# Bioconvection Unsteady Magnetized Flow in a Horizontal Channel with Dufour and Soret Effects

Authors: Muzamil Hussain; Umer Farooq; Gulfam Bano; Jifeng Cui; Taseer Muhammad
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

In the current decade, bioconvection phenomenon has received a lot of attention in research because of its applications in the biological polymer synthesis, biosensors and biotechnology, pharmaceutical industry, microbial enhanced oil recovery, environmentally friendly applications, and continuous refinements in mathematical modeling. Therefore, this article is prepared to address the unsteady mixed bioconvection in electrically conducting fluid flow between two infinite parallel plates with magnetic field and first-order chemical reaction impacts. Furthermore, the heat and mass transfer study has taken Dufour and Soret effects into account. The nonlinear coupled systems representing the continuity, momentum, energy, mass diffusion, and microorganisms’ equations are renewed into an ordinary differential equation (ODE) by employing the similarity renovation. The renovated ODEs are interpreted by the Homotopy Analysis Method (HAM). Impacts of the different emerging parameters, namely, magnetic field parameter (M), heat generation parameter (Q), Dufour number (Du), Soret number (Sr), Schmidt number (Sc), chemical reaction parameter (K0), Prandtl number (Pr), squeezing parameter (β), Peclet number (Pe), and Lewis number (Le) on the dimensionless velocity, temperature, concentration, and microorganism profiles as well as the frictional drag, Nusselt number, Sherwood number, and microorganisms mass flux are presented. The main outcomes of this investigation are that the velocity profile rises as the squeezing parameter is increased, and clear enhancement is noticed in the temperature profile for augmented estimations of chemical reaction, heat generation/absorption, and Dufour parameters. There is a significant downward trend in the concentration profile and microorganism density for elevated values of Dufour and Soret parameters.


---

## Body

## 1. Introduction

Bioconvection is initiated by the accumulated swimming of motile microorganisms in fluid. This effect happens because microorganisms are slightly denser than water in suspension and usually swim in the upward path. Bioconvection is a growing response due to its use in microfluidic devices, such as bioscience dispersions and bio galvanic devices, and in the investigation of a few thermophilic species existing in high-temperature springs, in microbial oil recovery, and in the formulation of oil and gas transporting sedimentary basins. A relatively new idea of bioconvection prompted by the insertion of microorganisms to a low concentration suspension of nanoparticles has drawn the consideration of researchers. Pal and Mondal [1] reported that bioconvection improves the stability of nanofluid flow. Kuznetsov and Avramenko [2] examined the bioconvection in fluid flow that contained gyrotactic microorganisms and nanoparticles. Khan et al. [3] addressed boundary layer nanofluid flow comprising microorganisms with Naiver slip across a vertical plate. Tham et al. [4] researched the bioconvection flow over a solid domain with the effects of gyrotactic microorganism density factor. Xu et al. [5] analyzed the completely developed flow using nanofluid having both gyrotactic microorganisms and nanoparticles in a horizontally placed channel. Raees et al. [6] studied bioconvective unsteady flow of Newtonian fluid with nanoparticles between two parallel plates. Mosayebidorcheh et al. [7] examined fluid flow including nanoparticles in a horizontal channel with gyrotactic microorganisms. Shen et al. [8] studied analytically bioconvective nanofluid flow carrying motile microorganisms across a stretched surface with radiation and velocity slip impacts by employing HAM. Kumar et al. [9] analyzed the unsteady bioconvective nanofluid flow with slip velocity, thermophoresis, and Brownian effects by employing the Keller-box method. Zhao et al. [10] explored an electrically conducting unsteady mixed bioconvection fluid flow between two plates. Tarakaramu and Satya Narayana [11] investigated the flow of bioconvection nanofluids in a rotating system with binary chemical reaction effects. Waqas et al. [12] visualized nanofluid flow with heat transfer rates and motile microorganisms across a stretching surface. Rashad and Nabwey [13] examined bioconvection and nanofluid flow over a horizontal cylinder. Shukla et al. [14] discussed heat transfer in bioconvective nanofluid flow with solar flux, radiation, and oblique magnetic field impacts.In the thermochemical process, convection by two distinct rates of diffusion is used in a variety of biomedical applications such as laser tumor therapy, improving oxygenated blood movement, polymeric liquids, and novel lubricants. The Soret (thermal diffusion) effect is the variation in mass flux induced by a temperature difference. However, the Dufour effect is usually defined as the heat flux caused by the concentration gradient. Soret impact is used to manage mixtures of gases with lighter and medium molecular weights. These phenomena have many practical applications in the fields of geoscience, chemical engineering, air pollution, isotope separation, purification of ground water, hydrology, etc. Researchers have paid extensive attention to these two aspects because of the above-mentioned applications. Cheng [15] explained natural convection with Soret (Sr) and Dufour (Du) effects on a fluid flow saturated in a porous medium. Hayat et al. [16] investigated viscoelastic fluid flow over a porous surface with a magnetic field, and Du and Sr impacts. Hayat and Nawaz [17] also studied Du and Sr effects for second-grade fluid flow. Unsteady MHD flow on a radiative porous plate along with binary chemical reaction, and Du and Sr effects was reviewed by Sharma et al. [18]. Hayat et al. [19] analyzed the Sr and Du impacts on 3-D viscoelastic fluid flow. Moorthy et al. [20] investigated MHD and convection flow across a porous surface with Du and Sr impacts. Sheri and Srinivasa Raju [21] examined the Soret effect on the time-dependent MHD fluid flow across a semi-infinite vertical surface with the consideration of viscous dissipation. Majeed et al. [22] explored the Du and Sr influences on second-grade fluid flow induced by an expanding cylinder with radiation effects. Liu et al. [23] utilized multirelaxation phenomena to develop a dual-diffusion natural convective flow with Du and Sr effects by using lattice Boltzmann theory. They showed that double-diffuse natural convective flow may be easily achieved via the use of the Du and Sr impacts. Sardar et al. [24] evaluated mixed convection processes in a Carreau fluid flow with Du and Sr influences that were confined by a wedge. Bilal Ashraf et al. [25] deliberated the mixed convective MHD viscoelastic fluid flow with Du and Sr impacts. Jiang et al. [26] demonstrated promising results in simultaneous heat and mass transfer processes with Du and Sr impacts. Hafeez et al. [27] studied the fluid flow over a disk with thermophoresis Du and Sr effects.Hannes Alfvén, in 1970, was the first person who introduced and developed Magnetohydrodynamics (MHD). MHD is a dynamics study in the presence of electrically conducting liquids with magnetic properties and its effects that has sufficient applications in biomedical sciences and engineering such as drug targeting, biowaste fluid transportation, cell separation, cancer tumor treatment, magnetic endoscopy, astrophysics, MHD pumps, metallurgy, ship propulsion, reduction of turbulent drag, jet printers, and fusion reactors. MHD flow across different geometries relevant to engineering is an attractive and appreciable field of science. The above-mentioned applications of MHD compel scientists to create innovative mathematical models in the fluid mechanics field [28–31]. Patel and Singh [32] analyzed the MHD, micropolar fluid flow with Brownian diffusion, and convective boundary condition. Aly and Pop [33] explored steady MHD hybrid nanofluid flow along the permeable flat plate. Rashid et al. [34] studied the MHD boundary layer flow over a porous shrinking surface with the radiation effects. Waini et al. [35] evaluated the steady fluid flow across a permeable wedge with magnetic field impacts. Naqvi et al. [36] addressed the chemical reaction and radiation impacts in MHD nanofluid flow over a radially stretching/shrinking disk. Raees et al. [37] examined mixed convection in a magnetized second-grade fluid flow over a stretched surface. Rizwana et al. [38] investigated MHD stagnation fluid flow with mixed convection across an oscillating plate.The motivation of the current study is to examine the unsteady bioconvective flow in a horizontal squeezing channel. Multiple results of the magnetic field, chemical reaction, and Dufour and Soret effects are employed into the account. The considered system of PDEs is transformed into the dimensionless ODEs by employing the similarity transformation Then, these converted ODEs are solved by adopting HAM (introduced by Liao [39]). We have successfully implemented HAM to solve the dimensionless form of momentum, energy, nanoparticles mass, and bioconvection (motile microorganism species) equations with suitable boundary conditions. Extensive graphical and tabulated results are presented with the aid of Mathematica software, and finally, the impacts of different emerging parameters on the velocity, temperature, nanoparticles fraction, and motile microorganism profiles along with the frictional drag, Nusselt number, Sherwood number, and microorganisms mass flux are examined and deliberated in detail. This research may be helpful for researchers and engineers who work on the industrial applications of bioconvection squeezed flow of nanofluid which is quite useful in the fields of polymer synthesis, biomedicine, lubrication, metal and polymer molding, foam production process geothermal system, and many others.


## 2. Mathematical Formulation

MHD and unsteady electrically conducting bioconvective nanofluid flow between two infinite parallel plates with first-order chemical reaction, and Dufour and Soret effects are considered. The coordinate system is considered in such a way that thex-axis is along the lower plate and y- axis is normal to the flow direction. Figure 1 depicts the geometry of the flow system. The plates are assumed at y=ht=ν1−at/b1/2 distance apart, and the top plate is assumed to move in towards or away directions of the lower plate with velocity vt=dh/dt. Here, t denotes the time, ν is the kinetic viscosity, and a,b are the positive numbers. Definitely, it is clear that 1−at>0, which shows that a<b2−4ac1/t. Clearly, a=0 indicates that both plates are static, 0<a<1/t illustrates that the upper plate is squeezed against the lower plate, and a<0 demonstrates that the upper plate is gone away from the lower one. Also, it is assumed that the upper and lower plates are kept at a constant chemical reaction concentration C2 and C1; constant temperatures and constant microorganisms’ concentration are T2,T1N2, and N1, respectively. The magnetic field t is considered along the y-axis.Figure 1
Flow configuration.The governing system under the above-mentioned assumptions are(1)∇.V=0,(2)∂V∂t+V.∇V=υ∇2V+1ρJ×B−1ρ∇p,(3)∂T∂t+V.∇T=α∇2T+QstρCpT−T0+J,(4)∂C∂t+V.∇C=D∇2C−ktC−C0+R,(5)∂N∂t=−∇.j.where V, B, J, ρ, p, υ, T0, T,α, Qst,D, C,C0, Kt, N, and j are the velocity vector, magnetic induction along y-direction, electric current density, fluid density, pressure, kinematic viscosity, reference temperature, temperature, thermal diffusivity, volumetric heat generation rate,  mass diffusivity, chemical reaction, reference concentrations, reaction rate, motile microorganism concentration, and flux of microorganisms, respectively.J is the heat flux, known as the “Dufour effect,” that can be given by Frick's law of diffusion as J=ρDKt/Cs∇2C, R is the flux concentration generated by the temperature difference known as “Soret effect,” from Fourier's law of heat conduction Rc=q./A=DKt/Tm∇2T. Tm,D, and Kt are the mean temperature, coefficient of diffusion species, and thermal diffusion ratio, respectively.Momentum Equation (2) in the component form of a two-dimensional channel flow can be written as(6)∂u∂t+u∂u∂x+v∂u∂y=−∂pρ∂x+υ∂2u∂x2+∂2u∂y2−σB2tuρ,(7)∂v∂t+u∂v∂x+v∂v∂y=−∂pρ∂y+υ∂2v∂x2+∂2v∂y2.To simplify equations (6) and (7), the given transformation is used:(8)ξ=∂v∂x−∂u∂y=−∇2ψ.By using the above information, the governing equations are transformed as follows:(9)∂u∂x+∂v∂y=0,(10)∂ξ∂t+u∂ξ∂x+v∂ξ∂y=υ∂2ξ∂x2+∂2ξ∂y2+σB2tρ∂u∂y,(11)∂T∂t+u∂T∂x+v∂T∂y=kρCp∂2T∂x2+∂2T∂y2+QstρCpT−T0+DktCsCp∂2C∂x2+∂2C∂y2,(12)∂C∂t+u∂C∂x+v∂C∂y=D∂2C∂x2+∂2C∂y2−KtC−C0+DktTm∂2T∂x2+∂2T∂y2,(13)∂N∂t++∂∂yNv˜+u∂N∂x+v∂N∂y+∂∂xNv˜=Dm∂2N∂x2+∂2N∂y2.Boundary conditions are as follows:(14)u=0,v=0,T=T1,C=C1,N=N1,aty=0,u=0,v=dhdt,T=T2,C=C2,N=N2,asy=ht,where u=∂ψ/∂yandv=−∂ψ/∂x are the velocity components, ξ=−∇2ψ is the vorticity function, N represents motile microorganisms density, Wc is the maximum cell speed, v˜  =  bcWc/C1−C0∂C/∂y is the microorganisms' average swimming velocity, bc is chemotaxis constant, and Dm is the diffusivity of microorganisms.Similarity transformations are the transformations that can be used to convert ann-independent variable partial differential system to a system with n−1 independent variables. When n  = 2, the situation is ideal since one is dealing with ODEs rather than PDEs. For the solution of nondimensional problem, the following similarity transformations and nondimensional quantities have been used:(15)ψx,y=bυ1−at1/2xfη,u=bx1−atf′η,v=−bυ1−at1/2fη,η=bυ1−at1/2y,θη=T−T0T1−T0,ϕη=C−C0C1−C0,wη=NN1.Using equation (14) into equations 9–12, we get the following nondimensional equations of momentum, energy, concentration, and microorganisms equations; the continuity equation (10) is satisfied identically as follows:(16)f″″−Mf″−βηf′″−3βf′−f″f′+f′″f=0,θ″+PrQθ−Prβηθ′+Prfθ′+DuPrϕ′=0,ϕ″−ScK0ϕ−Scϕ′βη+Scfϕ′+SCSrθ″,w″−Scβηw′+Scfw′−Pewϕ″−Peϕ′w′=0.According to equation (13), the transform boundary conditions are(17)f0=0,f′0=0,θ0=1,0=1,w0=1,f1=β,f′1=0,θ1=δθ,1=δϕ,w1=δw,where β=a/2b is the squeezing parameter, M=σ/υρB02 is the dimensionless magnetic field number, Pr=ρCPυ/k indicates the Prandtl number, Q=Q0/ρCp signifies the heat generation parameter, Du=DkTC1−C0/υCpT1−T0 and Sr=DkTT1−T0/υTmC1−C0 represents the Dufour and Soret numbers, Sc=υ/D is the Schmidt number, K0, Pe, and Le=α/D are the chemical reaction parameter, bio convection Peclet number, and Lewis number, respectively, δθ=T2−T0/T1−T0,δϕ=C2−C0/C1−C0, and δw=N2/N1 are the constants.where B0=b/υ1−at−3/4Bt and Q0=b/1−at−1Qst are the reference magnetic field and heat generation, respectively, while K0=b/1−at−1kt is the chemical reaction parameter.


## 3. Thermal Transport Analysis

The nonlinear ODEs (15–19) are tackled by employing HAM. This method is used to find the analytic solutions of the complicated nonlinear ordinary differential system. It has unique advantages as compared to other analytical approximation methods. Besides the other different analytical techniques, HAM delivers convenient results. It is investigated that the nonlinear boundary value problems (BVPs) of science, finance, and engineering can be analytically solved by applying HAM. It is developed on Mathematica and Maple softwares that can effectively be used on finite, semifinite, and infinite intervals to solve eigen value problems. Convergence of the resulting solution can be found with the help of a module in HAM by using the optimum value called the “convergence-control parameter,” which can be used at the minimum squared residual of the system of the governing equations in a certain order of approximations.Figure2 portrays the impacts of magnetic parameter M on the velocity profile f′η. It is seen that the velocity profile develops rapidly with the evolution of M, while the increment in M boosts the flow velocity in the vicinities of lower and upper plates, but decline in the flow velocity is observed between the plates. The fact behind this is that, the magnetic field interacts with the electrical conductivity of nanofluid and formed Lorentz forces which diminish the velocity of fluid. The influence of the squeezing parameter on the velocity profile is observed in Figure 3. It predicts that the velocity profile rises as the squeezing parameter (β) increases; this is because that the accelerated flow has a higher velocity. It is seen in Figure 4 that the temperature profile is greatly influenced by heat generation parameter Q, although continuously increasing Q raises the temperature profile rapidly. Physically, increasing Q increases the kinetic energy of the fluid particles, so the thickness of the thermal boundary layer rises, which leads to an increase in temperature profile. Figure 5 reveals that the temperature profile increases constantly when the chemical reaction parameter K0 rises. This occurred because of an increase in the concentration of interfacial nanoparticles.Figure 2
Magnetic parameterM effect on f′η.Figure 3
Squeezing parameterβ effect on f′η.Figure 4
Heat generation parameterQ effect on θη.Figure 5
Chemical reaction parameterK0 effect on θη.Figure6 demonstrates the effect of Dufour number Du on the temperature profile. Graph clearly portrays that Du has a great influence on temperature profile. It is noticed that on enhancing Du, the thermal profile is strengthened. This can be attributed to an increase in the Du, which leads to a rise in the concentration gradient, resulting in rapid mass diffusion. As a result, the rate of energy transfer between particles increases. Consequently, the temperature profiles rise. Figure 7 reveals the comportment of Soret number Sr on temperature profile θη. Rising values of Sr strengthens the thermal field.Figure 6
Dufour parameterDu effect on θη.Figure 7
Soret parameterSr effect on θη.Figures8–12 depict the impacts of heat generation/absorption (Q), chemical reaction parameters K0, Dufour (Du), Soret (Sr), and Schmidt (Sc) numbers on concentration profile ϕη. It is established that these physical parameters have a major impact on concentration. Figure 8 portrays the nanoparticle concentration profile for the heat generation parameter Q. The figure clearly shows that both the concentration and thickness of the concentration boundary layer are the decreasing functions of Q. Figure 9 illustrates the impact of chemical reaction parameter K0 on concentration profile ϕη. It is noted that ϕη and concentration boundary layer decline by raising the value of K0. This decline in ϕη is due to a decrease in molecular diffusivity as the number of chemical species increases. Figure 10 explores the conduct of Dufour parameter Du on the concentration profile of nanoparticles ϕη. It is clear from this graph that ϕη decreases for an increasing value of Du. Figure 11 explicates the effects of Soret number (Sr) on the nanoparticle’s concentration profile of ϕη. It represents a decreasing trend in ϕη for the uprising values of Sr. Figure 13 inspects the consequences of the Schmidt number Sc on the concentration field ϕη. It is observed that by increasing Sc, the concentration profile decays due to a reduction in mass diffusion.Figure 8
Heat generation parameterQ effect on ϕη.Figure 9
Chemical reaction parameterK0 effect on ϕη.Figure 10
Dufour parameterDu effect on ϕη.Figure 11
Soret parameterSr effect on ϕη.Figure 12
Schmidt numberSc effect on ϕη.Figure 13
Soret parameterSr effect on wη.Figures13–14 illustrate the effects of Dufour Du and Soret Sr on microorganism’s concentration profile. It is found that increasing estimations of Du and Sr numbers reduce the density of microorganism’s concentration wη. The effect of Peclet number Pe against wη is plotted in Figure 15. It illustrates that when Pe increases, the microorganism density decreases. Physically, Pe is a measurement of the relative strength of motile microorganisms' directional and random swimming. So, larger Pe values indicate increased directional movement of microorganisms, resulting in reduced wη profile.Figure 14
Dufour parameterDu effect on wη.Figure 15
Peclet numberPe effect on wη.Moreover, the impacts of some emerging parameters on the skin friction coefficient number (Cf), Nusselt number (Nux, Sherwood number (Sh) and microorganism mass flux (Qx) are described in Tables 1 and 2. Skin friction coefficient is a special parameter in the studies of heat transfer.Table 1
Skin friction coefficientCf for various values of magnetic field M,and squeezing parameter β and Nusselt number (Nux) for some distinct values of Prandtl number Pr and Dufour number Du.
MMCfRex−1/2PrDuNuxRex−1/20.5−1.1703910−0.651701−1.9217790.51−1.05671.55.1830761.5−3.008770.5−1.53527700.29023511.7993330.710.000541.55.2909211.5−0.225870.5−1.85503600.202161011.85503611−0.316211.55.4118441.5−0.84232Table 2
Sherwood number (Sh) for several values of Schmidth number Sc and Soret number Sr and microorganism mass flux (Qx) for some specific values of Schmidth number Sc and Peclet number Pe.
ScSrShRex−1/2)ScPeQxRex−1/20−1.171030.5−0.167980.50.5−1.315980.51−0.356031−1.511131.5−0.561740−1.329800.5−0.3725010.5−1.7051711−0.78721S1−2.449391.5−1.288130−1.477800.5−0.629691.50.5−2.198631.51−1.324821−3.831691.5−2.08003


## 4. Conclusions

The Soret and Dufour effects on a two-dimensional unsteady bioconvection squeezing flow incorporating motile gyrotactic microorganisms in a horizontal channel have been examined in the presence of a chemical reaction and a magnetic field. Obtained outcomes are exhibited graphically and in tabulated form. The summarization of results is as given below:(i)
The velocity profile is reduced in the center and slightly increases along the walls of channel.(ii)
Increasing values of squeezing parameter decrease the velocity profile.(iii)
For rising chemical reaction, heat generation/absorption parameters, and Dufour and Soret number values, the temperature profile is increased.(iv)
Chemical reaction parameter is critical for the concentration and motile microorganism’s profiles. When the chemical reaction parameter is increased, both profiles diminish.(v)
Fluid concentration drops when the heat generation parameter, Schmidth number, and Dufour and Soret numbers increase.(vi)
The concentration of microorganisms decreases for the rising values of Dufour and Soret, heat generation/absorption parameter, and bioconvection Peclet number.



---

*Source: 4771282-2022-01-29.xml*