Exploration of bioconvection flow of MHD thixotropic nanofluid past a vertical surface coexisting with both nanoparticles and gyrotactic microorganisms

This communication presents analysis of gravity-driven flow of a thixotropic fluid containing both nanoparticles and gyrotactic microorganisms along a vertical surface. To further describe the transport phenomenon, special cases of active and passive controls of nanoparticles are investigated. The governing partial differential equations of momentum, energy, nanoparticles concentration, and density of gyrotactic microorganisms equations are converted and parameterized into system of ordinary differential equations and the series solutions are obtained through Optimal Homotopy Analysis Method (OHAM). The related important parameters are tested and shown on the velocity, temperature, concentration and density of motile microorganisms profiles. It is observed that for both cases of active and passive control of nanoparticles, incremental values of thermophoretic parameters corresponds to decrease in the velocity distributions and augment the temperature distributions.

www.nature.com/scientificreports/ cases of hypolimnion and epilimnion. Waqas et al. 6 studied the solar radiation and joule heating in magnetohydrodynamic convective flow of thixotropic nanofluid, they found out that, increase in thixotropic parameters leads to increase in the fluid velocity. Other studies that reported boundary layer flow of thixotropic fluid can be found in [7][8][9] . A variety of approaches have been proposed to improve the heat transfer efficiency of various working fluids with low thermal conductivity. The thermal conductivity of the working fluids can be increased to improve heat transfer efficiency. Nanofluids, which assist to enhance the heat transfer efficiency of fluids in industrial processes, have been introduced as a novel notion by nanotechnology. Nanofluids are a novel type of solid-liquid composite material made up of solid nanoparticles (1−100 nm) suspended in a base liquid. Choi 10 . Because of their increased heat transfer efficiency in diverse thermal systems, nanofluids are employed in a wide range of engineering applications. Engine coolant, automatic transmission fluid, brake fluid, gear lubrication, engine oil, and greases are all possible applications of nanofluids 11 Senthilraja et al. Nanomaterials are also well designed as flavor additives, or carriers for food supplement (i.e. nanoencapsulation and nanoemulsion) Weir et al. 12 . Ahmed 13 presented the influence of slip boundary condition on Casson nanofluid flow over a stretching sheet in the presence of viscous dissipation and chemical reaction. With an increase in Brownian motion and thermophoresis parameter, the heat transfer rate is reduced, but the mass transfer rate is clearly enhanced, according to the study. Rauf et al. 14 studied mixed convection flow of couple stress nanofluid over oscillatory stretching sheet with heat absorption/generation effects. Entropy generation in second grade magnetohydrodynamic nanofluid flow over a convectively heated stretching sheet with nonlinear thermal radiation and viscous dissipation was studied by Mondal et al. 15 , they discovered that, thermophoresis parameter acts to increase the temperature of the fluid.
The process of pattern formation observed in the aqueous suspension of motile microorganisms when they respond to certain stimuli by swimming in certain directions termed taxes is referred to as bioconvection Platt 16 . In other words, bioconvection occurs when microorganisms with a higher density than the water swim upward towards the upper surface of the water; however, when the upper surface becomes too dense due to the accumulation of microorganisms, it becomes unstable, and microorganisms swim downward to the lower layer of the water, maintaining bioconvection. Ghorai and Hill 17 explained that gyrotaxis is swimming directed by the balance between the torque due to gravity acting on a bottom-heavy cell and the torque due to viscous forces arising from local shear flows. Raees et al. 18 reported that bioconvection in nanofluids has great potential in Colibri micro-volumes spectrometer and also to improve the stability of nanofluids. They further revealed the application of bioconvection in the field of microbial enhanced oil recovery, which involves injection of selected microorganisms into the reservoir and by their situ multiplication they reduce the residual oil left in the reservoir after secondary recovery is exhausted. Makinde and Animasaun 19 considered bioconvection in MHD nanofluid with nonlinear thermal radiation and quartic autocatalysis chemical reaction past an upper surface of a paraboloid of revolution. It was reported that velocity profiles and density of motile microorganisms are decreasing functions of Schmidt number for diffusing motile microorganism. Recently, Saleem et al. 20 discussed magneto Jeffrey nanofluid bioconvection over a rotating vertical cone due to gyrotactic microorganisms. It was discovered that when the bioconvection Peclet number and bioconvection Schmidt number increase, the rescaled density of motile microorganisms decreases.
Temperature and concentration differences, respectively, are known to be driving forces for heat and mass transfer. Furthermore, both heat and mass are considered to be transferred from the region of higher concentration to the region of lower concentration. The thermal relaxation time expression was initially incorporated into Fourier's law by Catteneo 21 . By adding the Oldroyd upper-convected derivative, Christov 22 improved on Catteneo's hypothesis. The Catteneo-Christov heat and mass theory is the name given to the theory currently (CCHmodel). Sarojamma et al. 23 analyzed the effects of an autocatalytic chemical reaction and a Catteneo-Christov heat flux on the dynamics of a micropolar fluid. They discovered that when the thermal relaxation parameter is set to zero, the temperature of the fluid declines and the Catteneo-Christov (CCH) heat flux model is simplified to the traditional Fourier's Law of conduction. Rasool and Zhang 24 considered Darcy-Forchheimer nanofluid flow manifested with Cattaneo-Christov theory of heat and mass flux over non-linearly stretching surface. It was discovered that the solutal relaxation time parameter exhibits mixed behavior in nanoparticles concentration distribution. Sandeep et al. 25 considered free convective MHD Cattaneo-Christov flow over three different geometries with thermophoresis and Brownian motion. It was reported that thermal relaxation parameter helps to enhance the heat transfer rate but lessened the mass transfer rate.
To the best of authors' knowledge, numerous studies have been reported on boundary layer flow of nanofluid in the presence of nanoparticles and gyrotactic microorganisms manifesting with the Catteneo-Christov heat and mass flux model (CCH) in the recent years. Furthermore, it is worth mentioning that there is no research on the comparison between the active and passive controls of nanoparticles of thixotropic fluid flow in the presence nanoparticles and gyrotactic microorganisms coexisting with Catteneo-Christov heat and mass flux model (CCH).

Mathematical formulation
In line with boundary layer theory a steady, buoyant convective boundary layer equation of thixotropic-nanofluid is explored to meet our objectives. The model is modified to a particular extent where the mathematical formulation of the problem can be justified. The flow is assumed to be in the x-direction, which is taken vertically upward along the plate, and the y-direction is normal to it. (T w ) , nanoparticle concentration (C w ) , and motile microorganism density (N w ) at the stretching surface are assumed to be constant and are considered to be greater than temperature (T ∞ ) , nanoparticles concentration (C ∞ ) , and motile microorganisms density (N ∞ ) at the free stream, respectively. The presence of nanoparticles is thought to have no influence on the direction and speed with which microorganisms swim. It is assumed that bioconvection flow occurs only in dilute nanoparticles www.nature.com/scientificreports/ suspensions. It is worth mentioning that the base fluid is water so that the microorganisms can survive. The active and passive controls of nanoparticles volume fraction at the boundary are both taken into account (on the solid wall). Under the foregoing assumptions with Boussinesq approximation, and following the works of Rehman et al. 26 and Shehzad 8 , the governing equations for mass, momentum, energy, and density of gyrotactic microorganisms in two-dimensional thixotropic-nanofluid may be expressed as follows; Continuity Equation

Momentum Equation
Energy equation of nanoparticles

Concentration equation of nanoparticles
Density of gyrotactic microorganisms equation The equations above are subjected to the following two sets of boundary conditions: (i) The actively controlled nanofluid model (ii) The passively controlled nanofluid model where u and v are velocity components in x and y directions respectively, c h is the chemotaxis constant, W c is the maximum cell swimming speed, m is the velocity power index, T f is the local fluid, T is the temperature, C is the nanoparticle, N is the density of motile micro-organisms, p is the pressure, ρ f , ρ p , ρ m are the density of nanofluid, nanoparticles and micro-organisms respectively, D B , D T , D m are the Brownian diffusion coefficient, thermophoresis diffusion coeffient and diffusivity of micro-organisms respectively, R 1 and R 2 are the non-Newtonian material constants, κ, σ are the thermal and electrical conductivity of the fluid respectively, α is the thermal diffusivities, τ is the ratio of the effective heat capacitance of the nanoparticle to that of the base fluid.
For the sake of non-dimensionalization and parameterization of Eqs. (2), (3), (4) and (5) subject to boundary conditions (6) and (7), the following two sets of similarity variables are considered corresponding to the following models: (i) In the actively controlled nanofluid model, In the dimensionless equations defined above, S cm is the Schimdt number for diffusing motile microorganisms, L e is the Lewis number, P r is the Prandtl number, G r is the Grashof number, N r is the buoyancy-ratio parameter, R b is the bioconvection Rayleigh number, N b is the Brownian motion parameter, N b is the thermophoresis parameter, P e is bioconvection Peclet number.
is the buoy- is the thermophoresis parameter, δ t = 1 a is the relaxation time parameter of temperature, L e = ϑ D B is the Lewis number, δ n = 2 a is the relaxation time parameter of nanoparticle volume fraction, ς = N ∞ (N w −N ∞ ) is gyrotactic microorganisms concentration difference parameter, S cm = ϑ D n is the Schmidth number for diffusing motile microorganisms and P e = bW c D n is the Peclet number. The shear stress, the local heat flux, the local mass flux, and the motile microorganisms flux on the surface are τ w , q w , q m and q n respectively. They are defined as; In dimensional practice these quantities can be written as where Re x = u w x ϑ denotes local Reynolds number.
Optimal homotopy analysis solutions. In many cases, by means of analyzing the physical background and the initial/boundary conditions of the nonlinear differential problem, we might know what kinds of base functions are proper to represent the solution, even without solving the given nonlinear problem. In view of the boundary conditions (18)  The linear operators L f , L θ , L φ and L w have the following properties In which C 1 , C 2 , C 3 , C 4 , C 5 , C 6 , C 7 , C 8 , and C 9 are constants. The linear operators were solved using Wolfram Mathematica to obtain the vales of skin friction coefficient f ′′ (0) , Nusselt number −θ ′ (0) , Sherwood number −φ ′ (0) and density number −w ′ (0) for various values of thermophoresis parameter N t

Results and discussion
The flow of thixotropic-nanofluid in a water-based suspension has been computed for various values of emerging parameters. On velocity profiles f ′ (η) , temperature profiles θ(η) , nanoparticles concentration profiles φ(η) , and density of motile microorganisms profiles w(η) , the effects of certain selected parameters have been evaluated and presented. The theoretical values of the governing parameters for both cases of actively controlled nanofluid model and passively controlled nanofluid model are M = G r = N r = R b = 0.5, N b = N t = ς = δ t = δ n = K 1 = K 2 = 0.1, S c = P e = 1.0 and P r = L e = 2.0 . The variations of local Nusselt numbers for both cases of passive and active controls of nanoparticles are depicted in Tables 1 and 2. In Table 1 it is revealed that the local Nusselt number is a decreasing function of P r , N b and N t = −θ ′ (0) for various values of P r , N b and N t when M = G r = N r = R b = 0.5, P e = 2.0, ς = δ n = δ t = K 1 = K 2 = 0.1.   Table 2 it is noticed that the local Nusselt number is a decreasing function of P r , N b and N t for active control of nanoparticles. The velocity of the thixotropic-nanofluid increases slightly in the event of active control of nanoparticles as seen in Fig. 1a. When η = 1.68 , all profiles merged and the fluid velocity dropped towards the free stream.
In Fig. 1b for case of passive control of nanoparticles, it is revealed that as thixotropic parameters increase, there is a noticeble decrease in the velocity of the fluid. Figure 2a shows that the temperature of the fluid decreases as the thixotropic parameters K 1 , K 2 increase in the case of active control of nanoparticles, while in Fig. 2b increamental values of K 1 , K 2 leads to rise in the temperature of the thixotropic-nanofluid in case of passive control of nanoparticles. In Fig. 3a it is observed that the velocity distribution declines with the incremental values of magnetic parameter M, while in Fig. 3b, the temperature distribution is enhanced as M increases. Physically, the introduction of magnetic field to the flow directions induces a Lorentz force, which has the ability to retard the flow of fluid and thereby reduce the velocity of fluid and enhance the temperature of fluid.
In Fig. 4a it is noticed that as the G r increases from 0.001 through 0.5, 1.0 to 1.5 for the case of active control of nanoparticles the velocity distribution slightly increases and decay towards η = ∞ and also in case of passive control of nanoparticles, there is rise in the velocity profile, as seen in Fig. 4b. The observed fluid velocity behavior is owing to the fact that gravitational force increases mobility on the vertical surface. It is revealed in Fig. 5a in case of active control of nanoparticles that as G r increases temperature profile decreases within the domain 0.67 ≤ η ≤ 2.50 at η = 2.52 all profiles merged together and decay towards η = ∞ , where as in Fig. 5b when nanoparticles are controlled passively, the temperature profiles decrease slightly, and all profiles fused together at η = 2.0 , and later decay towards the free stream. It is observed from Fig. 6a that as G r increases for the cases of active control of nanoparticles, concentration profile decreases along the wall. It is noticed from Fig. 6b for case of passive control of nanoparticles that as G r is increased all profiles merged at a distance very     www.nature.com/scientificreports/ close to the wall. It is later observed that at η = 0.58 the concentration of nanoparticles decline away from wall within the domain 0.58 ≤ η ≤ 3.75 and later on, the profiles merged together at η = 4.23 and thereafter, there is a substantial enhancement in the nanoparticles concentration. It is obvious that there is a noticeable difference in the behaviour of concentration of nanoparticles suspended in highly viscous thixotropic fluid for both cases of nanofluid model in consideration. Figures 7a, 8, 9, 10b depict the influence of the thermal relaxation time parameter δ t for K 1 = K 2 = 0 , which corresponds to Newtonian case. Figure 7a shows that δ t has a diminishing influence on the velocity profile, and Fig. 7b shows a similar pattern for passive control of nanoparticles. It is shown in Fig. 8a for case of active control of nanoparticles that as δ t increases both fluid temperature and thermal boundary layer thickness demonstrate a decelerating practice while similar behaviour is observed in Fig. 8b in the event of passive control of nanoparticles. Physically, due to thermal relaxation augmentation, material particles require more time for heat transfer to their neighboring particles. Figure 9a shows that for case of active control of nanoparticles, as δ t increases from 0.1 through 0.4, 0.7 to 1.0, a decreasing trend in the nanoparticles concentration profile with the solutal boundary layer thickness is observed. Whereas in case of passive control of nanoparticles, a mixed trend is observed in nanoparticles concentration profile for incremental values of δ t . Physically, relaxation time parameter δ t permits sufficient time to nanoparticles to dilute in base fluid which leads to an increasing trend with the passage of time. In Fig. 10a it is noticed that density of motile microorganisms profile decreases with increasing values of δ t for case of actively controlled nanofluid model. Physically, this corresponds to the fact that gyrotactic cells swim back to bottom layer of the fluid resulting in reduction in the density of the motile microorganisms. Figure 10b on the other hand shows how incremental values δ t result in a relatively slight increase in motile microorganisms   Fig. 11a,b. It is noticed in Fig. 11a that a mixed trend is observed for actively controlled model, it is further observed that concentration of nanoparticles is enhanced at the wall. For higher value of δ n concentration of nanoparticles rapidly increases to a peak value (which is more than 0.3). In case of passive control of nanoparticles, incremental values of δ n result in a decline in concentration profiles of nanoparticles, as shown in Fig. 11b. Figure 12a manifests that by varing the N t , non-dimensional nanoparticles concentration profile increases to peak value of 5.2 for case of actively controlled model. Likewise, it is revealed in Fig. 12b for case of passively controlled model that, as N t varies, the non-dimensional concentration profile increases to peak value of 4.8 and decay smoothly to the free stream. This observation is due to the fact that higher N t is expected to allow a deeper penetration of the concentration. Figure 13a,b communicate the implications of N b on the dimensionless temperature profiles respectively. Brownian motion N b can be described as the ratio of the nanoparticle diffusion to the nanofluid thermal diffusion Mahdy 26 . It is seen from Fig. 13a for case of the active control of nanoparticles that incremental values N b permits rise in temperature distribution and thermal boundary layer thickness. Physically, this is because presence of Brownian motion makes particles to be more closer together, allowing heat    www.nature.com/scientificreports/ to be transferred between them. Also, in Fig. 13b, it is observed that all particles merged together as a result of the strong intermolecular force binding the molecules of particles suspended in thixotropic-nanofluid. Beyond this reaction, all particles diverge and the temperature profile increases between the region 1.6 ≤ η ≤ 3.5 , owing to presence of Brownian motion which is motivated to warm the molecules of the particles and enhances the thermal conductivity, thereby causing the temperature of the fluid to be elevated. Brownian motion contributes to efficiency of heat transfer in thixotropic-nanofluids, as evidenced by the same behavior found in these two models.
Increasing values of ς correlate to reduction of the dimensionless velocity distributions for both active and passive control of nanoparticles, as shown in Fig. 14a,b. This is because density difference between the gyrotactic microorganisms and base fluid makes the surface of the fluid to become unstable as a result of cell agglomeration, causing the motile microorganisms to swim back to bottom layer of the fluid.
In case of active control of nanoparticles, an increase in ς causes decline in density of motile microorganisms profile, as shown in Fig. 15a. Similarly, in Fig. 15b in case of passive control of nanoparticles, it is revealed that density of motile microorganisms profile decreases near the wall with peak value estimated to be 6.29 for stronger ς. Figure 16a,b show that for both active and passive controls of nanoparticles, fluid velocity and momentum boundary layer thickness show a decelerating trend as P e increases. It shown in Fig. 17a that increase in the values of P e result in a slight increase in temperature of the fluid for active control of nanoparticles, whereas in case of passive control of nanoparticles, there is noticeable increase in temperature of the fluid and thermal boundary layer thickness, as shown in Fig. 17b. It is clearly observed in Fig. 18a in case of active control of nanoparticles that, incremental values of P e reflect decline nature of density of motile microorganisms distribution. It is observed in Fig. 18b for the passive control of nanoparticles that increasing values of P e is a dimensionless group that represents the ratio of heat transfer via fluid motion to heat transfer by thermal conduction; this is  www.nature.com/scientificreports/ the major reason why the temperature field increases while density of the motile microorganisms field decreases when P e value changes.

Conclusion
The Catteneo-Christov double-diffusion model is used to investigate the features of boundary layer flow of a water-based thixotropic-nanofluid containing gyrotactic microorganisms past a vertical surface. In the study, both actively and passively controlled nanofluid models were used. To convert the governing partial differential equations into a system of ordinary differential equations a set of similarity variables have been introduced. This investigation yielded the following significant findings 1. For both active and passive control of nanoparticles, dimensionless velocities and temperatures are decreasing functions of thermal relaxation δ t . Similarly, in case of active control of nanoparticles, dimensionless concentration nanoparticle is a decreasing function of relaxation parameter δ t , but in case of passive control of nanoparticles, a mixed trend is observed. 2. Increasing solutal parameter δ n reduces concentration of nanoparticles profiles in case of passive control of nanoparticles while a mixed trend is observed in case of active control of nanoparticles 3. Incremental values of thixotropic parameters K 1 , K 2 correspond to increase in dimensionless fluid velocity for both active and passive control of nanoparticles  for both cases of active and passive controls of nanoparticles. Inside thermal boundary layer, dimensionless temperature increases with thermophoretic parameter N t for both cases of active and passive controls of nanoparticles 6. Inside the thermal boundary layer, dimensionless temperature profiles increase with Brownian motion parameter N b for both cases of active and passive controls of nanoparticles.