Gyrotactic micro-organism flow of Maxwell nanofluid between two parallel plates

The present study explores incompressible, steady power law nanoliquid comprising gyrotactic microorganisms flow across parallel plates with energy transfer. In which only one plate is moving concerning another at a time. Nonlinear partial differential equations have been used to model the problem. Using Liao's transformation, the framework of PDEs is simplified to a system of Ordinary Differential Equations (ODEs). The problem is numerically solved using the parametric continuation method (PCM). The obtained results are compared to the boundary value solver (bvp4c) method for validity reasons. It has been observed that both the results are in best settlement with each other. The temperature, velocity, concentration and microorganism profile trend versus several physical constraints are presented graphically and briefly discussed. The velocity profile shows positive response versus the rising values of buoyancy convection parameters. While the velocity reduces with the increasing effect of magnetic field, because magnetic impact generates Lorentz force, which reduces the fluid velocity.


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www.nature.com/scientificreports/ viscoelastic term effect. The viscous nanoliquid flow over a stretching layer containing gyrotactic microorganisms was studied by Rehman et al. 22 . They discovered that increasing the values of Peclet numbers lowers the density of motile microorganisms. Khan et al. 23 inspected the bioconvection characteristics of nano-size particles in presence of chemical reaction and Lorentz force in non-Newtonian fluid over moving surface. Three dimensional nanoliquid flow passes via parallel plates with the effects of electric field and heat transfer characteristics using Matlab bvp4c and RK4 is analyzed by Shuaib et al. 24 . Acharya et al. 25,26 explored the bioconvection water-based nanofluid flow across a permeable surface in the presence of gyrotactic microorganisms. The flow study has been studied with the influence of the surface slip condition. The magnetic force in the field of hydrodynamics has several applications, especially the electro conducting fluid squeezing flow between two parallel plates have many applications. Which has been already explained in the above paragraphs, keep in view these uses of conducting squeezed flow; we extend the idea of Ferdows et al. 27 . Within the established mathematical model, incompressible squeezing flow of viscous fluid with heat transfer under magnetic field has been studied. The next section consists of formulation and discussion related to the problem.

Mathematical formulation
We consider incompressible, steady power law nanoliquid comprising gyrotactic microorganisms flow across parallel plates. The lower plate is at rest in this case. U w is the uniform velocity of the upper layer. The gap h * between the plates distinguishes them. Figure 1 illustrates the flow geometry. Both the horizontal y-axis and the vertical x-axis are subjected to a variable magnetic field. The lower plate is kept at temperature T 0 , while the upper plate is kept at a constant temperature T ∞ . The Lorentz force ⇀ J × ⇀ B is used to modify the equations for non-conducting plates. The continuity equation, electricity, and Maxwell can all be written as 27 under the above assumptions: www.nature.com/scientificreports/ The imposed boundary conditions for infinite rotating disk are as follows: Here, g, T ∞ , T , β , D T , D B , σ e , B 0 and C w is the acceleration, free stream temperature, fluid temperature, heat source/sink parameter, thermophoresis coefficient, Brownian diffusion coefficient, electrical conductivity, magnetic field strength and the concentration. Where, c p , C, τ , D n , W c , K, ρ and µ 2 is the specific heat, concentration, effective heat capacitance, diffusivity of the microorganisms, cell swimming speed, thermal diffusion ratio, density and magnetic permeability of the fluid respectively.
The structure of PDEs will be reduced into a function of single variable, using the following conversion 27 : The following system of ODE is formed by using Eq. (8) in Eqs. (1)- (7) The transform boundary conditions are spelled out as follows: here, R 1 = h 2 ν and R 2 = αh 2 2ν is the Reynolds number based on the speed of the plates. where, the magnetic strength, modified Prandtl number, Prandtl number, Peclet number, Brownian motion, Thermophoresis parameter, Lewis number, Bioconvection Lewis number, Buoyancy convection coefficient due to temperature, Buoyancy convection coefficient due to concentration and Batchlor number are defined as 27,28 : The skin friction C f , heat transmission rate Nu x , mass transmission rate Sh x and density of motile microorganism Nn x are given as: where, q w = −k ∂T/∂y y=o , τ w = −µ ∂u/∂y y=o , q n = −D n ∂n/∂y y=o and q m = −D B ∂C/∂y y=o is the heat flux, shear stress, flux of motile microorganism and mass flux at plate surface at the surface of the plate. Now using above terminologies, Eq. (16) yields: (16) x Nn x = −χ ′ (0).

Numerical solution
The following steps present the fundamental concept of applying the PCM method to an ODE system (9-13) with a boundary condition (14).
Step 1: The BVP system is being converted into a first-order system of ODE The following functions will be introduced: Using transformations (18) into the BVP (9-13) and (14), to get: The associated boundary conditions are: Step 2: The embedding term p is introduced as: We will thoroughly add the continuation parameter p in the system (19)(20)(21)(22)(23) to obtain an ODE system in a p-parametric family.
Step 4: For each element, assert the principle of superposition and define the Cauchy problem.
where U, W, and a denote unknown vector functions and blend coefficients, respectively. For each component, solve the two Cauchy problems listed below.
We get the approximate solution Eq. (32) by plugging it into the original Eq. (30).
Step 5: Solving the Cauchy problems This work employs a numerical implicit scheme, which is depicted below.
where the iterative form of the solution is obtained

Results and discussion
Velocity profile. Buoyancy convection parameter due to temperature 1 , concentration 2 and micro-  Fig. 3a,b respectively. Actually, the kinematic viscosity of fluid increases, and thermal diffusivity decreases with rising values of Prandtl number, that's why such trend has been observed. Brownian behavior is considered as the non-movement of fluid molecules over the plate's surface. Brownian motion induces heat by increasing the unspecific motion of liquid particles. As a result, the liquid temperature rises, as does the thickness of the thermal boundary layer as shown in Fig. 3c. Furthermore, as the thermophoresis component improves the smallest nanomaterials are escorted away from heated surface and toward the cold surface. As a consequence, as shown in Fig. 3d, a larger number of small nanoparticles are drawn away from the warm surface, increasing the liquid temperature.
(31) dζ i dτ , Gyrotactic microorganism profile. Prandtl number Pr, Lb and Pe effect on gyrotactic microorganism profiles χ(η) has been shown through Fig. 6a-c. The Gyrotactic Microorganism profile reduces versus the action of Pr, Lb and Pe respectively. With distinct Prandtl numbers, the thickness of the hydrodynamic boundary layer and the thickness of the thermal boundary layer are calculated physically. If Pr = 1, that means the thermal boundary layer's thickness is the same as the velocity boundary layer's thickness. As a result, it's the momentum-to-thermal-diffusivity ratio. That' why, fluid temperature reduces with growing value of Prandtl number as demonstrated in Fig. 6a. Similar trend has been observed of Gyrotactic Microorganism profile versus Bioconvection Lewis number Lb and Peclet number in Fig. 6c,d. Table 1 revealed the comparison of PCM technique with the existing literature, while varying n and M. The rest www.nature.com/scientificreports/ of parameters were chosen zero. Table 2 illustrate the numerical outcomes for skin friction and Nusselt number versus several physical constraints. The temperature of the sheet surface rises when the magnetic field parameter is increased, as shown in Table 2.

Conclusion
The current research investigates the movement of an incompressible, steady power law nanoliquid containing gyrotactic microorganisms between two parallel plates with heat transmission. Only one plate moves in relation to another at a time. Nonlinear partial differential equations have been used to model the problem (PDEs). Which reduced form (ordinary differential equations) is solved through the Parametric Continuation Method (PCM). The results are related to the boundary value solver (bvp4c) approach for validation and accuracy purposes. The below are the objectives:   www.nature.com/scientificreports/

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.