In fluid dynamics, bioconvection [1,2,3] occurs when microorganisms, which are denser than water, swim upwards. The upper surface of the fluid becomes thicker due to the assemblage of microorganisms. As a result, the upper surface becomes unstable and microorganisms fall down, which creates bioconvection. Bioconvection continues to be explored widely because of its enormous applications in the field of pharmaceutical industry, purification of cultures, microfluidic devices, mass transport enhancement and mixing, microbial enhanced oil recovery and enzyme biosensors. Bioconvection systems could be categorized based on the directional motion of different species of microorganisms. In particular, gyrotactic microorganisms are the ones whose swimming direction is dependent on a balance between gravitational and viscous torques [4,5]. Oyelakin et al.  pondered the impact of bioconvection and motile gyrotactic microorganisms on the Casson nanofluid past a stretching sheet and observed that the microorganism profile decreases as a result of an increment in the Peclet number. Saini and Sharma  explored the effects of bioconvection and gyrotactic microorganisms on the nanofluid flow over a porous stretching sheet. It is noted that the Lewis number escalates the bioconvection process. Dhanai et al.  explored the impact of bioconvection on the fluid flow over an inclined stretching sheet and assessed that the microorganism density profile is enhanced with an improvement in the bioconvection Schmidt number. Mahdy  pondered the effects of motile microorganisms on the fluid past a stretching wedge and noted that a positive variation in the Peclet number leads to an augmentation in the microorganism profile. Avinash et al.  pondered the impact of bioconvection and aligned magnetic field on the nanofluid flow over a vertical plate and concluded that the heat transfer rate increases with an improvement in the Lewis number. Makinde and Animasaun  studied the effects of magnetohydrodynamics (MHD), bioconvection, nonlinear thermal radiation and nanoparticles on fluid past an upper horizontal surface of a paraboloid of revolution and found that the Brownian motion boosts the concentration profile. Khan et al.  studied the impact of MHD, gyrotactic microorganisms, slip condition and nanoparticles on the fluid flow over a vertical stretching plate; it was observed that the magnetic field suppresses the dimensionless velocity inside the boundary layer. Later, the effects of different features of the gyrotactic microorganisms on the fluid flow are analyzed in various investigations [13,14,15].
Nanotechnology has been considered the most substantial and fascinating forefront area in physics, engineering, chemistry and biology. The thermal conductivity of a nanofluid is greater than that of the base fluid. The thermal conductivity of the fluid is considered to be enhanced by the nanoparticles present in the fluid. Buongiorno  established a model to examine the thermal conductivity of nanofluids. Baby and Ramaprabhu  analyzed the heat transport of fluids using graphene nanoparticles. They reported that the thermal conductivity of hydrogen-exfoliated graphene is enhanced with an increment in the volume fraction of the nanoparticles. Khan and Gorla  pondered the mass transfer of the nanofluid flow over a convective sheet using the Keller box scheme and noted that the heat transfer rate is high in the dilatant fluids compared with that in the pseudoplastic fluids. Das  discussed the rotating flow of a nanofluid with respect to the constant heat source. A boost in the volume fraction of nanoparticles was observed to cause an increment in the thermal boundary layer thickness. Gireesha et al.  considered the Hall impact on a dusty nanofluid and concluded that the skin friction coefficient decreases due to an improvement in the Hall current.
The experimental and the theoretical scientific studies of the non-Newtonian liquids together with MHD have achieved a considerable attention of researchers because of their adequate applications in the field of aeronautics, chemical, mechanical, civil and bio-engineering. The fluid becomes electrically conducting under the effect of MHD like ionized gases, plasmas and liquid metals such as mercury. The impact of MHD and nonlinear thermal radiation on the Sisko nanofluid flow over a nonlinear stretching surface is premeditated by Prasannakumara et al. . Rashidi et al.  pondered the MHD viscoelastic fluid together with the Soret and Dufour effects and observed that the velocity profile decreases with an improvement in the magnetic parameter. Kothandapani and Prakash  studied the effect of magnetic field on peristaltic tangent hyperbolic nanofluid past a asymmetric channel. Gaffar et al.  showed the tangent hyperbolic fluid flow over a cylinder together with the MHD and partial slip effects. Nagendramma et al.  analyzed the tangent hyperbolic fluid flow over a stretching sheet together with the MHD effect. Das et al.  investigated the impact of magnetic field, chemical reaction and double-diffusive convection on the Casson fluid flow past a stretching plate and noted that the skin friction coefficient decreases as a result of an augmentation in the Grashof number. Sravanthi and Gorla  examined the effect of the Maxwell nanofluid flow over an exponentially stretching sheet together with MHD, chemical reaction and heat source/sink.
Double-diffusion phenomena describe a form of convection driven by two different density gradients, holding distinct rate of diffusion. Double-diffusive convection occurs in a variety of scientific disciplines such as oceanography, biology, astrophysics, geology, crystal growth and chemical reactions . Nield and Kuznetsov  scrutinized the nanofluid past a porous medium along with the double-diffusive convection effect. The impact of double-diffusive convection on the fluid flow over a square cavity is analyzed by Mahapatra et al. . Gireesha et al.  discussed the Casson nanofluid past a stretching sheet along with the MHD and double-diffusive convection. Rana and Chand  explored the effect of double-diffusive convection on viscoelastic fluid and deduced that a Rayleigh number increases with an improvement in the Soret parameter. Gaikwad et al.  have monitored the fluid flow above a stretching sheet together with double-diffusive convection and found that an augmentation in the Nusselt number takes place with an improvement in the Dufour parameter. Kumar et al.  inspected the influence of nanoparticles and double diffusion on viscoelastic fluid and monitored that an increase in the velocity field occurs with an increment in the Dufour Lewis number.
Convection is a process common to particles, gases and vapours. Convection occurs when a fluid is in motion and that motion carries with it a material of interest such as the particles or the droplets of an aerosol. There are two types of convection: free convection and forced convection. In free convection or natural convection, the fluid motion cannot led by external sources such as fans, pumps, and suction devices etc. Gravity is the main driving force in the case of free convection. Free convection has various environmental and industrial applications such as plate tectonics, oceanic currents, formation of microstructures during the cooling of molten metals, fluid flows around shrouded heat dissipation fins, solar ponds and free air cooling without the aid of fans. In forced convection, the fluid motion is generated externally with the help of pumps, fans, suction devices, etc. This mechanism has enormous applications in our daily life such as heat exchangers, central heating system, steam turbines and air conditioning. Mixed convection is the situation in which both free convection and forced convection are of comparable order. Mixed convection is of great interest to researchers due to its enormous applications in the industrial and engineering sectors. Ibrahim and Gamachu  found the numerical solution of the mixed convective Williamson nanofluid past a stretching sheet by the Galerkin finite element method. Shateyi and Marewo  adopted the spectral quasi-linearization method to achieve the numerical solution of the mixed convective magneto Jeffrey fluid flow over an exponentially stretching sheet together with the thermal radiation and observed that the fluid velocity improves with an augmentation in the buoyancy parameter. Nalinakshi et al.  found the numerical solution of the mixed convective fluid past a vertical stretched plate using a nonlinear shooting method. El-Aziz and Tamer Nabil  gave the numerical solution for the problem of the MHD and Hall current effect on mixed convective fluid past a stretching sheet using the homotopy analysis method (HAM) and noted that a positive variation in the Hall current parameter leads to an increase in the velocity field. Beg et al.  employed an explicit finite difference scheme to yield the solution of the magneto mixed convection nanofluid flow over a stretchable surface under the effect of MHD and viscous dissipation. The numerical solution of the gravity-driven Navier–Stokes equation has been reported by Zhang et al. using a finite difference method . Pal and Chatterjee  studied the impact of the Soret and Dufour effects along with nonlinear thermal radiation on the double-diffusive convective fluid past a stretchable surface and achieved the numerical solution for problem using the Runge–Kutta–Fehlberg method along with the shooting scheme. They noted that the velocity field increases with an enhancement in the Grashof number.
The aim of this study was to construct a mathematical model that describes a form of convection driven by two different density gradients, which have different rates of diffusion (double-diffusive convection). So far, no reviews have been reported on the non-Newtonian fluid past a stretching sheet embedded with nanoparticles, double-diffusive convection and motile gyrotactic microorganisms.
2 Mathematical formulation
Figure 1 displays the effect of tangent hyperbolic nanofluid past a stretching sheet with stretching velocity uw = ax along the x-axis. When the Reynolds number is assumed to be small, the induced magnetic field can be neglected compared with the applied magnetic field B0, which is applied transversely to the surface. Tw, γw, Cw and Nw denote the temperature, solute concentration, concentration of nanoparticles and density of the motile gyrotactic microorganisms at the wall, respectively, whereas T∞, γ∞, C∞ and N∞ denote the ambient temperature, solute concentration, concentration of nanoparticles and density of the motile gyrotactic microorganisms, respectively. The fluid has further been assumed to contain the gyrotactic microorganisms. The microorganisms present in the fluid move towards light. The “bottom heavy” mass of the microorganisms orients its body and enables them to move against the gravity g, which is called as gyrotactic phenomena. The presence of microorganisms is considered to be beneficial for the suspension of the nanofluid. To maintain the stability of convection, the motion of microorganisms has been taken, independent of that of the nanoparticles. The double-diffusive fluid flow over a stretching sheet embedded with gyrotactic microorganisms has not been explored yet, and we want to rectify this problem in this study.
The governing equations include some important effects that have eminent involvement in the industries and engineering fields. The momentum equation includes bioconvection and MHD. MHD has been used in many engineering processes such as nuclear reactor, MHD power generation, in which heat energy is directly converted into electrical energy, Yamato-1 boat incorporating a superconductor cooled by liquid helium and microfluidics. A microorganism or microbe is an organism that is so small that it can be seen only through a microscope (invisible to the naked eye). The presence of microorganisms in the fluid becomes the core area of the research during the past decade. The presence of microorganisms in the base fluid causes a “stabilization” or “destabilization” in the motion of nanoparticles. The microorganisms have various applications in genetic engineering, wastewater engineering, agricultural engineering and chemical engineering. The temperature equation and concentration equations are embedded with nanoparticles and double-diffusive convection. Nanoparticles are used to enhance the thermal conductivity of the fluid and used in tissue engineering, mechanical engineering, nanomedicine, environmental engineering, etc. Double diffusion portrays the form of convection conducted by two different density gradients. There are various examples in environmental engineering such as Arctic Ocean study and Lake Kivu, in which magma, sand and materials of different densities are diffused with water. The same situation is applicable in our modelled problem, in which microorganisms and nanoparticles of different densities are diffused together in the fluid. The last governing equation tells us about the impact of gyrotactic microorganisms present in the fluid. Various types of microorganisms such as algae, fungi, protozoa and bacteria are suspended in the fluid. These microorganisms swim in the fluid under the combination of gravitational and viscous torques (gyrotactic) in fluid flow. The gyrotactic microorganisms have enormous contribution to genetic engineering, microbial engineering and soil engineering. Under the usual boundary layer approximations, the equations of conservation of mass, momentum, thermal energy, solute, concentration of nanoparticles and gyrotactic microorganisms take the following forms [11,12,13,31,32,34]:
Distinct physical parameters arising after the conversion of PDEs into ODEs are as follows:
The important quantities of interest like rate of shear stress Cf and heat as well as mass transfer rates Nux and Shx and Shx,n and Nnx are as follows:
3 Numerical scheme
The dimensionless system of equations (9)–(13) along with the boundary condition (14) should be handled with the help of the numerical scheme called the implicit finite difference method (Keller box technique) [42,43] for distinguished parameters that emerged during numerical simulation of the problem. Such type of differential equations in this article can usually be solved with the help of other numerical techniques such as shooting method, HAM and bvp4c [27,31,32,33,34,44,45,46,47,48,49]. In this study, the standard Keller box method has been used. This numerical technique is quite effective and flexible to solve the parabolic-type boundary value problems of any order, is unconditionally stable and attains remarkable accuracy. The Keller box scheme is numerically more stable and converges using less iterations compared with other numerical techniques. Figure 2 shows the flow chart procedure of the Keller box method. By adopting the new variables z1, z2, z3, z4, z5 and z6,
The transformed boundary conditions are as follows:
Figure 3 portrays the mesh structure for central difference approximations. The stepping procedure for the selection of the nodes in the case of domain discretization is as follows:
After linearization of the above-mentioned system of equations, the subsequent block-tridiagonal block structure:
To solve the problem numerically, the domain of the problem has been considered [0,ηmax] instead of [0,∞), where ηmax = 16 and the step size is hj = 0.01. All the numerical results achieved in this problem are subjected to an error tolerance of 10−5.
Table 1 displays the comparison analysis of the given numerical scheme results with Ibrahim .
4 Results and discussion
To discuss the outcomes, the behaviour of various pertinent parameters against the Nusselt number, the Sherwood number, motile density profile, velocity field, temperature field, mass fraction field and solute profile is monitored. Table 2 exhibits the behaviour of distinguished parameters on heat transfer at the boundary, mass fraction field and the motile microorganisms density profile for thermophoresis parameter (Nt) = 0.1, Prandtl number (Pr) = 6.2, Lewis number (Le) = 0.5, Dufour Lewis number (Ld) = 0.1 and mixed convection parameter Λ = 0.1. The heat transfer rate diminishes in the case of magnetic parameter M, Weissenberg number (We), modified Dufour parameter (Nd), power law index n, nanofluid Lewis number (Ln) and buoyancy ratio parameter (Nr), whereas an embellishment in the Nusselt number is seen for the Brownian motion parameter (Nb) and the bioconvection Rayleigh number (Nc). The Nusselt number has shown no variation in the case of microorganism concentration difference parameter σ, Peclet number (Pe) and bioconvection Lewis number (Lb). The mass fraction field depreciates in the case of M, Nc, nanofluid Lewis number (Ln) and buoyancy ratio parameter (Nr), but a positive variation is observed for Nb, We, Nd and n, whereas static behaviour is seen for σ, Pe and Lb. Furthermore, the number of motile microorganisms has been seen to increase in the case of positive variation in M, σ, Pe, Lb and Ln, but the situation is opposite in the case of Nr, Nc, n, We, Nd and Nb.
Figure 4 exhibits the effect of the magnetic parameter M on the velocity profile f′(η). It has been found that an increase in M decreases the velocity profile. Actually, the resistive force called the Lorentz force is generated due to the application of the magnetic field to the electrically conducting fluid. As a result, the velocity of the fluid reduces. Figure 5 indicates the effect of n on the velocity field f′(η). The parameter decides the viscosity of the fluid or how much viscous the fluid is. The fluid behaves like shear thinning for the case of n < 1, shear thickening for the larger values of n > 1 and Newtonian in the case of n = 1. The velocity of the fluid decreases in the case of n > 1, and as a result, the velocity field diminishes. Figure 6 depicts the effect of f′(η) on We. The Weissenberg number is defined as the ratio of viscous forces to the inertial forces. This parameter is important to study the fluid flow behaviour. The Weissenberg number actually depicts the elastic nature of the fluid. It is noted that the higher values of the Weissenberg number indicate the solid nature of the fluid, while lower values of the Weissenberg number depict the liquid nature of the fluid. It is clear that an augmentation in the Weissenberg number leads to a reduction in the velocity of the fluid. Figure 7 highlights the variation in the temperature profile θ(η) against the various values of M and observed that an electric current in the presence of magnetic field generates a Lorentz force. This force resists the motion of the fluid; hence, additional heat is produced, which enhances the fluid temperature. Figure 8 highlights the behaviour of temperature field θ(η) against Pr. The Prandtl number is a dimensionless quantity, which is defined as the ratio of momentum diffusivity to thermal diffusivity and has important application in the study of boundary layer concept. The thermal diffusivity dominates in the case of Pr ≪ 1, whereas momentum diffusivity dominates in the case of Pr ≫ 1. It is observed that the fluids with small Prandtl number are free flowing liquids with high thermal conductivity and favourable choice for heat conducting fluids. The thermal conductivity of the fluid decreases with an augmentation in the value of Pr, and the heat transfer decelerates, which decreases the temperature of the flow field, and as a result, a decrease in the temperature is observed.
Figure 9 portrays the effect of Brownian diffusion parameter (Nb) on the temperature distribution θ(η). Brownian motion is actually the random motion of the particles suspended in the fluid. The temperature of the fluid increases as a result of the random collision of particles suspended in the liquid, which further leads to an expected improvement in the temperature profile θ(η). Figure 10 explores the effect of the thermophoresis parameter (Nt) on the temperature distribution θ(η). In the thermophoresis process, smaller particles migrate from the region having high temperature to the region having low temperature, which ultimately causes an improvement in the fluid temperature.
Figure 11 shows the behaviour of temperature profile θ(η) against the different values of Nd. The situation in which heat and mass transfer happens simultaneously in a moving fluid affecting each other causes a cross-diffusion. The mass transfer caused by temperature gradient is called the Soret effect, whereas the heat transfer caused by concentration is called the Dufour effect. The Dufour number implies the effect of the concentration on the thermal energy flux in the flow. It is found that a variation in the modified Dufour number leads to a monotonic enhancement in the temperature field θ(η). Figure 12 highlights the effect of Nb on the mass fraction field. Brownian diffusion and thermophoresis parameters emerge as a result of an inclusion of nanoparticles into the fluid. Brownian diffusion and thermophoresis parameters help to understand the motion of the nanoparticles in the fluid. It is verified that the higher values of Nb are the root cause to boost the random motion among the nanoparticles present in the fluid. This results in the decrease in the concentration of the fluid.
Figure 13 describes the effect of Nt on the mass fraction field. It is observed that increasing values of Nt push nanoparticles away from the warm surface. The density of the concentration boundary layer upsurges due to an augmentation in the value of Nt, which leads to an embellishment in the mass fraction field. Figure 14 portrays the effect of the nanofluid Lewis parameter (Ln) on the mass fraction field. The Lewis number is defined as the ratio of thermal diffusivity to the mass diffusivity, and it is the prominent factor to study the heat and mass transfer. It is observed that the concentration profile decreases due to the dependence of the Lewis number on the Brownian diffusion coefficient, which means that an augmentation in the Brownian diffusion coefficient brings about a decrease in the concentration profile and the nanofluid Lewis number.
Figure 15 shows the effect of Peclet number (Pe) on the microrotation distribution χ(η). The Peclet number is the prominent factor to study the microorganisms swimming in the fluid. The Peclet number is defined as the ratio of maximum cell swimming speed to diffusion of microorganisms. Diffusion is the process in which a substance moves from an area of high concentration to an area of low concentration. It explains the movement of the substances in the fluid. It is found that diffusivity of microorganisms is decreased in the case of an augmentation in Pe. As a result, the microrotation distribution declines. Figure 16 depicts the effect of the bioconvection Lewis number (Lb) on the microrotation distribution. Similar to Figure 14, an augmentation in Lb results in a decrease in the diffusivity of microorganisms, which results in the reduction of the motile density profile.
Figure 17 portrays the effect of microorganism concentration difference parameter σ on the motile density profile. It is observed that by increasing the value of σ, the concentration of microorganisms in ambient fluid is decreased. Figure 18 delineates the effect of the regular Lewis number (Le) on the solute profile γ(η). The Lewis number is defined as the ratio of thermal diffusivity to mass diffusivity. As seen in Figure 13, the Lewis number is related to the Brownian diffusion coefficient. It is observed that a positive variation in Brownian diffusion leads to a decrease in the concentration of particles. Thus, a positive variation in the Lewis number (Le) leads to a decrease in the solute profile. Figure 19 portrays the relationship between the Dufour Lewis number (Ld) and the solute profile γ(η). The Dufour Lewis number depicts the influence of temperature gradient on the concentration field. It is perceived that the concentration gradient excites the flow with an enhancement in the thermal energy, which results in an increase in the solute profile. Figure 20 depicts the effect of mass fraction field on Nb for the distinguished values of the nanofluid Lewis number (Ln). It is also observed that due to the random collision of molecules, the heat transfer process escalates and nanoparticle diffusion reduces, which results in an increment in the Sherwood number.
Figure 21 elucidates the performance of Nt on the mass fraction field for various values of Ln. It is found that in the presence of the thermophoretic force, the nanoparticles present close to the hot boundary have been shifted towards the cold fluid, which decreases the thermal boundary layer and heightens the nanofluid Lewis number. An upsurge in Nt escalates nanofluid Lewis number (Ln) and further leads to an augmentation in the mass fraction field. Figure 22 presents the effect of microorganism concentration difference parameter σ on the density number of microrotation distribution for different values of Peclet number (Pe). A positive variation in σ lessens the thickness of the boundary layer and leads to an increment in the concentration of the motile gyrotactic microorganisms. Figure 23 elucidates the conduct of the Dufour Lewis number (Ld) on the solutal Sherwood number for different values of the Prandtl number. The Lewis number is defined as the ratio of thermal diffusivity to momentum diffusivity. It is observed that an enhancement in Lewis number drives more heat within the fluid, which brings about an augmentation in the Prandtl number. It is noteworthy that a positive variation in the Dufour Lewis parameter leads to an augmentation in the solutal Sherwood number. Figure 24 elaborates the effect of Lewis number (Le) on the solutal Sherwood number. It has been observed that the solutal Sherwood number increases with an augmentation in the Lewis number.
5 Concluding remarks
This article elaborates the effects of nanoparticles and double-diffusive convection along with motile gyrotactic microorganisms on the non-Newtonian fluid past a stretching sheet. To our knowledge, no model has been developed so far to see the impact of gyrotactic microorganisms and double-diffusive convection simultaneously on the non-Newtonian hyperbolic tangent nanofluid, and furthermore, a numerical technique (Keller box) has been used to achieve the numerical solution of the problem. A comparison with the previous literature was made to check the reliability of our proposed numerical scheme. The results are quite promising. Some of the key findings of the present investigation are as follows:
- An improvement in the Weissenberg number (We) leads to a decrease in the velocity profile.
- The mass fraction field shows an opposite behaviour as a result of variation in the nanofluid Lewis number (Ln).
- A positive variation in the Peclet number (Pe) leads to a decrease in the solute profile.
- The microrotation distribution profile declines with an improvement in the bioconvection Lewis number (Lb) and microorganism concentration difference parameter σ.
- The solute profile is decreased with an enhancement in the regular Lewis number (Le).
magnetic field strength
ambient solute concentration at the wall
skin friction coefficient
solute concentration at the wall
diffusivity of the microorganisms
bioconvection Lewis number
Dufour Lewis number
nanofluid Lewis number
power law index
ambient density of the motile microorganisms
modified Dufour parameter
density of the motile microorganisms at the wall
wall mass flux
wall heat flux
stretching velocity along the x-axis
cell swimming speed
mixed convection parameter
density of the fluid
density of the microorganisms
density of the nanoparticles
microorganism concentration difference
wall shear stress
volume fraction of the nanoparticles
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