# Unsteady MHD Bionanofluid Flow in a Porous Medium with Thermal Radiation near a Stretching/Shrinking Sheet

Authors: M. Irfan; M. Asif Farooq; A. Mushtaq; Z. H. Shamsi
Journal: Mathematical Problems in Engineering
 (2020)

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## 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 solverbvp4c. 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 A, magnetic parameter M, and porosity parameter Kp. In addition, we observe that the density of a motile microorganisms profile enhances larger values of the bioconvection Lewis number Lb and Peclet number Pe and decreases with the increasing values of an unsteady parameter A.


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## Body

## 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 Section1. 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 than1% 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 βo2 is taken along the normal to the sheet. It is also assumed that the sheet is stretching/shrinking with a velocity ue=ϵax1−A1t−1, ϵ>0 indicates the stretching sheet whereas ϵ<0 describes the shrinking sheet while ϵ=0 represents a stationary sheet. The configuration of the flow is given in Figure 1.Figure 1
Structural diagram of the flow problem.Under the above assumptions, the governing model of flow reads as follows [10, 51]:(1)∂u∂x+∂v∂y=0,(2)∂u∂t+u∂u∂x+v∂u∂y=ue∂ue∂x+∂ue∂t+ν∂2u∂y2−νk∗u−ue−σβo2ρu−ue,(3)∂T∂t+u∂T∂x+v∂T∂y=α∂2T∂y2+τ1DB∂T∂y∂C∂y+DTT∞∂T∂y2−1ρcp∂qr∂y+μρcp∂u∂y2+T−T∞Qρcp,(4)∂C∂t+u∂C∂x+v∂C∂y=DB∂2C∂y2+DTT∞∂2T∂y2−C−C∞Kc,(5)∂N∂t+u∂N∂x+v∂N∂y+bwcCw−C∞∂∂yN∂C∂y=Dn∂2N∂y2.However, the boundary conditions corresponding to the considered model is taken as follows:(6)t≤0:v=0,u=0,T=T∞,C=C∞,N=N∞,t>0:u=ϵuwx,t=ϵax1−A1t−1,withA1t≠1,v=0,T=Tw,C=Cw,N=Nwaty=0,u=uex,t=ax1−A1t−1,withA1t≠1,v=0,T=T∞,C=C∞,N=N∞asy⟶∞,where t is time, u, v are the velocity components in the x− and y− axes, respectively. Furthermore, T is a temperature of the fluid, C is the concentration, N is the density of the motile microorganisms, k∗ is the porosity of a porous medium, μ is the dynamic velocity of the fluid, σ is the electrical conductivity of the fluid, ρ is the density of the fluid, α is the thermal diffusivity, cp is the specific heat capacity at constant temperature, τ1 is the ratio of the effective heat capacity of the nanoparticle and the base fluid, DB is the Brownian diffusion coefficient, DT is thermophoretic diffusion coefficient, Dm is the diffusivity of the microorganisms, qr is the radiative heat flux, Q is the volumetric heat source, Kc is called a rate of chemical reaction between the base fluid and nanoparticles, Wc is the maximum cell swimming speed, and b is the chemotaxis constant. Moreover, Tw, Cw, and Nw are the temperature, nanoparticle concentration, and the density of the motile microorganisms at the wall and T∞, C∞, and N∞ are the temperature, nanoparticle concentration, and motile microorganisms far away from the sheet, respectively.Introducing the similarity solutions as follows:(7)η=aν1−A1ty,ψ=aν1−A1txfη,θη=T−T∞Tw−T∞,ϕη=C−C∞Cw−C∞,χη=N−N∞Nw−N∞.By inserting equation (7) into equations (1)–(5), we obtain the following transformed nonlinear ordinary differential equations:(8)f‴+ff″−f′2+1+A−Af′+η2f″−M+Kpf′−1=0,1+43Rdθ″+Prfθ′+Nbθ′ϕ′+Ntθ′2+PrEcf′′2+sθ−η2θ′A=0,ϕ″+NtNbθ′′+Le Prfϕ′−η2Le PrAϕ′−Le Pr Krϕ=0,χ″+Lb Prfχ′−Peϕ′χ′+χ+σ1ϕ″−η2Lb PrAχ′=0.Similarly, equations (7) reduces boundary condition (6) into(9)f0=0,f′0=ϵ,θ0=1,ϕ0=1,χ0=1,f′∞=1,θ∞=0,ϕ∞=0,χ∞=0,where A is an unsteadiness parameter, porous parameter Kp, magnetic parameter M, Prandtl number Pr, thermal radiation parameter Rd, Brownian motion parameter Nb, thermophoretic parameter Nt, Eckert number Ec, heat source parameter s, Lewis number Le, chemical reaction parameter Kr, bioconvection Lewis number Lb, Peclet number Pe , and bioconvection parameter σ1 are defined as follows:(10)A=A1a,Kp=ν1−A1tak∗,M=σBo21−A1tρa,Pr=να,Rd=4σT∞3k1k,Nb=τ1DBCw−C∞α,Nt=τ1DTTw−T∞T∞α,Ec=ue2cpTw−T∞,s=Q1−A1taρcp,Le=αDB,Kr=Kc1−A1ta,Lb=αDn,Pe=bwcDn,σ1=N∞Nw−N∞.The physical quantities of interest in this study are the local skin friction coefficientCfx, the local Nusselt number Nux, the local Sherwood number Shx , and the local density number of motile microorganisms Nnx are defined as follows:(11)Cfx=μ∂u/∂yy=0ρue2,Nux=−kx∂T/∂yy=0kTw−T∞,Shx=−DBx∂C/∂yy=0DBCw−C∞,Nnx=−Dnx∂N/∂yy=0DnNw−N∞.Inserting equation (7) into equation (11) yields the following expressions:(12)Rex1/2Cfx=f″0,Rex−1/2Nux=−1+43Rdθ′0,Rex−1/2Shx=−ϕ′0,Rex−1/2Nnx=−χ′0,where the local Reynolds number is defined as Rex=uex/ν.


## 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:(13)f=y1,f′=y2,f″=y3,f‴=y3′=−y1y3+y22−1−A+Ay2+η2y3+M+Kpy2−1,y4=θ,y5=θ′,θ″=y5′=−11+4/3RdPry1y5+Nby5y7+Nty52+PrEcy32+sy4−η2Ay5,y6=ϕ,y7=ϕ′,ϕ″=y7′=−NtNby5′−LePry1y7+LePrηA2y7+LePrKry6,y8=χ,y9=χ′,χ″=y9′=−LbPry1y9+Pey7y9+y8+σ1y7′+η2LbPrAy9.The converted form of boundary conditions into an initial condition for the shooting method is rewritten as follows:(14)y10=0,y20=ϵ,y40=1,y60=1,y80=1,y30=λ˜1,y50=λ˜2,y70=λ˜3,y90=λ˜4.


### 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:(15)y10=0,y20=ϵ,y40=1,y60=1,y80=1,y2∞=1,y4∞=0,y6∞=0,y8∞=0.


## 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:(13)f=y1,f′=y2,f″=y3,f‴=y3′=−y1y3+y22−1−A+Ay2+η2y3+M+Kpy2−1,y4=θ,y5=θ′,θ″=y5′=−11+4/3RdPry1y5+Nby5y7+Nty52+PrEcy32+sy4−η2Ay5,y6=ϕ,y7=ϕ′,ϕ″=y7′=−NtNby5′−LePry1y7+LePrηA2y7+LePrKry6,y8=χ,y9=χ′,χ″=y9′=−LbPry1y9+Pey7y9+y8+σ1y7′+η2LbPrAy9.The converted form of boundary conditions into an initial condition for the shooting method is rewritten as follows:(14)y10=0,y20=ϵ,y40=1,y60=1,y80=1,y30=λ˜1,y50=λ˜2,y70=λ˜3,y90=λ˜4.


## 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:(15)y10=0,y20=ϵ,y40=1,y60=1,y80=1,y2∞=1,y4∞=0,y6∞=0,y8∞=0.


## 4. Results and Discussion

A summary of the current and the reported findings is seen with a minimal disparity in Table1.Table 1
Comparison of the values off″0, −θ′0 and −ϕ′0 when ϵ=1, Le=2,M=Kp=A=Rd=Ec=s=Kr=Lb=Pe=0,Nt=Nb=0.5, and Pr=1.
Ibrahim et al. [51]Zaimi et al. [11]Naganthran et al. [10]Present result (SM)f″00000−θ′00.47670.4767370.4767370.4767−ϕ′01.04521.0451541.0451541.0452The data in Tables2 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 Cfx increases with increasing values of unsteady parameter A, magnetic parameter M, and porosity parameter Kp. However, decreasing trend is seen in the local Nusselt number Nux against an unsteady parameter A, radiation parameter Rd, Brownian motion parameter Nb, thermophoretic parameter Nt, Eckert number Ec, and heat source parameter s. The local Nusselt number enhances the increasing values of Prandtl number Pr. The local Sherwood number Shx increases for higher values of Prandtl number Pr, radiation parameter Rd, Brownian motion parameter Nb, Eckert number Ec, heat source parameter s, Lewis number Le, and chemical reaction parameter Kr. The local Sherwood number decreases for higher values of thermophoretic parameter Nt. For the local density number of motile microorganisms, Nnx shows decreasing trend for higher values of unsteady parameter A and thermophoretic parameter Nt is observed while it increases by enhancing the Prandtl number Pr, radiation parameter Rd, Brownian motion parameter Nb, Eckert number Ec, heat source parameter s, Lewis number Le , and chemical reaction parameter Kr. The local Sherwood number decreases for higher values of the thermophoretic parameter Nt. For the local density number of motile microorganisms, Nnx shows decreasing trend for higher values of unsteady parameter A and thermophoretic parameter Nt is observed while it increases by enhancing the Prandtl number Pr, radiation parameter Rd, and Brownian motion parameter Nb,Eckert number Ec, heat source parameter s, Lewis number Le, chemical reaction parameter Kr, the bioconvection Lewis number Lb, and Peclet number Pe.Table 2
Numerical values off″0,−θ′0, −ϕ′0, and −χ′0 for several values of the involved parameter A, M, Kp, Pr, Rd, Nb, Nt, Ec, s, Le, Kr, Lb, Pe with ϵ=0.5 and σ1=0.1 (shooting method (SM)).
SMSMSMSMAMKpPrRdNbNtEcsLeKrLbPef″0−1+4/3Rdθ′0−ϕ′0−χ′00.10.50.20.720.20.10.20.20.110.10.50.50.83640.51040.41080.58930.30.85760.46870.41090.56580.50.87840.42380.41210.54140.10.110.720.20.10.20.20.110.10.50.50.77490.50910.40820.58590.30.80620.50980.40950.58760.50.83640.51040.41080.58930.10.500.720.20.10.20.20.110.10.50.50.80620.50980.40950.58760.30.85120.51070.41150.59000.50.87990.51110.41270.59150.10.50.210.20.10.20.20.110.10.50.50.83640.58610.49190.692550.83641.10921.19711.5280100.83641.42711.76462.15530.10.50.20.7200.10.20.20.110.10.50.50.83640.43430.38120.57810.30.83640.54560.42320.59400.70.83640.67220.46110.60870.10.50.20.720.20.20.20.20.110.10.50.50.83640.48480.53860.64050.50.83640.41320.61380.67090.70.83640.36970.62720.67620.10.50.20.720.20.10.10.20.110.10.50.50.83640.52820.51160.62690.20.83640.51040.41080.58930.40.83640.47650.26850.54530.10.50.20.720.20.10.20.20.110.10.50.50.83640.51040.41080.58930.40.83640.48220.45180.61090.70.83640.43980.51320.64340.10.50.20.720.20.10.20.2010.10.50.50.83640.57010.33450.55080.10.83640.51040.41080.58930.20.83640.44680.49160.62990.10.50.20.720.20.20.20.10.70.10.50.50.83640.51500.26620.527810.83640.51040.41080.58931.30.83640.50730.52610.64020.10.50.20.720.20.10.20.20.110.10.50.50.83640.51040.41080.58930.40.83640.50750.61120.68610.80.83640.50470.82660.79180.10.50.20.720.20.10.20.110.10.50.50.83640.51040.41080.589310.83640.51040.41080.744620.83640.51040.41080.95780.10.50.50.83640.51040.41080.589310.83640.51040.41080.762930.83640.51040.41081.5188Table 3
Numerical values off″0,−θ′0, −ϕ′0, and −χ′0 for several values of involved parameter A, M, Kp, Pr, Rd, Nb, Nt, Ec, s, Le, Kr, Lb, Pe with ϵ=0.5 and σ1=0.1 (bvp4c).
bvp4cbvp4cbvp4cbvp4cAMKpPrRdNbNtEcsLeKrLbPef″0−1+4/3Rdθ′0−ϕ′0−χ′00.10.50.20.720.20.10.20.20.110.10.50.50.83640.51040.41080.58930.30.85760.46870.41080.56580.50.87840.42380.41210.54140.10.110.720.20.10.20.20.110.10.50.50.77490.50910.40820.58590.30.80620.50980.40950.58760.50.83640.51040.41080.58930.10.500.720.20.10.20.20.110.10.50.50.80620.50980.40950.58760.30.85120.51070.41150.59000.50.87980.51110.41270.59150.10.50.210.20.10.20.20.110.10.50.50.83640.58610.49190.692550.83641.10921.19711.5280100.83641.42711.76462.15530.10.50.20.7200.10.20.20.110.10.50.50.83640.43430.38120.57810.30.83640.54560.42320.59400.70.83640.67220.46110.60870.10.50.20.720.20.20.20.20.110.10.50.50.83640.48480.53870.64050.50.83640.41320.61380.67090.70.83640.36970.62720.67620.10.50.20.720.20.10.10.20.110.10.50.50.83640.52820.51160.62690.20.83640.51040.41080.58930.40.83640.47640.26850.54530.10.50.20.720.20.10.20.20.110.10.50.50.83640.51040.41080.58930.40.83640.48220.45180.61090.70.83640.43980.51320.64340.10.50.20.720.20.10.20.2010.10.50.50.83640.57010.33450.55080.10.83640.51040.41080.58930.20.83640.44670.49160.62990.10.50.20.720.20.20.20.10.70.10.50.50.83640.51500.26620.527810.83640.51040.41080.58931.30.83640.50730.52610.64020.10.50.20.720.20.10.20.20.110.10.50.50.83640.51040.41080.58930.40.83640.50750.61120.68610.80.83640.50470.82660.79180.10.50.20.720.20.10.20.110.10.50.50.83640.51040.41080.589310.83640.51040.41080.744620.83640.51040.41080.95780.10.50.50.83640.51040.41080.589310.83640.51040.41080.762930.83640.51040.41081.5189In Figures2 and 3, we present velocity profile results against parameters M and Kp with ϵ=−0.5,0.5 corresponding to shrinking and stretching sheets. In both cases, the boundary layer thickness decreases.Figure 2
Velocity profilef′η for different M.Figure 3
Velocity profilef′η for different Kp.Figures4–6 illustrate the impact of the Brownian motion parameter Nb on the temperature, concentration, and the density of motile microorganisms profiles for the case of stretching sheet ϵ=−0.5 and shrinking sheet ϵ=−0.5, respectively. Figure 4 gives an incremental thermal boundary layer thickness results as Nb 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 Nb, 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 Nb, 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.Figure 4
Temperature profileθη for different Nb.Figure 5
Concentration profileϕη for different Nb.Figure 6
Microorganisms profileχη for different Nb.The impact of the thermophoresis parameterNt 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 Nt. Figures 8 and 9 indicate that the concentration and density of motile microorganisms increases by increasing thermophoresis parameter Nt, respectively.Figure 7
Temperature profileθη for different Nt.Figure 8
Concentration profileϕη for different Nt.Figure 9
Microorganisms profileχη for different Nt.Figure10 depicts the behavior of a radiation parameter Rd on the temperature profile. We observe that by increasing radiation parameter, thermal boundary layer thickness increases in both stretching and shrinking sheet cases.Figure 10
Temperature profileθη for different Rd.Figure11 characterizes the influence of Eckert number Ec on temperature distribution. We conclude that increment in Eckert number Ec enhances the temperature profile.Figure 11
Temperature profileθη for different Ec.Figure12 scrutinizes the impact of the heat source parameter s on the temperature profile. It is seen that for higher values of the heat source parameter s, the temperature profile increases.Figure 12
Temperature profileθη for different s.Figure13 examines the effect of the Prandtl number Pr on the temperature profile. We analyzed that enhancement in Prandtl number Pr causes a reduction in thermal boundary layer thickness.Figure 13
Temperature profileθη for different Pr.Figure14 is drawn to perceive the impact of bioconvection Lewis number Lb on the density of motile microorganisms profile. It is observed that higher values of bioconvection Lewis number Lb lower the boundary layer thickness of motile microorganisms profile.Figure 14
Microorganisms profileχη for different Lb.Figure15 represents the influence of the Peclet number Pe on the density of motile microorganisms profile. It is validated the fact that increment in Peclet number Pe causes a reduction in motile microorganisms boundary layer thickness.Figure 15
Microorganisms profileχη for different Pe.Figures16 and 17 portray the impact of the Lewis number Le and the chemical reaction Kr on the concentration profile. It is analyzed that by increasing both the parameter Lewis number Le and chemical reaction Kr, the concentration boundary layer thins.Figure 16
Concentration profileϕη for different Le.Figure 17
Concentration profileϕη for different Kr.Figure18 depicts the skin friction coefficient against the porosity parameter Kp with variations A and M. The skin friction seems to increase with the porosity parameter and with the increasing values of A and M.Figure 18
The skin friction coefficient with variations of A and M.


## 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 parameterA, magnetic parameter M , and porosity parameter Kp.(2)
The increment in the Brownian motion parameterNb, thermophoresis parameter Nt, thermal radiation parameter Rd, Eckert number Ec, heat source parameter s causes enhancement in thermal boundary layer thickness while an increase in Prandtl number Pr causes a reduction in thermal boundary layer thickness.(3)
The concentration boundary layer thickness increases for the thermophoresis parameterNt , whereas it decreases for higher values of the Brownian motion parameter Nb, Lewis number Le , and chemical reaction parameter Kr.(4)
The increment of the Brownian motion parameterNb, bioconvection parameter Lb , and Peclet number Pe reduces the density of motile microorganisms while it increases for larger values of the thermophoresis parameter Nt.(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 parameterKP as it can be seen through tables and graphical representation.



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