# Heat and Mass Transfer in Three-Dimensional Flow of an Oldroyd-B Nanofluid with Gyrotactic Micro-Organisms

Authors: M. Sulaiman; Aamir Ali; S. Islam
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper discusses the three-dimensional flow of the gyrotactic bioconvection of an Oldroyd-B nanofluid over a stretching surface. Theory of microorganisms is utilized to stabilize the suspended nanoparticles through bioconvection induced by the effects of buoyancy forces. Analytic solution for the governing nonlinear equations is obtained by using homotopy analysis method (HAM). The effects of involved parameters on velocity, temperature, nanoparticles concentration, and density of motile microorganisms are discussed graphically. The local Nusselt, Sherwood, and motile microorganisms numbers are also analyzed graphically. Several known results have been pointed out as the particular cases of the present analysis. It is found that the non-Newtonian fluid parameters, i.e., relaxation time parameterβ1 and retardation time parameter β2, have opposite effects on the velocity profile. The velocity of the fluid and boundary layer thickness decreases for increasing relaxation time while it decreases for increasing retardation time effects.


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## Body

## 1. Introduction

In recent years the significance of the study of non-Newtonian fluids has remarkably increased due to its applications in engineering, industry, and biological sciences, e.g., in material processing, food, and cosmetic industries [1]. These applications generally involve complex geometries and require that the response of the fluid to given temperature conditions is controlled. The basic equations that governs the flow of non-Newtonian fluids are highly nonlinear. The mathematical description of a non-Newtonian fluid requires a constitutive equation that determines the rheological properties of the fluid. Due to the complexity, the aforementioned fluids are generally divided into three types, namely, the differential type, the rate type, and the integral type. The simplest example of rate type fluid is Maxwell fluid model which narrates the properties of relaxation time. But it cannot predict the retardation time effects. The Oldroyd-B fluid is one subclass of the rate type fluid which describes the properties of both the relaxation time and the retardation time. There are several investigations in this direction for two-dimensional flow configurations with different features of the non-Newtonian fluid models [2–10]. However, in practical applications sometimes, the flow is three-dimensional. In view of such motivation, researchers have studied the three-dimensional flow for different flow geometries (see ref. [11–17]).The problem of heat and mass transfer analysis over a stretched boundary layer is of great interest to researchers due to its engineering and industrial applications like glass fiber, manufacture and drawing of plastics and rubber sheets, paper production, cooling of continuous stripes and an infinite metallic sheet, the continuous casting, polymer extrusion process, heat treated materials travelling on conveyer belts, food processing, and many others. Sakiadis [18] has done the pioneer work in the analysis of the boundary layer flow on a moving surface, which was later extended by Crane [19] by considering the linearly stretching surface. Rashad and Kabeir [20] studied the heat and mass transfer in transient flow by mixed convection boundary layer over a stretching sheet embedded in a porous medium with chemically reactive species. Sandeep and Sulochana [21] studied the generalized model for momentum and heat transfer behavior of Jeffrey, Maxwell, and Oldroyd-B nanofluids over a stretching surface with nonuniform heat source/sink. After that, several investigations have been done by considering different flow configurations [22–28].With the advent of nanoscience, nanofluids have become a focus of attention in the study of fluid flow in the presence of nanoparticles. Nanofluids can be used to increase the thermal conductivity of the fluids and are stable fluids having much better writing, dispersion, and spreading properties on solid surfaces. Choi and Eastman [29] initially presented the idea of suspension of nanoparticles in base fluids. Later, Buongiorno [30] extended the concept by considering Brownian motion and thermophoresis movement of nanoparticles in view of application in hybrid power engine, thermal management, heat exchanger, domestic refrigerators, etc. They have enhanced heat conduction power as compared to base fluids and are useful in cancer therapy and medicine. With the presentation of a reliable model of nanofluid given by Buongiorno [30], nanofluids have become a subject of great interest for researchers in the last few decades [31–34]. Recently, Nabwey et al. [35] studied the group method analysis of mixed convection stagnation-point flow of non-Newtonian nanofluid over a vertical stretching surface.Bioconvection is a phenomenon that is used to describe the unstructured pattern and instability produced by the microorganisms, which are swimming to the upper part of a fluid and has a lesser density. Due to swimming upwards, these involved microorganisms such as gyrotactic microorganisms like algae tend to concentrate in the upper part of the fluid layer, causing unstable top heavy density stratification (see [36]). Nanofluid bioconvection phenomenon describes the unstructured density stratification and formation of patterns due to synchronized interaction of the nanoparticles, thicker self-propelled microorganisms and buoyancy forces. Brownian motion and thermophoresis effect produce the motion of these particles, which is independent of the movement of motile microorganisms. That is why the combined interface of bioconvection and nanofluids becomes significant for microfluidic appliances. Uddin et al. studied the computational investigation of Stefan blowing and multiple-slip effects on buoyancy-driven bioconvection nanofluid flow with microorganisms [37] and numerical solutions for gyrotactic bioconvection in nanofluid saturated porous media with Stefan blowing and multiple slip effects [38]. Different types of microorganisms, theoretical biconvection models, have been studied [39–51].In present study, we have entered a new concept in the problem of boundary layer flow passing a stretching sheet. We have considered the three-dimensional bioconvection flow of an Oldroyd-B nanofluid which contains self-impelled motile gyrotactic microorganisms, over a bidirectional stretching surface. This work is an extension of the work of W. A. Khan et al. [16] by considering the gyrotactic microorganisms in the three-dimensional flow of Oldroyd-B nanofluid. The problem is formulated in terms of governing equations of motion with appropriate boundary conditions, which are then transformed into a system of ordinary differential equations using similarity transformations. The analytical solution is found using Homotopy Analysis Methods (HAM). The effects of involved physical parameters on the flow fields are analyzed with the help of graphs.


## 2. Problem Formulation

We have considered the steady, incompressible three-dimensional flow of an Oldroyed-B nanofluid containing gyrotactic microorganisms past a stretching surface. The surface is maintained at constant temperatureTw, the nanoparticle volume fraction Cw, and the number of motile microorganisms nw; the ambient values of temperature, nanoparticle volume fraction, and number of motile microorganisms are denoted by T∞, C∞, and n∞, respectively, while the reference values are by T0, C0, and n0, respectively. The microorganisms are taken into the nanofluid to stabilize the nanoparticles due to bioconvection. The mechanical analog for the Oldroyd-B model and the geometry of the present analysis is shown in Figure 1. The boundary layer form of the continuity, momentum, energy, nanoparticle concentration, and density of microorganisms for three-dimensional Oldroyd-B fluid is as follows ([16, 37, 50]):(1)∂u∂x+∂v∂y+∂w∂z=0,(2)u∂u∂x+v∂u∂y+w∂u∂z+λ1u2∂2u∂x2+v2∂2u∂y2+w2∂2u∂z2+2uv∂2u∂x∂y+2vw∂2u∂y∂z+2uw∂2u∂x∂z=ν∂2u∂z2+λ2u∂3u∂x∂z2+v∂3u∂y∂z2+w∂3u∂z3-∂u∂x∂2u∂z2-∂u∂y∂2v∂z2-∂u∂z∂2w∂z2+1ρfρfβ1-C∞T-T∞-ρp-ρfC-C∞-γρm-ρfn-n∞g,(3)u∂v∂x+v∂v∂y+w∂v∂z+λ1u2∂2v∂x2+v2∂2v∂y2+w2∂2v∂y2+2uv∂2v∂x∂y+2vw∂2v∂y∂z+2uw∂2v∂x∂z=ν∂2v∂z2+λ2u∂3u∂x∂z2+v∂3u∂y∂z2+w∂3u∂z3-∂u∂x∂2u∂z2-∂u∂y∂2v∂z2-∂u∂z∂2w∂z2,(4)u∂T∂x+v∂T∂y+w∂T∂z=αm∂2T∂z2+τDB∂C∂z∂T∂z+DTT∞∂T∂z2,(5)u∂C∂x+v∂C∂y+w∂C∂z=DB∂2C∂z2+DTT∞∂2T∂z2,(6)u∂n∂x+v∂n∂y+w∂n∂z+bw0ΔC∂∂zn∂C∂z=Dm∂2n∂z2,and the boundary conditions are as follows:(7)u=uwx=Uw=ax,v=vwy=Vw=by,w=0,T=Tw=T0+b1x,C=Cw=C0+d1x,n=nw=n0+e1x,atz=0,u→0,v→0,T→T∞,C→C∞,n→n∞asz→∞,in which u, v, and w are the velocity components in the x, y, and z directions, respectively, ν is the kinematic viscosity, λ1 and λ2 are the relaxation time and retardation time, ρf is the density of nanofluid, ρp is the density of nanoparticles, ρm is the density of microorganisms particles, g is the gravity, τ=(ρc)p/(ρc)f is the heat capacity ratio, Uw and Vw are the stretching velocities, a,b>0 are the stretching rate, and b1,d1,e1 are the dimensional constant. Let us introduce the following transformations:(8)η=aνz,u=axf′η,v=ayg′η,w=-aνfη+gη,θη=T-T∞Tw-T0,ϕη=C-C∞Cw-C0,Nη=n-n∞nw-n0,The continuity (1) is identically satisfied and (2) to (6) takes the following forms:(9)f′′′+f+gf′′-f′2+β12f+gf′f′′-f+g2f′′′+β22f′′+g′′f′′-f+gfiv+λθ-Nrϕ-RbN=0,(10)g′′′+f+gg′′-g′2+β12f+gg′g′′-f+g2g′′′+β22f′′+g′′g′′-f+ggiv=0,(11)θ′′+Prf+gθ′-Prf′θ+Nbϕ′θ′+Ntθ′2=0,(12)ϕ′′+Lef+gϕ′-Lef′ϕ+NtNbθ′′=0,(13)N′′+Lbf+gN′-Lbf′N-Peϕ′N′+N+Ωϕ′′=0,and the boundary conditions (7) is reduced to the following:(14)f0=0,g0=0,f′0=1,g′0=α,θ0=1,ϕ0=1,N0=1,f′∞=0,g′∞=0,θ∞→0,ϕ∞→0,N∞→0,where prime is the differentiation with respect to η. Furthermore, β1,β2, λ, Nr, Rb, Pr, Nb, Nt, Le, Lb, Pe, Ω, and α are the Deborah number, the mixed convection parameter, the buoyancy ratio parameter, the bioconvection Rayleigh number, the Prandtl number, the Brownian motion parameter, the thermophoresis parameter, the Lewis number, the bioconvection Lewis number, the bioconvection Peclet number, the microorganisms concentration difference parameter, and stretching ratio parameter which are the nondimensional parameters and are defined as follows:(15)β1=λ1a,β2=λ2b,λ=gβρf1-C∞Tw-T0x3/ν2Uw2x2/ν2=GrxRex2,Nr=ρp-ρfCw-C0ρf1-C∞Tw-T0β,Rb=γρm-ρfnw-n0gρf1-C∞Tw-T0β,Pr=ναm,Nb=τDBCw-C0αm,Nt=τDTTw-T0αmT∞,Le=νDB,Lb=νDm0,Pe=bwm0Dm0,Ω=n∞nw-n0,α=ba.The nondimensional ordinary differential (9)-(13) subject to the boundary conditions (14) are solved analytically by using homotopy analysis method (HAM). HAM technique was first introduced by Liao [52] in his Ph.D. thesis. He devised a parameter that controls convergence of the series solution. The method has been successfully applied in the literature to compute analytic solution of various problems ([53–56] and references therein). The details of the method are given in the appendix. Homotopy analysis method ensures the convergence of derived series solution. The auxiliary parameter h plays a vital role in controlling and adjusting the convergence region of series solutions. We have plotted the h- curves shown in Figure 2 and found that the admissible values of auxiliary parameters hf, hg, hθ, hϕ and hN are -1.45≤hf≤-0.05, -1.25≤hg≤-0.1, -1.45≤hθ≤-0.05, -1.3≤hϕ≤-0.15, and -1.5≤hN≤-0.2, respectively. The expressions for local Nusselt number, local Sherwood number, and local density number of the motile microorganisms are defined as follows:(16)Nu=xqwkTw-T0,Sh=xqmDBCw-C0,Nn=xqnDnnw-n0,where(17)qw=-k∂T∂zz=0,qm=-DB∂C∂zz=0,qn=-Dm∂n∂zz=0.The dimensionless forms of local Nusselt number, local Sherwood number, and local motile microorganism are as follows(18)NuRex-1/2=-θ′0,ShRex-1/2=-ϕ′0,NnRex-1/2=-N′0,where Rex=xUw/ν is the Reynolds number.Figure 1
Geometry of the problem.Figure 2
h -curves for (a) f, (b) g, (c) θ, (d) ϕ, and (e) N.
(a) (b) (c) (d) (e)


## 3. Results and Discussion

This section explores the results of three-dimensional flow of an Oldroyed-B nanofluid over a stretching sheet in the presence of gyrotactic micr-organisms. The analytic results for the problem are presented graphically to study the behavior of some physical parameters on the velocitiesf′, g′, temperature θ, concentration of nanoparticles ϕ, and concentration of microorganisms N. In Figure 3, the effects of Deborah numbers β1 and β2 on the velocity component f′ are examined for fixed values of other parameters. The opposite behavior of velocity profile is observed for increasing values of β1 and β2. Nonzero values of β1 and β2 correspond to elastic effects which retards the flow and hence the boundary layer will be thinner which is noted. The similar behavior for β1 and β2 can be observed for the velocity component g′ in Figure 4.Figure 3
Influence of (a)β1 (b) β2 on f′.
(a) (b)Figure 4
Influence of (a)β1 (b) β2 on g′.
(a) (b)The influence of stretching ratio parameterα on the velocity component g′ is shown in Figure 5. It is observed that when the values of α are increased the velocity at the wall, η=0 is increased.Figure 5
Influence of stretching ratioα on velocity component g′.In Figure6, the impact of (a) bioconvection Rayleigh number Rb and (b) buoyancy ratio parameter Nr on the velocity component f′ is shown. The graphical results show that when the values of Rb and Nr are increasing, a rapid decrease in the velocity profile is observed.Figure 6
Influence of (a)Rb (b) Nr on f′.
(a) (b)The combined effects of the thermophoresis parameterNt and Brownian motion parameter Nb on the temperature profile are shown in Figure 7(a). The temperature profile increases when we increase the values of Nt and Nb, since increasing the magnitude of Brownian motion on the particles and thermal diffusivity of the nanoparticles will accelerate the temperature of the nanofluid. The impact of Prandtl number Pr on temperature profile is shown in Figure 7(b). The increasing values of Prandtle number decrease the temperature of the system. Figure 7(b) clearly depicts that the larger value of Prandtl number Pr corresponds to the lower heat and thermal boundary layer thickness. The Prandtl number indeed plays an essential role in heat transfer; it controls the relative thickness of the thermal boundary layer. In order to characterize the mixed convection flow, it is also important to investigate the effect of the Prandtl number on the temperature profile. We observe that the temperature decreases with increasing Prandtl number. This may be explained by the fact that increasing the Prandtl number is equivalent to reducing the thermal diffusivity, which characterizes the rate at which heat is conducted.Figure 7
Influence of (a)Nb and Nt (b) Pr on θ.
(a) (b)To analyze the concentrationϕ of the nanoparticles, the values of Brownian diffusion parameter and thermophoresis diffusion parameter are varied for fixed values of other parameters as shown in Figure 8(a); for larger values of Nt and Nb, the concentration profile ϕ increases. Increase in the values of Nt and Nb refers to increase in number of nanoparticles. Thus higher concentration of nanoparticles corresponds to higher Brownian and thermophoresis diffusion which can also be seen from Figure 8(a). The effect of Lewis number can be seen in Figure 8(b). If we increase the value of Lewis number, the concentration of the nanoparticle decreases.Figure 8
Influence of (a)Nb and Nt (b) Le on ϕ.
(a) (b)The influence of dimensionless bioconvection Lewis numberLb on concentration of microorganisms N is shown in Figure 9(a). The graphs illustrate a rapid decline in the profile, because the bioconvection Lewis number opposes the motion of the fluid. The relation of microorganisms concentration difference parameter Ω and motile density of the microorganisms N is discussed in Figure 9(b). For higher values of Ω, a decrease noticed in the value of N.Figure 9
Influence of (a)Nb and Nt (b) Ω on N.
(a) (b)The effects of Peclet numberPe on N are shown in Figure 10, which shows the buoyancy parameter is found to be more pronounced for a fluid with greater values of bioconvection Péclet number Pe.Figure 10
Influence ofPe on N.Numerical values for-f′′(0) and -g′′(0) are compared with the existing literature in the absence of non-Newtonian effects are shown in Table 1, and an excellent agreement has been found, which validate the present results. The numerical values of local Nusselt number, local Sherwood number, and local density of motile microorganisms are presented in Table 2 for different values of α, λ, Nb, Nt, and Ω. We can observe that the magnitude of local Nusselt number is increased by increasing α and λ while it decreased by increasing Nb and Nt. In other words, heat flux at the surface is a decreasing function of nanoparticle properties. In other words, heat flux at the surface reduces in the presence of nanoparticles and can be controlled by varying the quantity and quality of nanoparticles. Similarly, the magnitude of local Sherwood number and local density of motile microorganisms is increasing by increasing values of α, λ, and Nb while it decreases for increasing Nt. In other words, local Sherwood number and local density of motile microorganisms are decreasing function of both the thermophoretic diffusion and Brownian diffusion parameter for all values of other parameters.Table 1
Comparison for the velocity gradient for different values ofα when β1=β2=0.
-f′′(0) - g ′ ′ ( 0 ) α HPM result [15] Exact result [15] Present result HPM result [15] Exact result [15] Present result  0.0 1.0 1.0 1.0 0 0 0 0.1 1.02025 1.020259 1.020253 0.06684 0.066847 0.066849 0.2 1.03949 1.039495 1.039498 0.14873 0.148736 0.148730 0.3 1.05795 1.05794 1.057959 0.24335 0.243359 0.243360 0.4 1.07578 1.075788 1.075789 0.34920 0.349208 0.349212 0.5 1.09309 1.093095 1.093093 0.46520 0.465204 0.465206 0.6 1.10994 1.109946 1.109944 0.59052 0.590528 0.590527 0.7 1.12639 1.126397 1.126396 0.72453 0.724531 0.724529 0.8 1.14248 1.142488 1.142489 0.86668 0.866682 0.866683 0.9 1.15825 1.158253 1.158256 1.01653 1.016538 1.016541 1.0 1.17372 1.17372 1.173723 1.17372 1.173720 1.173723Table 2
Numerical values of local Nusselt numberNuRex-1/2, local Sherwood number ShRex-1/2, and local density of motile microorganisms NnRex-1/2 for different values of α, λ, Nb, Nt, and Ω.
α λ N b N t Ω - θ ′ ( 0 ) - ϕ ′ ( 0 ) - N ′ ( 0 ) 0.0 0.2 0.3 0.3 0.2 0.861666 0.522014 1.024791 0.5     0.929823 0.571493 1.118607 1.0     1.002058 0.634386 1.227815 0.5 0.0 0.3 0.3 0.2 0.918579 0.554933 1.104269 0.5    0.943804 0.592014 1.136527 1.0    0.960169 0.616448 1.158511 0.5 0.3 0.1 0.3 0.2 1.010617 - 0.700285 1.129376 0.5   0.864386 0.832249 1.125671 0.9   0.738619 0.997418 1.127016 0.5 0.3 0.3 0 0.2 0.998064 1.124014 1.124014 0.5  0.896472 0.263520 1.125716 1.0  0.812228 - 0.390333 1.127555 0.5 0.2 0.3 0.3 0 0.929823 0.571493 1.118607 0.5 0.929823 0.571493 1.118607 1.0 0.929823 0.571493 1.118607


## 4. Concluding Remarks

In this paper, we have discussed the three-dimensional bioconvection flow of an Oldroyd-B nanofluid over a stretching surface with gyrotactic microorganisms. Analytic technique is applied to investigate the problem and series solution is computed by using homotopy analysis method. The velocity, temperature, concentration, and microorganisms profiles are plotted to analyze the effects of various physical parameters. The main observations are as follows:(i)
The Deborah numberβ1 decreases the velocity profile, while β2 increases the velocity profile.(ii)
The velocity componentf′ decreases for increasing values of Rb and Nr.(iii)
The stretching ratio parameterα increases the velocity component g′.(iv)
The combined effects of Brownian diffusion parameterNb and thermophoretic diffusion parameter Nt decrease the temperature and concentration profiles.(v)
The Prandtl numberPr and Schmidt number Sc also decrease the temperature and concentration profiles.(vi)
The concentration of microorganismN has decreasing behavior for increasing Lb, Pe, and Omega.(vii)
The local Nusselt number and local Sherwood number are decreasing functions of nanoparticle properties.(viii)
The density of motile microorganisms is an increasing function of bioconvection Lewis number and bioconvection Peclet number.



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