#### 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 parameter and retardation time parameter , 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.

#### 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 temperature , the nanoparticle volume fraction , and the number of motile microorganisms ; the ambient values of temperature, nanoparticle volume fraction, and number of motile microorganisms are denoted by , , and , respectively, while the reference values are by , , and , 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]):and the boundary conditions are as follows:in which , , and are the velocity components in the , , and directions, respectively, is the kinematic viscosity, and are the relaxation time and retardation time, is the density of nanofluid, is the density of nanoparticles, is the density of microorganisms particles, is the gravity, is the heat capacity ratio, and are the stretching velocities, are the stretching rate, and are the dimensional constant. Let us introduce the following transformations:The continuity (1) is identically satisfied and (2) to (6) takes the following forms:and the boundary conditions (7) is reduced to the following:where prime is the differentiation with respect to . Furthermore, , , , , , , , , , , , and 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: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 plays a vital role in controlling and adjusting the convergence region of series solutions. We have plotted the curves shown in Figure 2 and found that the admissible values of auxiliary parameters , , , and are , , , , and , respectively. The expressions for local Nusselt number, local Sherwood number, and local density number of the motile microorganisms are defined as follows:whereThe dimensionless forms of local Nusselt number, local Sherwood number, and local motile microorganism are as followswhere is the Reynolds number.

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#### 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 velocities , , temperature , concentration of nanoparticles , and concentration of microorganisms . In Figure 3, the effects of Deborah numbers and on the velocity component are examined for fixed values of other parameters. The opposite behavior of velocity profile is observed for increasing values of and . Nonzero values of and correspond to elastic effects which retards the flow and hence the boundary layer will be thinner which is noted. The similar behavior for and can be observed for the velocity component in Figure 4.

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The influence of stretching ratio parameter on the velocity component is shown in Figure 5. It is observed that when the values of are increased the velocity at the wall, is increased.

In Figure 6, the impact of (a) bioconvection Rayleigh number and (b) buoyancy ratio parameter on the velocity component is shown. The graphical results show that when the values of and are increasing, a rapid decrease in the velocity profile is observed.

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The combined effects of the thermophoresis parameter and Brownian motion parameter on the temperature profile are shown in Figure 7(a). The temperature profile increases when we increase the values of and , 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 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 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.

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To analyze the concentration 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 and , the concentration profile increases. Increase in the values of and 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.

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The influence of dimensionless bioconvection Lewis number on concentration of microorganisms 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 and motile density of the microorganisms is discussed in Figure 9(b). For higher values of , a decrease noticed in the value of .

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The effects of Peclet number on 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 .

Numerical values for and 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 , , , , and . We can observe that the magnitude of local Nusselt number is increased by increasing and while it decreased by increasing and . 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 , , and while it decreases for increasing . 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.

#### 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 number decreases the velocity profile, while increases the velocity profile.(ii)The velocity component decreases for increasing values of and .(iii)The stretching ratio parameter increases the velocity component .(iv)The combined effects of Brownian diffusion parameter and thermophoretic diffusion parameter decrease the temperature and concentration profiles.(v)The Prandtl number and Schmidt number also decrease the temperature and concentration profiles.(vi)The concentration of microorganism has decreasing behavior for increasing , , and .(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.

#### Appendix

#### A. Homotopy Analysis Method (HAM)

The initial guesses and auxiliary linear operators for the dimensionless momentum, energy, concentration of nanoparticles, and concentration of motile microorganisms equations are denoted by and and are defined as follows:and withwhere are the arbitrary constant. The zeroth and -th order deformation problems are as follows.

##### A.1. Zeroth-Order Problem

withandwhere represents the embedding parameter and , , , , and are the nonzero auxiliary parameters.

##### A.2. th-Order Problem

withand for and , and we can writewhen varies from to , , , , , and vary from the initial solutions , , , , and to the final solutions , , , and , respectively. By Taylor’s series, we haveThe values of auxiliary parameter are chosen in such a way that the series (A.19) converge at , i.e.,The general solutions of (A.11)-(A.12) in terms of special solutions are given by the following:where the constants through the boundary conditions (A.12) are as follows:

#### Nomenclature

Dimensional coordinates () | |

Nanofluid temperature () | |

Dimensional velocity components | |

Nanoparticles volume fraction (−) | |

Number of motile micro-organisms | |

: | Relaxation time () |

: | Retardation time |

Kinematic viscosity | |

: | Density of nanofluid |

: | Density of nanoparticle |

: | Density of microorganisms particle |

: | Ambient fluid temperature (−) |

: | Ambient nanoparticles volume fraction (−) |

: | Ambient number of motile microorganisms (−) |

Gravitational acceleration | |

Heat capacity ratio (−) | |

: | Brownian diffusion coefficient |

: | Thermophoretic diffusion coefficient |

: | Thermal diffusivity of nanofluid |

: | Diffusivity of microorganisms |

: | Stretching velocities |

: | Surface temperature () |

: | Surface nanoparticles volume fraction |

: | Surface number of motile microorganisms (−) |

Stretching rate | |

: | Reference temperature () |

: | Reference nanoparticles volume fraction |

: | Reference number of motile micro-organisms (−) |

: | Constants |

Dimensionless variable (−) | |

Dimensionless fluid temperature | |

Dimensionless nanoparticles volume fraction (−) | |

Dimensionless number of motile microorganisms | |

, : | Dimensionless Deborah numbers (−) |

Mixed convection parameter | |

Buoyancy ratio parameter (−) | |

Bioconvection Rayleigh number | |

Prandtl number (−) | |

Brownian motion parameter | |

Thermophoresis motion parameter (−) | |

Schmidt number | |

Bioconvection Lewis number | |

Bioconvection Peclet number | |

Micro-organisms concentration difference parameter (−) | |

Stretching ratio parameter (−). |

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.