer on the asymmetric flow in a permeable channel with changing walls. The arrangement of consistently extending or contracting channel in a semi-infinite rectangular permeable channel was researched by Mohyud-Din et al. (2010). A mathematical solution was explored by Magalakwe and Khalique (2013) for the flow and heat transfer between gradually expanding or contracting channel walls. Xinhui et al. (Xinhui et al., 2014; Si et al., 2013) characterized the flow of non-Newtonian flow in a permeable channel with extending or contracting walls.
In engineering problems, solving nonlinear ordinary differential equations and partial differential equations has always become a difficult task for the researchers. Therefore, with the passage of time, many techniques developed to solve this issue. The most prominent knowledge regarding the current issue is addressed in the form of literature related to analytical methods, semi-analytical methods and numerical methods. In the field of heat transfer, main structures of the problems are often in the form of nonlinear ordinary differential equations. Many researchers solved the nonlinear ODEs by utilizing analytical and semi-analytical methods such as least square method (Sheikholeslami et al., 2013; Jafaryar et al., 2014), differential transformation (Hatami, Sheikholeslami, Hosseini and Ganji, 2014; Sheikholeslami et al., 2015), homotopy perturbation method (Mohammadian et al., 2015; Majidian et al., 2014), homotopy analysis method (Srinivas et al., 2017) and perturbation method (Akinshilo and Sobamowo, 2017) for the different topological structures.
Recently, many authors investigated the close form of the solutions on different fluid flow conditions (Ali, Sheikh, Khan and Saqib, 2017; Ali, Sheikh, Saqib and Khan, 2017; Sheikh et al., 2017; Khan et al., 2017; Saqib et al., 2016; Singh et al., 2016; Raza et al., 2016a, b, c, 2017).
However, some of the problems cannot be solved analytically, especially when there exists more than one solution, and the two of them are very close to each other. To overcome this problem, many researchers used numerical techniques. Shooting method, Runge‒Kutta‒Fehlberg method, implicit finite difference, Keller box method and finite element method are most common numerical techniques. Recently, Singh et al. (2016) used Runge‒Kutta‒Fehlberg method with shooting method to solve the problem of MHD unsteady nanofluid flow between parallel plates with slip effects. Raza et al. (2016a, b, c, 2017) and Raza, Rohni, Omar and Baig (2016) successfully solved the fluid flow problems in a channel with shooting method. Volume finite method was used by Mebarek-Oudina (Mebarek-Oudina and Bessaïh, 2007, 2010, 2014, 2016; Mebarek-Oudina, 2017) to solve numerically the differential equations of the fluid problem. Hatami, Sheikholeslami and Ganji (2014) investigated the flow of nanofluid between expanding or contracting disks. Runge‒Kutta‒Fehlberg method was used in order to solve nonlinear ordinary differential equations numerically.
The validity of the literature review cited above ensures that the current study has never been reported before and it is quite new; therefore, in case of validity of the results, a three-stage Lobatto III-A formula is implemented in Matlab 15 by built in routine “bvp4c,” and it is found to be in an excellent agreement with the literature published before (Tables I and II).
When the flow of an unsteady, laminar and incompressible electrically conductive nanofluid is considered in a porous channel, it is found that the channel walls are variant in the direction of y-axis and can be expanded or contracted with respect to the time-dependent rate . Moreover, both the channel walls are assumed to be porous, having the same permeability. For the uniform wall suction/injection, we assume that fluid is symmetric about y-axis, as shown in Figure 1. Governing equations of:
Using the Rosseland approximation for radiation, we have:
Therefore, Equation (4) becomes:
These physical quantities can be described mathematically, as given by Brinkman (Dauenhauer and Majdalani, 2003):
Fluid can be injected or sucked with uniform velocity vw at the channel walls. Moreover, the injection/suction coefficient , which appears in Equation (10), is a measure of wall permeability.
The steam function is introduced:
Subject to the boundary conditions:
For self-similar solution, we consider by the transformation introduced by Uchida and Aoki (1977) and Dauenhauer and Majdalani (2003). This can lead us to consider that α is a constant and f = f(η). Therefore, fηηt = 0. So Equation (16) becomes:
Subject to the boundary conditions:
At distance η from the wall, the fluid temperature is expressed as:
With boundary condition:
Physical quantities of our interest
In the current case, the non-dimensional Nusselt number is obtained as:
Similarly, the skin friction coefficient can be expressed as:
Equations (19) and (23) are highly nonlinear ordinary differential equations. It is quite difficult to find the exact solution of these equations. Therefore, we employed a numerical technique known as a three-stage Lobatto III-A formula. The numerical approach of bvp4c can be seen as a collocation with C1 piece-wise cubic polynomial S(x). Many researchers studied this formula to solve highly nonlinear problems (Shampine et al., 2003; Sharma et al., 2014, 2016). In this perspective, by introduction of the new variables, the nonlinear differential equations are first reduced into a system of first-order differential equations. Mesh selection and error control are based on the residual of the continuous solution. In this study, the relative error tolerance is fixed at 10−8. The solution is returned by bvp4c as structure called sol.y. The appropriate mesh selection is generated and returned in the field sol.x. However, solution values can be fetched from the array named sol.y, corresponding to the field sol.x (Figure 2).
Results and discussions
The influence of various parameters, φ (solid volume fraction), α (wall expansion ratio), R (Reynolds number), M (magnetic parameter), N (radiation parameter), m (power law index) and Pr (Prandtl number) is graphically discussed for velocity and temperature profiles for different types of nanofluid particles. Moreover, validations of numerical results are presented in the form of table also. The results were obtained by a three-stage Lobatto III-A formula, as shown in Table V.
Figure 3 presents the effect of Reynolds number R on the velocity profile f′(η) of nanofluid. It is depicted from this graph that at a low Reynolds number R, the velocity profile is symmetric with center line. Moreover, as Reynolds number R increases, the velocity profile f′(η) is shifted toward the solid wall where shear stresses are larger.
The influence of Reynolds number on the temperature profile is shown in Figure 4 for the different sizes of the nanoparticles. It is observed that temperature of the nanofluid decreases monotonically by increasing the numerical values of Reynolds number R. The effect of wall expansion ratio on the velocity profile is presented in Figure 5. It is depicted from this profile that an increase in the values of α velocity profile f′(η) increased from lower wall to the center of the channel; however, a totally reverse phenomenon is observed from 0.5⩽η⩽1 due to the wall injection of the fluid. Physically, it can be assumed that as α>0, the distance between the two walls enhances and the fluid can easily pass through it. Furthermore, the velocity of the fluid increases near the center and graph is shifted toward the lower wall of the channel because upper wall is under the influence of fluid injection. Particularly, the fluid viscosity decreases and nanoparticles can easily flow within the passage due to expansion of the walls. So, it is important to highlight that through expansion of walls, we can easily control the nanoparticles’ fluid flow. This result is highly obliged in order to understand the mechanics of the fluid flow in duck which leads us to control the flow.
Similarly, Figure 6 shows the effects of wall expansion on the temperature profile and it is observed that thermal boundary layer thickness decreases as the wall expansion ratio increases for every shape of the nanoparticles. The effect of magnetic parameter M on the velocity profile is demonstrated in Figure 7. Velocity profile f′(η) is decreased in the region 0.15<η<0.6 by the enhancement in the strength of magnetic parameter. However, a totally reverse trend is observed afterwards. It is because of the fact that magnetic parameter M is applied in transverse direction to the channel walls; thus, the effect of magnetic field M is dominant near the walls. From physical point of view, it can be assumed that when magnetic field is applied to the MoS2 nanoparticles, then the fluids’ apparent viscosity decreases due to the chain formation of the fluid. The chain-like structure retards the flow and accelerates the motion. This results into the fact that the flow of nanofluid can be controlled very efficiently by applying varying magnetic fields, which results into many control-based applications. Figure 8 elucidates the effects of magnetic parameter M on the temperature profile for different sizes of the nanoparticles. It is depicted from this graph that the thermal boundary layer thickness is increased as the strength of the magnetic parameter is enhanced; however, the effects are asymptotically close to each other. Figure 9 represents the effect of power law index on the temperature profile of nanofluid. It is observed from this graph that the temperature profile decreases gradually for all shapes of nanoparticles by increasing the values of power law index. The effects of radiation parameter N on the temperature profile are plotted in Figure 10. Moreover, the thermal boundary layer thickness decreases by increasing the values of the radiation parameter N. The effects of solid volume fraction on the temperature profile are shown in Figure 11. It can be seen from this profile that the temperature profile increases gradually by the enhancement in the strength of solid volume fraction; therefore, thermal boundary layer thickness increases. Physically, it can be observed that the temperature of the nanofluid can be controlled by adjusting the values of solid volume fraction concentration.
The variation of the skin friction for various values of wall expansion ratio and Reynolds number is shown in Figure 12. It can be observed that as the Reynolds number increases, the skin friction increases monotonically for the case of different shapes of nanoparticles. Moreover, with the increase in the Reynolds number and wall expansion ratio, skin friction at the wall of the channel increases; therefore, velocity of the nanofluid adjacent to the lower wall of the channel tends to decrease. We eventually explore the impact of solid volume fraction on skin friction coefficient for the variation of magnetic parameter in Figure 13. This figure shows that skin friction coefficient increases strictly monotonically by increasing the magnitude of the magnetic parameter for different sizes of the nanoparticles (Cylindrical, Spherical and Laminar). It is due to the combined effect of magnetic parameter and the strength of the solid volume, which offer great reduction in the fluids particles, and therefore skin friction at the lower wall of the channel is increased gradually.
Table V presents numerical results of Nusselt number against the values of radiation parameter, solid volume fraction and power law index for different shapes of nanoparticles. It is depicted from these values that local Nusselt number increases gradually by increasing the values of radiation parameter and power law index. Moreover, spherical nanoparticles have a significant change in the local Nusselt number as far as the rest of the nanoparticles are considered. To validate the current numerical solution, we compared our results with other works reported in literature (Xinhui et al., 2011). They are in a very good agreement, as shown in Table VI.
In this paper, a numerical method was utilized to investigate the influence of molybdenum disulfide (MoS2) nanoparticle shapes on MHD flow of nanofluid in a channel with expanding walls. The major findings of the current study are stated below:
an increase in the value of the wall expansion ratio α caused the velocity profile f′(η) to increase from the lower wall to the center of the channel and to decrease afterwards;
the heat transfer rate was enhanced when the solid volume fraction was increased;
spherical nanoparticles boosted the local Nusselt number more than the other considered nanoparticles; and
the evaluation of the results by a three-stage Lobatto III-A formula and their comparison with previously published results illustrate that the approach is well-matched, accurate and efficient to solve the problem.
Summary of the semi-analytical methods for nanofluid
Summary of the numerical methods for nanofluid
Thermophysical properties of water and nanoparticle
|Properties||ρ/kg.m−3||Cp/j.kg−1.k||k/W.m−1.k||β × 105/K−1|
|MoS2||5.06 × 103||397.21||904.4||2.8424|
Source: Shafie et al. (2016)
Nanoparticle shape depends upon m
Impact of radiation, solid volume fraction and power law index on Nusselt number for different shapes of nanoparticles
Comparison between present and Xinhui et al. (2011) results when R=5, M=0, φ=0, m=3
|η||Present results||Xinhui et al. (Hatami, Sheikholeslami and Ganji, 2014) results|
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