Magnetohydrodynamic time-dependent bio-nanofluid flow in a porous medium with variable thermophysical properties

Research Article

Magnetohydrodynamic Time-Dependent Bio-Nanofluid Flow in a Porous Medium with Variable Thermophysical Properties

M. Irfan ,

M. Asif Farooq ,

A. Aslam,

A. Mushtaq ,

and Z. H. Shamsi

1Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, 44000, Islamabad, Pakistan

2Seksjon for Matematikk, Nord Universitet, 8026 Bodø, Norway

3Department of Mathematics, University of the Punjab, 54590 Lahore, Pakistan

Correspondence should be addressed to A. Mushtaq; asif.mushtaq@nord.no

Received 29 October 2020; Revised 12 March 2021; Accepted 29 March 2021; Published 14 April 2021

Academic Editor: Hafiz Muhammad Ali

Copyright © 2021 M. Irfan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this work, a theoretical model with a numerical solution is brought forward for a bio-nanofluid with varying fluid features over a slippery sheet. The partial differential equations (PDEs) involving temperature-dependent quantities have been translated into ordinary differential equations (ODEs) by using similarity variables. Numerical verifications have been done in three different methods: finite difference method, shooting method, and bvp4c. To figure out the influence of parameters on the flows, the graphs are plotted for the velocity, temperature, concentration, and microorganism curves. The boundary layer thickness of the mi- croorganism profile reduces with the Schmidt number and Peclet number. In addition to adding radiative heat flux, we added heat generation, rate of chemical reaction, and first-order slip. Adding these parameters brought new aspects to the underlying profiles.

Moreover, the obtained data of the skin friction coefficient, the local Nusselt number, the local Sherwood number, and the local density of motile microorganisms are tabulated against various parameters for the physical parameters. From the results, it is apparent that the local Nusselt number decreases with the Brownian and thermophoretic parameters. The data obtained for physical parameters have a close agreement with the published data. Finally, the graphs for slip conditions are significantly different when the comparison is drawn with no-slip condition.

1. Introduction

Thermal analysis has attracted attention from the scientific community because of its role in our daily lives. The ap- plications of heat transfer range from electrical devices and power plants to the heating and cooling devices, boiler, condenser, and evaporators within houses where efficiency of these devices plays a key role. The efficient devices not only reduce energy consumption but also give additional life to it.

Nanofluid is a mixture of a base fluid with 100 nm-size nanoparticles. Since the work by Choi and Eastman [1] on nanofluid, the research in this direction took a huge stride.

The thermal conductivity is significantly higher than that of the traditional fluids as it was reported in Lee et al. [2]. There are many applications in the field of nanofluids including

lubricants, automation, electronics, and biomedicine. For the list of references which took multiple paths considering nanofluid in their study, one is referred to in [3–7].

Bioconvection is another phenomenon which occurs due to the density difference of the fluid. Raees et al. [8] recorded the homotopy analysis method (HAM) solution for an un- steady bioconvection flow in a channel and showed that the velocity component decreases with the increase in time.

Uddin et al. [9] discussed bioconvection nanofluid over a wavy surface with slip flow in application to nano-biofuel cells. Khan and Makinde [10] explored bioconvection flow due to gyrotactic microorganisms. They noticed that, with rising the values of the convective variable, the dimensionless temperature on the surface rises. Uddin et al. [11] investigated Stefan blowing with multiple slip effects in bioconvection. For finding similarity transformation, they used Lie group

Volume 2021, Article ID 6666863, 16 pages https://doi.org/10.1155/2021/6666863

analysis. The resources for further study on this topic can be found in [12–16].

One of the ways through which heat transfer occurs is thermal radiation. It has diverse technological applications in combustion, furnace design, turbines, and solar collectors. The thermal radiation with variable fluid properties is reported in [17]. The author found that the skin friction coefficient in- creases with viscosity parameter. RamReddy and Naveen [18]

reported results for activation energy and thermal radiation.

Aziz et al. [19] discussed free convection flow in nanofluids with microorganisms. They discovered that the bioconvection parameter affects heat transfer rate. Mutuku and Makinde [20]

discussed hydromagnetic fluid flow in microorganisms. Sk et al. [21] presented multiple slip effects in the presence of microorganisms. Anwar et al. [22] discussed MHD flow in a porous channel with generalized conditions. For further study on this topic, one is referred to in [23–25].

The magnetic field has many applications including geothermal energy extractions, plasma studies, chemical engineering, and magnetic resonance imaging (MRI) equipment [26]. Ali et al. [27] discussed hybrid nanofluid with slip conditions for Jeffrey fluid. Mburu et al. [28] reported magnetic and thermal radiation effects over an inclined cylinder. Mabood et al. [29] combined electrical and magnetic flows for non-Newtonian nanofluids over a thin needle.

The viscosity of the fluid generally depends on pressure and temperature. However, less effect in fluid flow is observed with pressure. Therefore, viscosity is dependent on the temperature variation. Fatunmbi and Adeniyan [30] reported nonlinear thermal radiation in fluid flow with variable properties.

Dandapat et al. [31] discussed thin film unsteady flow with variable fluid properties. Vajravelu et al. [32] discussed un- steady convective flow in a vertical surface with variable fluid properties. Shahsavar et al. [33] investigated the impact of variable fluid properties in hybrid nanofluid. Naganthran et al.

[34] found results of the stretching and shrinking sheet with

variable fluid properties. They discussed dual solutions in this rotating disk. Salahuddin et al. [35] discussed variable fluid properties for viscoelastic fluid between two rotating plates.

Shafiq et al. [36] presented second-grade bioconvective nanofluid flow and computed the solution from the shooting method. In another study, Rasool and Shafiq [37] discussed Powell–Eyring nanofluid flow in a porous medium over a nonlinear surface. The porosity factor is enhancing the drag force. In another work by Shafiq et al. [38], the numerical solution of the bioconvective tangent hyperbolic nanofluid was found. The effect of temperature-dependent viscosity, thermal radiation, and gyrotactic phenomenon in a convection flow over a cylinder has been discussed in [39–41], respectively. In another work by Khan et al. [42], the bioconvection flow was discussed in a truncated cone. For critical review about nanofluid and its effects on viscosity along with thermal con- ductivity, the reader is referred to in [43–46]. For experimental investigation on nanofluids, the reader is referred to in [47].

Most theoretical studies mentioned above are focused on the idea of constant fluid properties in fluid flows. The vis- cosity of a fluid, however, relies heavily on temperature than on other factors, such as pressure. It comes out that the use of variable properties offers distinct effects on fluid flow motion.

This paper is ordered in the following way: the flow model is presented in Section 2. The numerical procedure for the so- lution is presented in Section 3. Results and discussion are given in Section 4. Conclusion of the paper is drawn in Section 5.

2. Flow Model

Consider the movement of a nanofluid containing gyrotactic microorganisms past a stretching sheet with variable physical properties. The magnetic fieldβ²_ois applied normal to the surface. Due to low magnetic Reynolds number, the induced magnetic field is assumed negligible. The stretching velocity isU_w�ax(1− A₁t)⁻¹. The governing model is [48]

zu􏽢₁ zx +zv􏽢₁

zy �0, (1)

zu􏽢₁

zt +u􏽢₁zu􏽢₁

zx +v􏽢₁zu􏽢₁ zy � 1

ρ_∞ z

zy μ􏼐T􏽢₁􏼑zu􏽢₁

􏼠 zy􏼡− σβ²₀

ρ_∞u􏽢₁− μ(T)􏽢

ρ_∞k^∗u􏽢₁, (2)

zT􏽢₁

zt +u􏽢₁zT􏽢₁

zx +v􏽢₁zT􏽢₁ zy � 1

ρ_∞c_p z

zy k􏼐T􏽢₁􏼑zT􏽢₁

􏼠 zy􏼡+τ₁ D_B(c)zT􏽢₁ zy

zC􏽢₁ zy +

DT􏽢₁ T_∞

zT􏽢₁

􏼠zy􏼡

2

⎛⎝ ⎞⎠

− 1 ρ∞c_p

zq_r

zy+μ􏼐T􏽢₁􏼑 ρ∞c_p

zu􏽢₁

􏼠zy􏼡

2

+

T􏽢₁− T_∞

􏼐 􏼑Q

ρ∞c_p +σB²₀u􏽢₁²

ρ∞c_p +μ􏼐T􏽢₁􏼑 􏽢u₁² c_pk^∗ ,

(3)

zC􏽢₁

zt +u􏽢₁zC􏽢₁

zx +v􏽢₁zC􏽢₁ zy � z

zy D_B􏼐C􏽢₁􏼑zC􏽢₁

􏼠 zy􏼡+

DT􏽢₁ T_∞

z²T􏽢₁

zy² − 􏼐C􏽢₁− C_∞􏼑K_c, (4) zN􏽢₁

zt +u􏽢₁zN􏽢₁

zx +v􏽢₁zN􏽢₁

zy + bw_c C_w− C_∞

z zy

N􏽢₁zC􏽢₁

􏼠 zy􏼡

􏼠 􏼡� z

zy D_m􏼐C􏽢₁􏼑zN􏽢₁

􏼠 zy􏼡, (5)

and the boundary condition corresponding to the consid- ered model is taken as

􏽢

u₁�U_w(x, t) +N₁zu􏽢₁

zy, v�0, T􏽢₁�T_w(x, t) +D₁zT􏽢₁

zy, C􏽢₁�C_w,N􏽢₁�N_w, aty�0,

􏽢

u₁⟶0,T􏽢₁⟶T_∞,

C􏽢⟶C_∞,N􏽢₁⟶N_∞, asy⟶ ∞,

(6)

where all the variables are defined in the glossary.

The similarity variables are defined as

η�

���������

a ] 1− A₁t􏼁

􏽳

y,

ψ�

������

a]

1− A₁t

􏽲

xf(η),

θ(η) �T􏽢₁− T_∞ T_w− T_∞, ϕ(η) �C􏽢₁− C_∞

C_w− C_∞, χ(η) �N􏽢₁

N_w.

(7)

Inserting equation (7) into equations (1)–(6), we get

μ􏼐T􏽢₁􏼑 μ_∞ f″

⎛⎝ ⎞⎠^′− f′²+ff″− A f′+η 2f″

􏼒 􏼓− M+Kp μ(T)􏽢 μ_∞

􏼠 􏼡

􏼠 􏼡f′�0, (8)

k􏼐T􏽢₁􏼑 k_∞ θ′

⎛⎝ ⎞⎠^′+4

3Rdθ″+Nb D_B(C)􏽢 D_B,∞

􏼠 􏼡θ′ϕ′+Ntθ′²

+Pr_∞ fθ^′− ηA

2 θ^′+Ec μ􏼐T􏽢₁􏼑 μ_∞

⎛

⎝ ⎞⎠f′′²+MEcf^′2+EcKp μ(T)􏽢 μ_∞

􏼠 􏼡f′²+sθ

⎛

⎝ ⎞⎠�0,

(9)

D_B(C)􏽢 D_B,∞ ϕ^′

􏼠 􏼡^′+ Nt

Nbθ^″+Sc fϕ^′− Aη

2 ϕ^′− Krϕ

􏼒 􏼓�0, (10)

D_m(C)􏽢 D_m,∞χ^′

􏼠 􏼡^′+Sb fχ^′− Aη 2 χ^′

􏼒 􏼓− Pe ϕ^′χ^′+χϕ^″􏼁�0, (11)

f(0) �0,

f′(0) �1+δf′′(0), θ(0) �1+cθ^′(0), ϕ(0) �1,

χ(0) �1, f′(∞) �0, θ(∞) �0, ϕ(∞) �0, χ(∞) �0.

(12)

Following Amirsom et al. [48], the physical quantities consisting of viscosity, thermal conductivity, nanoparticle, and microorganism diffusivities are written as

μ􏼐T􏽢₁􏼑�μ_∞􏽨1+h₁􏼐T_∞− T􏽢₁􏼑􏽩�μ_∞ 1+h₂− θh₂􏼁, k􏼐T􏽢₁􏼑�k_∞􏽨1+h₃􏼐T_∞− T􏽢₁􏼑􏽩�k_∞ 1+h₄θ􏼁, D_B􏼐C􏽢₁􏼑�D_B,∞􏽨1+h₅􏼐C􏽢₁− C_∞􏼑􏽩�D_B,∞ 1+h₆ϕ􏼁, D_m􏼐C􏽢₁􏼑�D_m,∞􏽨1+h₇􏼐C􏽢₁− C_∞􏼑􏽩�D_m,∞ 1+h₈ϕ􏼁.

(13)

Equation (13) when used into equations (8)–(11), one can get

1+h₂− h₂θ􏼁f^‴− h₂θ^′f″− f′²+ff″− A f′+η 2f″

􏼒 􏼓− M+Kp 1+h₂− h₂θ􏼁􏼁f′�0, 1+h₄θ+4

3Rd

􏼒 􏼓θ^″+h₄θ^′2+Nb 1+h₆ϕ􏼁θ^′ϕ^′+N_tθ^′2 +Pr∞ fθ^′− Aη

2 θ^′+Ec 1+h₂− θh₂􏼁f′′²+MEcf^′2+KpEc 1+h₂− h₂θ􏼁f′²+sθ

􏼒 􏼓�0,

1+h₆ϕ􏼁ϕ^″+h₆ϕ^′2+Sc fϕ^′− Aη

2ϕ^′− Krϕ

􏼒 􏼓+Nt

Nbθ^″�0, 1+h₈ϕ􏼁χ^″+h₈ϕ^′χ^′+Sb fχ^′− Aη

2 χ^′

􏼒 􏼓− Pe ϕ^′χ^′+χϕ^″􏼁�0.

(14)

All these parameters are grouped into

A�A₁ a,

Kp�]_∞ 1− A₁t􏼁

ak^∗ , M�σB²_o 1− A₁t􏼁

ρ_∞a , Pr_∞�]_∞

α_∞, Rd�4σT³_∞

k₁k_∞,

Nb�τ₁D_B,∞ C_w− C_∞􏼁

α_∞ ,

Nt�τ₁D_T T_w− T_∞􏼁 T_∞α , Ec� u²_w

c_p T_w− T_∞􏼁, s�Q 1− A₁t􏼁

a , Sc� ]_∞

D_B,∞, K_r �K_c 1− A₁t􏼁a,

Sb� ]_∞ D_m,∞, Pe� bwc

D_m,∞, δ�N₁ a

]_∞ 1− A₁t􏼁

􏼠 􏼡

(1/2)

,

c�D₁ a ]_∞ 1− A₁t􏼁

􏼠 􏼡

(1/2)

.

(15)

The physical quantities of the interest in this study are the local skin friction coefficient C_fx, the local Nusselt number Nux, the local Sherwood number Shx, and the local density number of motile microorganisms Nnx defined as

C_fx�μ(T)(zu/zy)_y�0 ρu²_w , Nux�−k(T)x(zT/zy)_y�0

k(T) T_w− T_∞􏼁 , Shx�−D_B,∞x(zC/zy)_y�0 D_B,∞ C_w− C_∞􏼁 , Nnx�−D_m,∞x(zN/zy)_y�0

D_m,∞N_w

􏼐 􏼑 .

(16)

Inserting equation (7) into equation (13) yields the following expressions:

Re^1/2_x C_fx� − 1+h₂ϕ􏼁f″(0), Re⁻_x^1/2Nux� − 1+4

3Rd

􏼒 􏼓θ′(0),

Re⁻_x^1/2Shx� −ϕ′(0),

Re⁻x^1/2Nnx� −χ^′(0),

(17)

where the local Reynolds number is defined as Rex� (U_wx/]).

3. Numerical Process

3.1. Shooting Method. A boundary value problem ((8)–(12)) can be solved with the shooting method. The stable iterative scheme, Newton–Raphson method, has been used in lo- cating the roots followed by obtaining the solution from the fifth-order Runge–Kutta solver. The system of first-order ODEs is

f�y₁, f′�y₂, f′′�y₃, f^‴�y₃′� 1

1+h₂− y₄h₂􏼁􏼐h₂y₅y₃+y²₂− y₁y₃􏼑A y₂+η 2y₃

􏼒 􏼓+ M+Kp 1+h₂− h₂y₄􏼁y₂􏼁, y₄�θ, y₅�θ′,θ″�y₅′� −1

1+h₄y₄+(4/3)Rd􏼁

· h₄y²₅+Nb 1+h₆y₆􏼁y₅y₇+Nty²₅+Pr_∞ y₁y₅− η

2Ay₅+Ec 1+h₂− y₂y₄􏼁y²₃+sy₄+MEcy²₂+KpEc 1+h₂− h₂y₄􏼁y²₂

􏼒 􏼓,

􏼒

y₆�ϕ, y₇�ϕ^′,ϕ^″�y₇′

� −1 1+h₆y₆􏼁

N_t

N_by₅′+h₆y²₇+Scy₁y₇+ScηA

2 y₇− ScKry₆

􏼠 􏼡,

y₈�χ, y₉�χ^′,χ^″�y₉′� −1

1+h₈y₆􏼁 h₈y₇y₉− Sb ηA

2y₉− y₁y₉

􏼒 􏼓− Pe y₇y₉+y₈y₇′􏼁

􏼒 􏼓.

(18)

The results’ verification is achieved from the bvp4c solver. For details on bvp4c, the reader is referred to in [49].

3.2. Finite Difference Method. In this section, we present the finite difference method to solve boundary value problem (8)–(12). The spatial discretization is given by first defining f′�Fin the momentum equation:

1+h₂− h₂θi􏼁 F_i+2− 2F_i+1+F_i (Δη)²

􏼠 􏼡− h₂θ′i F_i+1− F_i

􏼠 Δη 􏼡− F²_i+f_i F_i+1− F_i

􏼠 Δη 􏼡

− A F_i+η 2

F_i+1− F_i

􏼠 Δη 􏼡

􏼠 􏼡− M− Kp 1+h₂− h₂θ_i􏼁􏼁F_i�0,

· 1+h₄θi+4 3Rd

􏼒 􏼓 θ_i+2− 2θ_i+1+θi

(Δη)²

􏼠 􏼡+ h₄+Nt􏼁 θ_i+1− θi

􏼠 Δη 􏼡

2

+Nb 1+h₆ϕ_i􏼁 θ_i+1− θi

􏼠 Δη 􏼡 ϕ_i+1− ϕi

􏼠 Δη 􏼡+Pr_∞ f_i θ_i+1− θi

􏼠 Δη 􏼡− Aη 2

θ_i+1− θi

􏼠 Δη 􏼡+Ec 1+h₂− h₂θi􏼁

􏼠

· F_i+1− F_i

􏼠 Δη 􏼡

2

+MEcF²_i+KpEc 1+h₂− h₂θi􏼁F²_i+sθi�0,

1+h₆ϕi􏼁 ϕ_i+2− 2ϕ_i+1+ϕi

(Δη)²

􏼠 􏼡+h₆ ϕ_i+1− ϕi

􏼠 Δη 􏼡

2

+Sc f_i ϕ_i+1− ϕi

􏼠 Δη 􏼡− Aη 2

ϕ_i+1− ϕi

􏼠 Δη 􏼡− Krϕi+Nt Nb

θ_i+2− 2θ_i+1+θi

(Δη)²

􏼠 􏼡�0,

􏼠

· 1+h₈ϕi􏼁 χ_i+2− 2χ_i+1+χi

(Δη)²

􏼠 􏼡+h₈ ϕ_i+1− ϕi

􏼠 Δη 􏼡 χ_i+1− χi

􏼠 Δη 􏼡+Sb f_i− Aη

􏼒 2􏼓 χ_i+1− χi

􏼠 Δη 􏼡

− Pe ϕ_i+1− ϕi

􏼠 Δη 􏼡 χ_i+1− χ_i

􏼠 Δη 􏼡+χ_i ϕ_i+2− 2ϕ_i+1+ϕi

(Δη)²

􏼠 􏼡

􏼠 􏼡�0,

(19)

and the boundary conditions are

f₀�0,

F₀�1+δ F₁− F₀

􏼠 Δη 􏼡,

θ₀�1+c θ₁− θ₀

􏼠 Δη 􏼡,

ϕ₀�1,χ₀�1, F_∞�0,θ_∞�0,ϕ_∞�0,χ_∞�0. (20)

4. Results and Discussion

An excellent agreement with published results is obtained for a comparison of the skin friction coefficient −f″(0) which is shown in Tables 1–3.

The data in Table 4 show computational results for the local Nusselt number, the local Sherwood number, and the local density number of motile microorganisms obtained with bvp4c. The local Nusselt number Nuxis reduced against Brownian motion parameter Nb, thermophoretic parameter Nt, Eckert number Ec, heat source parameters, and thermal conductive parameterh₄.

With increasing values of Prandtl number Pr∞ and radiation parameter Rd, the local Nusselt number shows an upward trend.

The physical parameter, the local Sherwood number Shx, depicts an upward trend against Brownian motion pa- rameter Nb, thermophoretic parameter Nt, Schmidt number Sc, and chemical reaction parameter Kr. However, a de- creasing trend for the local Sherwood number is observed for rising values of mass diffusivity parameterh₆.

Finally, the values of the local density number of motile microorganisms Nnx decline with the increase of mass diffusivity parameterh₆ and microorganism diffusivity pa- rameter h₈. However, there is an upsurge for increasing values of the bioconvection Schmidt number Sb and Peclet number Pe.

Figures 1 and 2 illustrate the effects of magnetic pa- rameterMand porosity parameter Kp on the velocity profile with and without hydrodynamic slip. The boundary layer thickness reduces with increasing values ofMand Kp. When fluid flow encounters the Lorentz forces, the velocity of the fluid decelerates which affects the boundary layer thickness.

The same argument holds for Kp.

Figure 3 is plotted to perceive the effect of Prandtl number Pr_∞on the temperature profile. It is noted that an enhancement in Prandtl number Pr_∞ causes reduction in the temperature distribution. The smaller values of Pr_∞ correspond to the increase in thermal conductivities which causes reduction in a thermal boundary layer. For Prandtl number (Pr≥1), the momentum diffusivity is dominant in fluid behavior. Thus, less thermal diffusivity contributes to lowering the thermal boundary layer thickness.

Figure 4 depicts the influence of radiation parameter Rd on the temperature profile. It is seen that an increase in Rd enhances the temperature of the fluid. Larger values of

radiation parameter transfer more heat to the fluid which overall increases the temperature and its profile.

Figure 5 reports the influence of Eckert number Ec on the temperature profile. The higher values of Eckert number Ec cause an increase in the thermal boundary layer thick- ness. The Eckert number Ec enhances kinetic energy, which increases fluid’s temperature.

Figure 6 illustrates the impact of heat source parameters on the temperature distribution. It is observed that tem- perature of the fluid increases with an increment in the heat generation parameter. The higher values ofsprovide more heat to the fluid resulting in the rise of the temperature of the fluid.

Figure 7 examines the effect of temperature-dependent thermal conductivity parameter h₄ on temperature. It is noted that the thermal boundary layer thickness increases by increasing parameter h₄.

Figures 8 and 9 are drawn to perceive the effect of Brownian motion parameter Nb on the temperature and concentration profiles. It is revealed in the figure that, by increasing Brownian motion parameter Nb, thermal boundary layer thickness rises, while concentration boundary layer thickness declines. The Brownian parameter appears due to the presence of nanoparticles’ concentration.

Figures 10 and 11 convey the impacts of thermophoresis parameter Nt on temperature and concentration distribu- tions. The temperature and concentration profile rise for rising values of Nt. The thermophoresis term appears due to the temperature gradient in particulate flows. Larger values of Nt transmit more temperature to the fluid along with the concentration profile.

Figure 12 portrays the influence of chemical reaction parameter Kr on the concentration profile. The rising values of Kr suppress diffusion which lowers the concentration boundary layer.

Figure 13 depicts the effects of Schmidt number Sc on the concentration distribution. The rise in Sc causes reduction in the concentration profile. The higher the Schmidt number, the lower the mass diffusivity which is the reason for re- duction in the concentration boundary layer thickness.

Figure 14 presents the influence of mass diffusivity parameterh₆on the concentration profile. One can observe that rise in mass diffusivity parameter h₆ results in an in- crease of the concentration profile.

Figure 15 describes the influence of Peclet number Pe on the density of motile microorganism profile. The incre- mental values of Peclet number Pe cause reduction in motile microorganisms’ boundary layer thickness. The Peclet number appears in the study of transport processes. It measures the importance of convection over diffusion. For larger values of the Peclet number, the convection is dominant and diffusion is negligible which is happening here in the motile microorganisms’ boundary layer thickness.

Figure 16 investigates the impact of bioconvection Schmidt number Sb on the density of motile microorganism profile. It is shown that rising values of bioconvection Schmidt number Sb lower the boundary layer thickness of

the motile microorganism profile. In high values of Sb, the particles are giant which means these diffuse slowly.

Figures 17 and 18 are drawn to perceive the effect of mass diffusivity parameter h₆ and microorganism diffusivity

parameterh₈. Increasing the values of mass diffusivity pa- rameter and microorganism diffusivity parameter elevates the boundary layer thickness of the motile microorganism profile.

Table1: Comparison of skin friction coefficient−f″(0)for different values ofMwhen Pr_∞�1 and Kp�δ�c�h₂�h₄�h₆�h₈�0.

M Hayat et al. [50] Mabood and Mastroberardino [51] Amirsom et al. [48] Shooting method bvp4c

0 1.0000 1.000008 1.0000002 1.0000 1.0001

1 1.41421 1.4142135 1.41422211 1.4142 1.4142

5 2.44948 2.4494897 2.4494901 2.4495 2.4495

10 3.31662 3.3166247 3.3166229 3.3166 3.3166

50 7.14142 7.1414284 7.1414279 7.1414 7.1414

100 10.04987 10.049875 10.049868 10.0499 10.0499

500 22.38302 22.383029 22.383031 22.3830 22.3830

1000 31.63858 31.638584 31.638578 31.6386 31.6386

Table2: Comparison of skin friction coefficient−f″(0)for different values ofMwhen Pr_∞�1 and Kp�δ�c�h₂�h₄�h₆�h₈�0.

M Hayat et al. [50] Mabood and Mastroberardino [51] Amirsom et al. [48] FDM

0 1.0000 1.000008 1.0000002 1.0001

1 1.41421 1.4142135 1.41422211 1.4142

5 2.44948 2.4494897 2.4494901 2.4495

10 3.31662 3.3166247 3.3166229 3.3166

50 7.14142 7.1414284 7.1414279 7.1414

100 10.04987 10.049875 10.049868 10.0499

500 22.38302 22.383029 22.383031 22.3830

1000 31.63858 31.638584 31.638578 31.6386

Table3: Comparison of skin friction coefficient−f″(0)for different values ofδwhen Pr_∞�1 and Kp�M�c�h₂�h₄�h₆�h₈�0.

δ Andersson [52] Hamad et al. [53] Amirsom et al. [48] Shooting method bvp4c

0 1.0000 1.00000000 1.00000000 1.0000 1.0001

0.1 0.8721 0.87208247 0.87204247 0.8721 0.8722

0.2 0.7764 0.77637707 0.77593307 0.7764 0.7765

0.5 0.5912 0.59119548 0.59119589 0.5912 0.5913

1.0 0.4302 0.43015970 0.43016000 0.4302 0.4303

2.0 0.2840 0.28397959 0.28398932 0.2840 0.2841

5.0 0.1448 0.14484019 0.14464015 0.1448 0.1449

10.0 0.0812 0.08124198 0.08124091 0.0812 0.0813

20.0 0.0438 0.04378834 0.04378790 0.0438 0.0439

50.0 0.0186 0.01859623 0.01857868 0.0186 0.0186

100.0 0.0095 0.00954997 0.00954677 0.0095 0.0096

Table4:NumericalvaluesofNux,Shx,andNnxforseveralvaluesofinvolvedparametersPr∞,Rd,Nb,Nt,Ec,s,Sc,Kr,Sb,Pe,h2,h4,h6,andh8withA�0.1,M�0.5,Kp�0.2,δ�0,and c�1(bvp4c). Pr∞RdsEcNbNtScKrSbPeh2h4h6h8bvp4cbvp4cbvp4c −(1+(4/3)Rd)θ′(0)−ϕ′(0)−χ′(0) 40.50.10.10.10.250.2210.10.10.10.10.45781.63062.0202 50.51321.61232.0097 6.80.57451.59482.0020 6.810.10.10.10.250.2210.10.10.10.10.72761.60182.0002 1.50.83621.61652.0072 20.90541.63302.0171 6.80.500.10.10.250.2210.10.10.10.10.67711.47371.8969 0.10.57451.59472.0020 0.20.41831.77132.1544 6.80.50.10.10.10.250.2210.10.10.10.10.57451.59472.0020 0.150.44681.75802.1461 0.20.31981.91982.2889 6.80.50.10.10.50.250.2210.10.10.10.10.49741.64522.0171 10.40001.65052.0179 1.50.30401.65142.0170 6.80.50.10.10.1150.2210.10.10.10.10.50801.76462.3215 1.50.46262.15632.7886 20.41412.78113.4623 6.80.50.10.10.10.230.2210.10.10.10.10.57801.10721.6118 50.57451.59472.0020 100.57122.44382.7096 6.80.50.10.10.10.250210.10.10.10.10.57631.14911.6338 0.50.57292.07652.4069 10.57162.65842.9040 6.80.50.10.10.10.250.20.510.10.10.10.10.57451.59471.6129 10.57451.59471.7596 30.57451.59472.2000 6.80.50.10.10.10.250.220.50.10.10.10.10.57451.59471.3577 10.57451.59472.0020 30.57451.59474.6914 6.80.50.10.10.10.250.2210.10.10.10.10.57451.59472.0020 0.50.55731.63362.0454 0.90.53861.66902.0830 6.80.50.10.10.10.25210.10.10.10.10.57451.59472.0020 0.50.54931.61012.0127 0.90.52581.62292.0213 6.80.50.10.10.10.250.2210.10.10.10.10.57451.59472.0020 0.50.57151.27881.7325 0.90.56861.08451.5698 6.80.50.10.10.10.250.2210.10.10.10.10.57451.59472.0020 0.50.57451.59471.5143 0.90.57451.59471.2232

f′ (η)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.5 1 1.5 2 2.5 3 3.5

0

η δ = 0 (M = 0)

δ = 0 (M = 0.5) δ = 0 (M = 1)

δ = 1 (M = 0) δ = 1 (M = 0.5) δ = 1 (M = 1) Pr_∞ = 6.8, Kp = Nt = Kr = 0.2, Rd = 0.5, γ = 1, s = A = Nb = Ec = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure1: Velocity profilef′(η)for differentM.

f′ (η)

0.5 1 1.5 2 2.5 3 3.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δ = 0 (M = 0) δ = 0 (M = 1.5) δ = 0 (M = 3)

δ = 1 (M = 0) δ = 1 (M = 1.5) δ = 1 (M = 1) Pr_∞ = 6.8, Nt = Kr = 0.2, M = Rd = 0.5, γ = 1, s = A = Nb = Ec = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure2: Velocity profilef′(η)for different Kp.

θ (η)

1 2 3 4 5 6

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Pr_∞ = 4) γ = 0 (Pr∞ = 5) γ = 0 (Pr∞ = 6.8)

γ = 1 (Pr∞ = 4 γ = 1 (Pr∞ = 5) γ = 1 (Pr∞ = 6.8) Kp = Nt = Kr = 0.2, M = Rd = 0.5, δ = 0, s = A = Nb = Ec = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure3: Temperature profileθ(η)for different Pr_∞.

θ (η)

1 2 3 4 5 6

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Rd = 1) γ = 0 (Rd = 1.5) γ = 0 (Rd = 2)

γ = 1 (Rd = 1) γ = 1 (Rd = 1.5) γ = 1 (Rd = 2) Pr_∞ = 6.8, Kp = Nt = Kr = 0.2, M = 0.5, δ = 0, s = A = Nb = Ec = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure4: Temperature profileθ(η)for different Rd.

θ (η)

1 2 3 4 5 6

0

η 0

0.2 0.4 0.6 0.8 1 1.2 1.4

γ = 0 (Ec = 0.1) γ = 0 (Ec = 0.3) γ = 0 (Ec = 0.5)

γ = 1 (Ec = 0.1) γ = 1 (Ec = 0.3) γ = 1 (Ec = 0.5) Pr_∞ = 6.8, Kp = Nt = Kr = 0.2, Rd = M = 0.5, δ = 0, s = A = Nb = Ec = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure5: Temperature profileθ(η)for different Ec.

θ (η)

0.5 1 1.5 2 2.5 3 3.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (s = 0) γ = 0 (s = 0.1) γ = 0 (s = 0.2)

γ = 1 (s = 0) γ = 1 (s = 0.1) γ = 1 (s = 0.2) Pr_∞ = 6.8, Kp = Nt = Kr = 0.2, Rd = M = 0.5, δ = 0, A = Nb = Ec = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure6: Temperature profileθ(η)for differents.

θ (η)

0.5 1 1.5 2 2.5 3

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (h₄ = 0.1) γ = 0 (h₄ = 0.5) γ = 0 (h₄ = 0.9)

γ = 1 (h₄ = 0.1) γ = 1 (h₄ = 0.5) γ = 1 (h₄ = 0.9) Pr_∞ = 6.8, Kr = Kp = Nt = 0.2, Rd = M = 0.5, γ = 0, s = A = Ec = Nb = h₂ = h₆ = h₈ = 0.1, Pe = Sc = Sb = 5

Figure7: Temperature profileθ(η)for differenth₄.

θ (η)

0.5 1 1.5 2 2.5 3 3.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Nb = 1.5) γ = 0 (Nb = 2) γ = 0 (Nb = 2.5)

γ = 1 (Nb = 1.5) γ = 1 (Nb = 2) γ = 1 (Nb = 2.5) Pr_∞ = 6.8, Kp = Nt = Kr = 0.2, Rd = M = 0.5, δ = 0, s = A = Ec = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure8: Temperature profileθ(η)for different Nb.

φ (η)

0.5 1 1.5 2 2.5 3 3.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Nb = 0.1) γ = 0(Nb = 0.2) γ = 0 (Nb = 0.5)

γ = 1 (Nb = 0.1) γ = 1 (Nb = 0.2) γ = 1 (Nb = 0.5) Pr_∞ = 6.8, Kp = Nt = Kr = 0.2, Rd = M = 0.5, δ = 0, s = A = Ec = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure9: Concentration profileϕ(η)for different Nb.

φ (η)

0.5 1 1.5 2 2.5 3 3.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Kr = 0) γ = 0 (kr = 0.5) γ = 0 (Kr = 1)

γ = 1 (Kr = 0) γ = 1 (Kr = 0.5) γ = 1 (Kr = 1) Pr_∞ = 6.8, Kp = Nt = 0.2, Rd = M = 0.5, δ = 0, s = A = Ec = Nb = h₂ = h₄ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure12: Concentration profileϕ(η)for different Kr.

φ (η)

0.5 1 1.5 2 2.5 3 3.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Nt = 0.1) γ = 0 (Nt = 0.2) γ = 0 (Nt = 0.3)

γ = 1 (Nt = 0.1) γ = 1 (Nt = 0.2) γ = 1 (Nt = 0.3) Pr_∞ = 6.8, Kp = Nt = Kr = 0.2, Rd = M = 0.5, δ = 0, s = A = Ec = h2 = h4 = h6 = h8 = 0.1, Sc = Sb = Pe = 5

Figure11: Concentration profileϕ(η)for different Nt.

θ (η)

0.5 1 1.5 2 2.5 3 3.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Nt = 1) γ = 0 (Nt = 1.5) γ = 0 (Nt = 2)

γ = 1 (Nt = 1) γ = 1 (Nt = 1.5) γ = 1 (Nt = 2) Pr_∞ = 6.8, Kp = Kr = 0.2, Rd = M = 0.5, δ = 0, s = A = Ec = Nb = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = Pe = 5

Figure10: Temperature profileϕ(η)for different Nt.

φ (η)

0.5 1 1.5 2 2.5 3 3.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Sc = 3) γ = 0(Sc = 5) γ = 0 (Sc = 10)

γ = 1 (Sc = 3) γ = 1 (Sc = 5) γ = 1 (Sc = 10) Pr_∞ = 6.8, Kr = Kp = Nt = 0.2, Rd = M = 0.5, δ = 0, s = A = Ec = Nb = h₂ = h₄ = h₆ = h₈ = 0.1, Sb = Pe = 5

Figure13: Concentration profileϕ(η)for different Sc.

φ (η)

0.5 1 1.5 2 2.5 3 3.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (h₆ = 0.1) γ = 0 (h₆ = 0.5) γ = 0 (h₆ = 0.9)

γ = 1 (h₆ = 0.1) γ = 1 (h₆ = 0.5) γ = 1 (h₆ = 0.9) Pr_∞ = 6.8, Kr = Kp = Nt = 0.2, Rd = M = 0.5, γ = 0, s = A = Ec = Nb = h2 = h4 = h8 = 0.1, Pe = Sc = Sb = 5

Figure14: Concentration profileϕ(η)for differenth₆.

χ (η)

0.5 1 1.5

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Pe = 1) γ = 0 (Pe = 2) γ = 0 (Pe = 3)

γ = 1 (Pe = 1) γ = 1 (Pe = 2) γ = 1 (Pe = 3) Pr_∞ = 6.8, Kr = Kp = Nt = 0.2, Rd = M = 0.5, δ = 0, s = A = Ec = Nb = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Sb = 5

Figure15: Microorganisms’ profileϕ(η)for different Pe.

χ (η)

0.2 0.4 0.6 0.8 1 1.2 1.4

0

η 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ = 0 (Sb = 0) γ = 0 (Sb = 5) γ = 0 (Sb = 10)

γ = 1 (Sb = 0) γ = 1 (Sb = 5) γ = 1 (Sb = 10) Pr_∞ = 6.8, Kr = Kp = Nt = 0.2, Rd = M = 0.5, δ = 0, s = A = Ec = Nb = h₂ = h₄ = h₆ = h₈ = 0.1, Sc = Pe = 5

Figure16: Microorganisms’ profileϕ(η)for different Sb.