Abstract
This paper deals with two-dimensional steady boundary layer flow, heat, and mass transfer characteristics of micropolar nanofluid past on exponentially stretching/shrinking surface. The effect of different physical parameters like magnetic field, buoyancy, thermal radiation, and connective heat transfer are examined. Furthermore, similarity solutions are obtained by similarity transformation on the governing system of partial differential equations. The shooting method with help of the Maple software is used to achieve the numerical solutions of the equations. For the different ranges of the applied parameters, triple solutions are obtained for both cases of the surface. In view of the triple solutions, stability analysis is performed by bvp4c in the MATLAB software, where only first solution is found feasible which is discussed. The main findings of the first solution indicate the skin friction, drag force, heat, and mass transfer rates are increasing for the and decreasing for as the is enhanced. The velocity profiles decrease with increase in magnetic, slip velocity, and suction parameters. The temperature profiles increase with increase in magnetic, thermophoresis, thermal radiation, and Brownian motion parameters, whereas concentration profiles reduce with increase in Schmitt number and Brownian motion.
1. Introduction
The study of non-Newtonian fluid flows have gained much importance, because of the common Newtonian fluids may not completely satisfy the properties of the fluid flow in many industrial applications, examples of such fluids are biological fluids, polymeric fluids, fluids containing additives, liquid crystals, and paint colloidal solutions. Moreover, the class of the non-Newtonian fluids containing different kinds of complex properties are Casson fluids, Maxwell fluids, and micropolar fluids. The micropolar fluid introduced by Eringen [1] possesses a microscopic effect due to the microstructure and micromotion of particles present in fluid. These microstructure particles are of different shapes which rotate independently to the motion of the fluid perticles (Anwar et al. [2]). The system of the micropolar fluid flow equations contains a microrotating vector besides the classical velocity vector. These fluids contain smaller rigid particles which rotate about the centroid of volume particles that predicts the flow behaviors at rotation and microscale independently that is defined by a microrotation vector. Therefore, micropolar fluids are very important in fluid dynamics, especially in studying some flows around some important surfaces such as stretching surfaces or shrinking surfaces. In this regard, MHD micropolar fluid flow on the inclined plate was investigated by Kasim et al. [3]. The micropolar fluid flow on the inclined surface with different physical parameters was also studied by Das [4]. Srinivasacharya and Bindu [5] examined the entropic generation of the micropolar fluid with parallel plates on the inclined channel. Rahman et al. [6] examined the heat flux process in the micropolar fluid with incorporation different kinds of the fluid properties. Dero et al. [7] examined MHD micropolar nanofluid flow on exponential shrinking/stretching surface with radiation, Brownian motion, and thermophoresis effects. Srinivasacharya et al. [8] examined double diffusion impacts on the micropolar fluid flow on a slanted sheet. In [9], the authors studied peristaltic flow of nanofluids through a tapered channel with a porous medium.
Whereas magnetohydrodynamics (MHD) is concerned with physical and mathematical scaffold which shows magnetic dynamics in the electrical conducting fluid, the applications of magnetohydrodynamics are especially used in the modern industrial and engineering areas, such as in drawing of the plastic wires and films, polymer extrusion in melt spinning process, crystal growth, paper production and glass fiber, fluid film in the condensations processes, manufacturing in foods, electronic chips, electrochemical process, flow through the filtering devices, and thermal energy storage. Several investigators have worked on magnetohydrodynamics flow overstretching surfaces by using different physical parameters [10].
Moreover, nanotechnology has achieved much attention of the investigators because of having many applications in industrial and biological sciences. In industries, clay nanocomposites are used for the production of the impermeable wrapping films, silver nanoparticles are particularly used for productions of bins storage, while particles of the zinc oxides can be used for protection of the materials from the ultraviolet radiations and the carbon nanotubes pervaded graphite are used for the productions of light tennis rackets. In biological science, the investigators are busy developing the nanocapsules for the injections replacement that can be passed through the stomach and can accessed in the blood stream easily, whereas nanoparticles’ antibacterial properties make them fit to cut and stitch the bandages. The nanofluid is a subclass of such a widely growing research field. Actually, nanofluids are engineered manufacturing fluids that are made by suspensions of the smallest sized particles (10−9 to 10−11 m) in common traditional fluids. Choi [11] called such flu as the fluids because of the addition of the smallest nanosized particles. Numerous engineering problems are studied by many researchers for more than two decades on the progress of nanofluids [12, 13], whereas the extensive convective rate of the heat transfers in nanofluids was studied by Buongiorno [14] by developing a two-phase nanofluid model. Furthermore, the boundary layer nanofluid flow phenomena on the shrinking/stretching surfaces have been discussed by many researchers [15, 16]. The nanofluid stagnation point flow was considered by Nazar et al. [17]. Kuznetsov and Nield [18] considered the boundary layer flow of the nanofluid through a vertically moving plate with buoyancy effect, while Chamkha et al. [19] considered mixed convective magnetohydrodynamics nanofluid flow with buoyancy effect by using Buongiorno’s (2006) model.
Motivated from the abovementioned work, two-dimensional laminar magnetohydrodynamic boundary layer flow of micropolar nanofluid on exponentially shrinking/stretching surface with mixed convection, thermal radiation, magnetic, Brownian motion, and thermophoresis parameters has been studied. Furthermore, convective heat transfer and porous medium are also taken into account. There has been extension on the work done by Bidin and Nazar [20]. Their work was on the viscous fluid with unique solution while the present work is much updated. The graphical achieved results are used to study the influence of various physical parameters on the velocity, microrotation, temperature, and concentration profiles. Few other interesting studies in this direction are given in [21–24].
To the best of the authors’ knowledge, such an attempt on MHD mixed convection flow of a micropolar nanofluid in the existence of the thermal radiation and magnetic field on an exponentially vertical stretching/shrinking surface by obtaining triple solutions with stability analysis has not been tried before. The results are demonstrated graphically with detailed discussion. It can be expected that the present research work will prove helpful for the researchers which are interested to study nanofluids particularly micropolar-based nanofluids as well as the importance of the stability analysis in case of the occurrence of multiple solutions, where stability analysis decides about real and extraneous solutions.
2. Problem Formulation
There is considered two-dimensional laminar magnetohydrodynamic (MHD) mixed convection flow, heat, and the mass transfer of the micropolar nanofluid on the permeable exponentially vertical shrinking/stretching surface with radiation effect. An electrically conducting nanofluid is considered with influence of the magnetic field which is used perpendicular to the surface. The value of the Reynolds number is negligible that is ignored. The flow is considered along the -axis perpendicular to the -axis. The surface velocity is considered as at . A graphical presentation of flow model along the coordinate system is presented in Figure 1.
Applying boundary layer approximations, the continuity, momentum, microrotation, energy, and concentration equations for the micropolar nanofluid flow of present study are written as where velocity components in the directions of the and are symbolized by and , respectively. is an electrical conduction, is the specific heat of the constant pressure, is the kinematic viscosity, is the constant of the magnetic field , is the thermal expansion coefficient, is concentration expansion coefficient, and is the density of the fluid. is the microrotation, is the vortex viscosity, represents the spin gradient viscosity, is the ratio of microinertia, is thermal diffusion, and thermal conduction of the fluid. is the constant whose range is . Where at , there will be that represents strong concentration that show no rotation due to microelements nearest to the solid surface, while represents weaker concentration. Furthermore, shows turbulences in flows of fluid (Ishak et al. [21]). Several researchers have taken where is the material parameter and . is the temperature of the boundary layer flowing fluid, is the surface temperature, is the ambient fluid temperature, is the nanoparticle volumetric fraction, is the radiative thermal flux, and and are Brownian and thermophoresis diffusions coefficients, respectively. The radiative heat flux is written by
Here, is a mean absorptions coefficient and is the Stefan Boltzmann constant. At smaller difference in the temperature of the flow, the will be described as the function of . Using Taylor’s series to and by ignoring higher terms due to smallest values, we have
By using Equations (5) and (6) in Equation (3), we get
The considered boundary conditions of this problem are
Magnetic field for exponential surface is written as
Furthermore, the following exponential type of the similarity transformation is applied to achieve the similarity solutions: here, is an stream function that in velocity form is defined as and . Thus, we get
Whereas using Equation (15) in Equations (2) to (5), we get following system of equations:
Boundary conditions take the form: here, prime represents derivative along the , is the non-Newtonian parameter, is the Prandtl number, is the radiation parameter, is the Schmidt number, is the thermophoresis parameter, is the parameter of Brownian motion, is the shrinking/stretching surface, here denotes shrinking case and denotes stretching case and denotes the suction of mass by the surface due to porosity, and is a velocity slip parameter. where is ratio between buoyancy and inertia forces that is used as the criterion to create dominant regions of the forced and free convections. The stands for mixed (free and forced) convection and stands for mass convective parameters. The skin friction coefficient , Nusselt number , and the Sherwood number are written as using Equation (15), we get
where is the Reynolds number.
3. Stability Analysis
To check the stability, there are taken unsteady case for Equations (2)–(5) with addition of the new time dependent variable as mentioned by Roşca and Pop [22].
To get the similarity equations, the similarity transformation (15) takes the form as follows:
Using Equation (33) in Equations (25)–(28),the following is obtained:
The boundary conditions given in Equation (21) are written as
For the stability of steady flow solutions , , , and which placate the problem given in Equations (17)–(21), we further define
where denotes unknown eigenvalues and , , , and are smaller functions related to the , , , and respectively. By substitution of Equation (43) in Equations (34)–(42), we have
For to be achieved steady solution, there will be taken, so, we get
Boundary conditions are
Solutions of the eigenvalue problem of Equations (45)–(48) that satisfy Equations (49) are achieved by the solver functions of bvp4c through MATLAB software. These solutions provide a set of eigenvalues . According to Harris et al. [23], the required smallest eigenvalue can be obtained by relaxing one of boundary condition to initial condition. In this regard, in the present problem, we have relaxed as in the form of the . Where the smallest positive values of the demonstrates initial deterioration of the disturbance, the solutions concerned to the flow possessing smallest positive eigenvalues are considered physically feasible and stable. Converse to it, the negative demonstrates initial growth of the disorder; therefore, solutions of the fluid flow are considered not practically feasible and unstable.
4. Numerical Method
There are presented a short view of the shooting technique (Meade et al. [24]) that is used in Equations (17)–(20) that follow the boundary conditions given in Equation (21). Actually, this method converts the BVPs into the IVPs. So, we have
The boundary conditions are where , , , and are the unknown applied initial conditions. Therefore, shooting values of , , , and are carefully taken which satisfy the specified boundary conditions of the problem properly. Furthermore, it is noticed that computations of this problem are done by the help of Maple software where shootlib function is added.
5. Result and Discussion
Equations (17)–(20) that subject to the boundary conditions given in Equations (21) are solved numerically by a shooting method in Maple software for the distinct values of used physical parameters. Triple solutions are obtained for different initial guesses at various ranges of specified parameters. Furthermore, for authentication of results, our obtained results have been compared with previously obtained results by the Bidin and Nazar [20], where the comparison indicates a good agreement with our obtained results which are shown in Table 1.
The numerical computations demonstrate the effects of different used physical parameters on skin friction coefficientcouple stress coefficient, local Nusselt number , and Sherwood number as well as on the velocity, microrotation, temperature, and concentration profiles of the micropolar nanofluid which are graphically presented in Figures 2–22. Stability analysis is performed due to occurrence of triple solutions. Therefore, stability analysis is performed by bvp4c in MATLAB programming. The achieved eigenvalues are presented in Table 2, while result of the stability shows that the first solution is the stable and the physically feasible which possesses smallest positive eigenvalues, while negative eigenvalues are found for two other solutions which are declared as unstable and not physically feasible. Therefore, in this article, the result of only first solutions which is physically feasible solutions are presented.
The influence of the material parameter on the skin friction coefficient, couple stress coefficient, Nusselt number, and Sherwood number with variation of stretching/shrinking parameter are shown in Figures 2–5, where triple solutions are found for and unique solution for , while second and third solutions are merged into one another at the point and only one (first) solution remained continuous for variation of the . Furthermore, Figure 2 and Table 3 show that an increase in with variation in increase the skin friction coefficient for the and decrease for in the first (stable) solution. Table 3 and Figure 3 indicate that an increase in along the variation in , the couple stress rises for the and reduces for the . Furthermore, Table 3 defines the numerical results related to the Figures 2 and 3.
Figures 4 and 5 along Table 4 indicate that an increment in the values of with variation of , the Nusselt number, and the Sherwood number increase for and reduce for in the first solution throughout the flow of the micropolar nanofluid.
While the effect of magnetic parameter on velocity profile is shown in Figure 6, it can be seen that velocity decreases with increase in the parameter. That occurs due to rising Lorentz forces transverse magnetic field that becomes a reason to decelerate the velocity of the fluid. Figure 7 indicates the influence of thermal convective parameter on velocity profile. It is seen that the velocity of micropolar nanofluid declines as is increased. Figure 8 shows the influence of the suction parameter on velocity profile. It is examined that the increasing rate of the suction decreases the velocity and momentum boundary layer thickness in flow of the micropolar nanofluid. It is because of the flow approaches nearer to solid surface, so the suction takes place due to porosity of the medium which becomes a reason to decrease the velocity and its boundary layer thickness. Figure 9 illustrates the influence of the velocity slip parameter on the velocity profile . It is observed that by rising value of the parameter , the velocity profile along the velocity boundary thickness decrease throughout the flow. Actually, the increasing value of develops a resistance forces at the surface that becomes the reason to decline the velocity profile, while the effect of parameter is found opposite to the influence of the parameter on velocity profile of micropolar nanofluid that is shown in Figure 10. Likewise, influence of parameter on angular velocity profile is shown in Figure 11. The figure shows that any increment in rises the velocity of the bulk particles that possesses microrotations during the flow. Figure 12 indicates the impact of the parameter on angular velocity profile . It is seen that an increment in decelerate the angular velocity related to bulk particles. Figure 13 specifies the effect of the parameter on the angular velocity profile . The result of this figure shows that the angular velocity of bulk particles reduces during the flow. The influence of various physical parameters on temperature profiles of micropolar nanofluid are shown by Figures 14–19. Figure 14 indicates that any increment in the parameter rises the temperature of the micropolar nanofluid during flow. Against it, Figure 15 shows that the increasing rate of suction decreasing the thermal boundary thickness along the temperature profile of the micropolar nanofluid. Actually, this occurs due to rise in suction rate which raises the resistance force between the fluid layers and solid surface. Such sorts of results are observed in Figure 16, when the value of the Prandtl number (Pr) is increased. Actually, higher values of the Prandtl number decreases thermal diffusion of fluid particles which result in the decrease temperature of the fluid. Therefore, increasing the value of the decreases the temperature of the all types of the fluids during the flow. An increase in thermal radiative parameter raises temperature and boundary layer thickness during the flow of the micropolar nanofluid that is shown in Figure 17. Actually, the existence of the thermal radiation leads to enhancing the thermal boundary layer thicknesses. Practically, enlarging radiations yields a major enhancement in temperature of fluid. Figures 18 and 19 show the influence of rising the Brownian motion parameter and thermophoresis parameter on the temperature profilesrespectively. The results show that any increment in the mentioned parameters increases the temperature of the particular nanofluid and its boundary layer thicknesses. This takes place because of and increases the temperature gradient at the solid adjacent surface.
Figures 20–22 show the influence of the Brownian motion parameter , Schmitt number, and thermophoresis parameter on nanoparticle concentration profiles . Figure 20 shows that an increase in deceases the concentration of the nanoparticles in the micropolar nanofluid throughout the flow. Figure 21 shows the influence of the Schmitt number () on the concentration profile . This figure shows that the concentration of the nanoparticles decreases with increase in the . It is due to increasing the value of Sc which decreases the mass diffusion; the larger diffusion becomes a reason to decrease in the concentration field. Moreover, it also decreases the thickness of the concentration boundary layers, while influence of the looks opposite to the influence of Sc on the concentration profile, where increasing rate of the raises concentration of nanoparticles throughout the flow of micropolar nanofluid.
6. Conclusions
The effects of the various types of physical flow parameters on MHD mixed convection boundary layer flow, heat, and mass transfer of the micropolar nanofluid on exponentially vertical stretching/shrinking surface are studied here. The similarity solutions are achieved by applying similarity transformation from partial differential equations into the ordinary differential equation. The shooting method is implemented to obtain the numerical solution of the equations with variation of the used physical parameters. At various ranges of used parameters, triple solutions are obtained, so the stability is performed by help of the MATLAB through bvp4c. Effects of various used physical parameters such as the magnetic parameter, thermophoresis parameter, microrotation parameter, suction parameter, radiation parameter, Prandtl number, Brownian motion parameter, thermal slip parameter, and Schmidt number are examined, discussed, and also presented graphically. The main results of this study are described below: (1)Skin friction coefficient, couple stress, Nusselt, and Sherwood numbers are increasing for and decreasing for with increase in material parameter(2)Triple solutions are found at the different ranges of the flow parameters, and from stability analysis, it is found that only first solution is stable as well as physically feasible(3)The buoyancy parameter and high suction are the leading causes for existence of triple solutions in this problem(4)The velocity field of the micropolar nanofluid decelerates with increase in the suction parameter, magnetic parameter, and velocity slip parameter, while increases for increase in thermal convection parameter and stretching parameter (5)An increase in parameters, , , and raises the temperature profiles of the micropolar nanofluid, while it decrease with increase in and(6)Any increment in rises the concentration boundary layer thickness and its profile, while an increase in and decreases the concentration boundary layer thickness and its profiles
Nomenclature
| : | Cartesian coordinates (m) |
| : | Velocity components (m·s-1) |
| : | Shrinking and stretching velocity |
| : | Material parameter |
| : | Microrotation |
| : | Gravitational force |
| : | Temperature (K) |
| : | Ambient temperature |
| : | Variable temperature at sheet (K) |
| : | Dimensionless temperature (K) |
| : | Concentration (kg·m-3) |
| : | Ambient concentration (kg·m-3) |
| : | Variable concentration at sheet |
| : | Thermal expansion coefficients |
| : | Concentration expansion coefficients |
| : | Kinematic viscosity (m2·s-1) |
| : | Vortex viscosity (m2·s-1) |
| : | Spin gradient viscosity (m2·s-1) |
| : | Microinertia per unit mass (kg·m-3) |
| : | Stefan-Boltzmann constant |
| : | Thermal diffusivity (m2/s) |
| : | Stream function |
| : | Transformed variable |
| : | Radiative heat flux (w/m2) |
| : | Magnetic field (kg s-2A-1) |
| : | Thermal radiation |
| : | Magnetic parameter |
| Pr: | Prandtl number |
| : | Brownian diffusion (kg·m-1·s-1) |
| : | Thermophoretic diffusion (kg·m-1·s-1·K-1) |
| : | Suction/injection velocity (m·s-1) |
| : | Electrical conductivity |
| : | Thermophoresis parameter |
| : | Brownian motion parameter |
| : | Suction/injection parameter |
| : | Shrinking/stretching parameter |
| Sc: | Schmidt number |
| : | Thermal conductivity |
| : | Constant |
| : | Skin friction coefficient |
| : | Local Nusselt number |
| : | Mean absorption coefficient |
| : | Local Sherwood number |
| : | Local Reynolds number |
| : | Unknown eigenvalue |
| : | Smallest eigenvalue |
| : | Dynamic viscosity |
| : | Stability transformed variable. |
Data Availability
No data was needed to perform this research.
Conflicts of Interest
The authors declare that they have no conflict of interest.