Abstract
This research aims at providing the theoretical effects of the unsteady MHD stagnation point flow of heat and mass transfer across a stretching and shrinking surface in a porous medium including internal heat generation/absorption, thermal radiation, and chemical reaction. The fundamental principles of the similarity transformations are applied to the governing partial differential equations (PDEs) that lead to ordinary differential equations (ODEs). The transformed ODEs are numerically solved by the shooting algorithm implemented in MATLAB, and verification is done from MATLAB built-in solver bvp4c. The numerical data produced for the skin friction coefficient, the local Nusselt number, and the local Sherwood number are compared with the available result and found to be in a close agreement. The impact of involved physical parameters on velocity, temperature, concentration, and density of motile microorganisms profiles is scrutinized through graphs. It is analyzed that the skin friction coefficient enhances with increasing values of an unsteady parameter , magnetic parameter , and porosity parameter . In addition, we observe that the density of a motile microorganisms profile enhances larger values of the bioconvection Lewis number and Peclet number and decreases with the increasing values of an unsteady parameter .
1. Introduction
Nanofluids have been in demand because of its use in energy efficient devices due to its high performance contribution in thermal conductivity compared to a traditional fluid [1–3]. Nanofluids have recently been used in detergent, vehicle coolant, sensing in microelectromechanical systems (MEMS), and thermal energy storage [4]. Thus, it can be used in heating and electronic devices to make it more cost effective by minimization of energy lost in heat transfer process. There are a number of applications where nanofluids have been used such as in biomedical engineering, fluid power, mechanical and manufacturing industry, hydraulics, etc. The nanofluids are a composite solution containing nanoparticles and the base fluid [5].
The scope of nanofluid has been further enlarged by coalescing nanoparticles with blood to cultivate comprehension of biological sciences as well. Such a fluid is ordinarily known as bionanofluid. Recent applications of bionanofluid in medical sciences, such as medicine, cancer therapy, etc., have generated interest in investigating the bionanofluid flow. Moreover, the bionanofluid has instigated research in nanotechnology, biomedical engineering (applying biological in medical innovation), bioengineering (applying engineering principle to biology), and medical devices, etc.
Bioconvection is a process in which microorganisms convection occur in the fluid [6]. Khan and Makinde [7] investigated nanofluids in motile gyrotactic microorganisms. In [8], analytical solution of bioconvection of oxytactic bacteria was found. Mutuku and Makinde [9] discussed hydromagnetic bioconvection due to microorganisms and solution is obtained numerically. Recently, Naganthran et al. [10] applied extrapolation technique in time dependent bionanofluid. Zaimi et al. [11] discussed stagnation point flow not only containing nanoparticles but also gyrotactic microorganisms. Ali and Zaib [12] discussed unsteady flow of an Eyring-Powell nanofluid near a stagnation point. Zeng and Pedley [13] discussed gyrotactic microorganisms in complex three-dimensional flow. Shah et al. [14] have developed a fractional model in discussing a natural convection of bionanofluids between two vertical plates. Amirsom et al. [15] have discussed melting bioconvection nanofluid with second-order slip and thermal physical properties. Khader et al. [16] performed experimental study to determine the thermal and electrical conductivity to develop a new correlation in bionanofluid. For other details in this direction, see [17–22].
The thermal radiation plays an important role in industrial and engineering processes. Thermal radiation is a phenomenon in which energy is transported through indirect contact. Izadi et al. [23] discussed thermal radiation in a micropolar nanoliquid in a porous chamber. They applied the Galerkin finite element method to compute the numerical solution. Daniel et al. [24] presented a theory on entropy analysis for EMHD nanofluids considering thermal radiation and viscous dissipation. Muhammad et al. [25] obtained numerical solutions via the shooting method and bvp4c for the significant role nonlinear thermal radiation played in 3D Eyring-Powell nanofluid. Sohail et al. [26] described entropy analysis of Maxwell nanofluid in gyrotactic microorganisms with thermal radiation. Gireesha et al. [27] provide hybrid nanofluid flow across a permeable longitudinal moving fin with thermal radiation.
Eid [28] presents two-phase chemical reactions over a stretching sheet. Tripathy et al. [29] research chemical reactive flow over a moving vertical plate. In Pal and Talukdar [30], chemical reaction effects in a mixed convection flow have been covered. Katerina and Patel [31] reported results on radiation and chemical reaction in Casson fluid over an oscillating vertical plate. The works of Shah et al. [32], Rasool et al. [33], Khan et al. [34], and Khan et al. [35] contain chemical reactions as well as entropy generation over a nonlinear sheet. Khan et al. [36] present results on axisymmetric Carreau nanofluid along with chemical reaction. Gharami et al. [37] provide an unsteady flow of tangent nanofluid with a chemical reaction. Hamid et al. [38] simultaneously presented work on chemical reaction and activation energy in the unsteady flow of Williamson nanofluid. Reddy et al. [39] report results on nanofluid over a rotating disk with a chemical reaction. For other references on this topic, the reader is referred to [40–50].
In aforementioned literature studies, the chief emphasis has been made on various physical situations to find an in-depth understanding of physics but the route of bionanofluid along with other situations of unsteady effect in a free stream flow is mostly absent from the literature.
The paper is written in the following order. Introduction of the paper is given in Section 1. Problem formulation is presented in Section 2. Numerical method is presented in Section 3. The results and discussion of the work are discussed Section 4. Conclusion is drawn at the end in Section 5.
2. Problem Formulation
Assuming an unsteady two-dimensional MHD stagnation point flow of bionanofluid in the presence of thermal radiation, chemical reaction, and internal heat generation/absorption adjacent to a stretching sheet with thermal radiation, a water-based nanofluid containing nanoparticles and gyrotactic microorganisms is considered. It is assumed that the presence of nanoparticles has no effect on the swimming direction of microorganisms and on their swimming velocity. This assumption holds only for less than concentration of nanoparticles. The magnetic Reynolds number of the flow is taken to be very small, so that the induced magnetic field is presumed to be negligible. The applied magnetic field is taken along the normal to the sheet. It is also assumed that the sheet is stretching/shrinking with a velocity , indicates the stretching sheet whereas describes the shrinking sheet while represents a stationary sheet. The configuration of the flow is given in Figure 1.
Under the above assumptions, the governing model of flow reads as follows [10, 51]:
However, the boundary conditions corresponding to the considered model is taken as follows:where is time, , are the velocity components in the and axes, respectively. Furthermore, is a temperature of the fluid, is the concentration, is the density of the motile microorganisms, is the porosity of a porous medium, is the dynamic velocity of the fluid, is the electrical conductivity of the fluid, is the density of the fluid, is the thermal diffusivity, is the specific heat capacity at constant temperature, is the ratio of the effective heat capacity of the nanoparticle and the base fluid, is the Brownian diffusion coefficient, is thermophoretic diffusion coefficient, is the diffusivity of the microorganisms, is the radiative heat flux, is the volumetric heat source, is called a rate of chemical reaction between the base fluid and nanoparticles, is the maximum cell swimming speed, and is the chemotaxis constant. Moreover, , , and are the temperature, nanoparticle concentration, and the density of the motile microorganisms at the wall and , , and are the temperature, nanoparticle concentration, and motile microorganisms far away from the sheet, respectively.
Introducing the similarity solutions as follows:
By inserting equation (7) into equations (1)–(5), we obtain the following transformed nonlinear ordinary differential equations:
Similarly, equations (7) reduces boundary condition (6) intowhere is an unsteadiness parameter, porous parameter , magnetic parameter , Prandtl number , thermal radiation parameter , Brownian motion parameter , thermophoretic parameter , Eckert number , heat source parameter , Lewis number , chemical reaction parameter , bioconvection Lewis number , Peclet number , and bioconvection parameter are defined as follows:
The physical quantities of interest in this study are the local skin friction coefficient , the local Nusselt number , the local Sherwood number , and the local density number of motile microorganisms are defined as follows:
Inserting equation (7) into equation (11) yields the following expressions:where the local Reynolds number is defined as .
3. Numerical Procedure
3.1. Shooting Method
The physical model of ODEs alongside boundary conditions quantitatively evaluated by the shooting method implemented in MATLAB. The shooting approach involves two stages: Converting the boundary value problem (BVP) into an initial value problem (IVP) and the higher-order ODEs into a system of first-order ODEs. We employed the Newton–Raphson approach in locating roots. The Runge–Kutta method of order five is implemented in determining the solution of the IVP. The system of first-order ODEs reads as follows:
The converted form of boundary conditions into an initial condition for the shooting method is rewritten as follows:
3.2. bvp4c
Having found numerical results from the shooting method, we verify these results using MATLAB built-in solver bvp4c [52, 53]. The bvp4c is a collocation solver which uses Gauss–Lobatto points to compute accurate results. In bvp4c, the first-order system of ODEs remains the same as discussed in Section 3.1. However, the boundary conditions implemented in MATLAB are as follows:
4. Results and Discussion
A summary of the current and the reported findings is seen with a minimal disparity in Table 1.
The data in Tables 2 and 3 show computational results for the skin friction coefficient, the local Nusselt number, the local Sherwood number, and the local density number of motile microorganisms obtained with the shooting method and the bvp4c. In Table 2, it is revealed that the skin friction coefficient increases with increasing values of unsteady parameter , magnetic parameter , and porosity parameter . However, decreasing trend is seen in the local Nusselt number against an unsteady parameter , radiation parameter , Brownian motion parameter , thermophoretic parameter , Eckert number , and heat source parameter . The local Nusselt number enhances the increasing values of Prandtl number . The local Sherwood number increases for higher values of Prandtl number , radiation parameter , Brownian motion parameter , Eckert number , heat source parameter , Lewis number , and chemical reaction parameter . The local Sherwood number decreases for higher values of thermophoretic parameter . For the local density number of motile microorganisms, shows decreasing trend for higher values of unsteady parameter and thermophoretic parameter is observed while it increases by enhancing the Prandtl number , radiation parameter , Brownian motion parameter , Eckert number , heat source parameter , Lewis number , and chemical reaction parameter . The local Sherwood number decreases for higher values of the thermophoretic parameter . For the local density number of motile microorganisms, shows decreasing trend for higher values of unsteady parameter and thermophoretic parameter is observed while it increases by enhancing the Prandtl number , radiation parameter , and Brownian motion parameter ,Eckert number , heat source parameter , Lewis number , chemical reaction parameter , the bioconvection Lewis number , and Peclet number .
In Figures 2 and 3, we present velocity profile results against parameters and with corresponding to shrinking and stretching sheets. In both cases, the boundary layer thickness decreases.
Figures 4–6 illustrate the impact of the Brownian motion parameter on the temperature, concentration, and the density of motile microorganisms profiles for the case of stretching sheet and shrinking sheet , respectively. Figure 4 gives an incremental thermal boundary layer thickness results as increases. The thermal boundary layer thickness for the Brownian motion parameter with the stretching sheet is lower than the shrinking sheet. From Figure 5, it is observed that by increasing the Brownian motion parameter , the concentration boundary layer thickness reduces in both stretching and shrinking sheet cases. Figure 6 exhibits that for higher values of the Brownian motion parameter , the density of motile microorganisms decreases. This decrease in the density of motile microorganisms is higher in the shrinking sheet case as compared to the stretching sheet case.
The impact of the thermophoresis parameter on temperature, concentration, and density of motile microorganisms can be seen in Figures 7–9. Figure 7 reveals that the thermal boundary layer thickness increases for larger values of the thermophoresis parameter . Figures 8 and 9 indicate that the concentration and density of motile microorganisms increases by increasing thermophoresis parameter , respectively.
Figure 10 depicts the behavior of a radiation parameter on the temperature profile. We observe that by increasing radiation parameter, thermal boundary layer thickness increases in both stretching and shrinking sheet cases.
Figure 11 characterizes the influence of Eckert number on temperature distribution. We conclude that increment in Eckert number enhances the temperature profile.
Figure 12 scrutinizes the impact of the heat source parameter on the temperature profile. It is seen that for higher values of the heat source parameter , the temperature profile increases.
Figure 13 examines the effect of the Prandtl number on the temperature profile. We analyzed that enhancement in Prandtl number causes a reduction in thermal boundary layer thickness.
Figure 14 is drawn to perceive the impact of bioconvection Lewis number on the density of motile microorganisms profile. It is observed that higher values of bioconvection Lewis number lower the boundary layer thickness of motile microorganisms profile.
Figure 15 represents the influence of the Peclet number on the density of motile microorganisms profile. It is validated the fact that increment in Peclet number causes a reduction in motile microorganisms boundary layer thickness.
Figures 16 and 17 portray the impact of the Lewis number and the chemical reaction on the concentration profile. It is analyzed that by increasing both the parameter Lewis number and chemical reaction , the concentration boundary layer thins.
Figure 18 depicts the skin friction coefficient against the porosity parameter with variations and . The skin friction seems to increase with the porosity parameter and with the increasing values of and .
5. Conclusions
The current analysis focuses on the unsteady MHD stagnation point flow of bionanofluid with internal heat generation/absorption in a permeable medium with thermal radiation and chemical reaction into account over a stretching and shrinking sheet. The significant findings of the problem are summarized as follows:(1)The skin friction coefficient enhances for higher values of the unsteady parameter , magnetic parameter , and porosity parameter .(2)The increment in the Brownian motion parameter , thermophoresis parameter , thermal radiation parameter , Eckert number , heat source parameter causes enhancement in thermal boundary layer thickness while an increase in Prandtl number causes a reduction in thermal boundary layer thickness.(3)The concentration boundary layer thickness increases for the thermophoresis parameter , whereas it decreases for higher values of the Brownian motion parameter , Lewis number , and chemical reaction parameter .(4)The increment of the Brownian motion parameter , bioconvection parameter , and Peclet number reduces the density of motile microorganisms while it increases for larger values of the thermophoresis parameter .(5)Different trends have been seen for boundary layer thickness through graphs. Graphs describe that boundary layer thickness is different in the stretching sheet case when compared to the shrinking sheet case.(6)The skin friction coefficient increases with the increase in porosity parameter as it can be seen through tables and graphical representation.
Nomenclature
| : | Positive constant () |
| (, ): | The velocity components () |
| (, ): | Cartesian coordinates (m) |
| : | Unsteadiness parameter |
| : | Dimensionless parameter |
| : | Applied magnetic field () |
| : | The coefficient of viscosity () |
| : | The density of fluid () |
| : | The electrical conductivity of the fluid () (S is siemens) |
| : | Magnetic parameter |
| : | Porosity parameter |
| : | Stretching/Shrinking parameter |
| : | Fluid temperature (K) |
| : | Constant temperature at wall (K) |
| : | The ambient fluid temperature (K) |
| : | The thermal conductivity () |
| : | The thermal diffusivity () |
| : | Mean absorption coefficient () |
| : | Stefan–Boltzman constant () |
| : | The specific heat capacity () |
| : | The radiative heat flux () |
| : | Rate of heat generation/absorption |
| : | Skin friction coefficient |
| : | Local Nusselt parameter |
| : | Local heat source/sink parameter |
| : | Thermal radiation parameter |
| : | Ambient Prandtl number |
| : | Brownian diffusion coefficient () |
| : | Thermophoretic diffusion coefficient () |
| : | Diffusivity of microorganisms () |
| : | Diffusivity coefficient () |
| : | Ratio of effective heat capacitance of the nanoparticle to the base fluid |
| : | Nanoparticle heat capacity |
| : | Brownian motion parameter |
| : | Thermophoresis parameter |
| : | The concentration |
| : | The concentration at the wall |
| : | The ambient fluid concentration |
| : | The concentration of microorganisms |
| : | Microorganisms at the wall |
| : | Microorganisms far from the wall |
| : | Lewis number |
| : | Bioconvection Lewis number |
| : | Chemotaxis constant (m) |
| : | Maximum cell swimming speed |
| : | Peclet number |
| : | Local Sherwood parameter |
| : | Local density parameter of the motile microorganisms. |
Data Availability
No experimental data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
ZHS would like to thank the Department of Mathematics, University of the Punjab, Lahore, for the partial support. AM would like to acknowledge the support of Mathematics Teaching and Learning, Research Groups within the Department of Mathematics, FLU, Nord University, Bodø.