Parametric estimation of gyrotactic microorganism hybrid nanofluid flow between the conical gap of spinning disk-cone apparatus

The silver, magnesium oxide and gyrotactic microorganism-based hybrid nanofluid flow inside the conical space between disc and cone is addressed in the perspective of thermal energy stabilization. Different cases have been discussed between the spinning of cone and disc in the same or counter wise directions. The hybrid nanofluid has been synthesized in the presence of silver Ag and magnesium oxide MgO nanoparticulate. The viscous dissipation and the magnetic field factors are introduced to the modeled equations. The parametric continuation method (PCM) is utilized to numerically handle the modeled problem. Magnesium oxide is chemically made up of Mg2+ and O2- ions that are bound by a strong ionic connection and can be made by pyrolyzing Mg(OH)2 (magnesium hydroxide) and MgCO3 (magnesium carbonate) at high temperature (700–1500 °C). For metallurgical, biomedical and electrical implementations, it is more efficient. Similarly, silver nanoparticle's antibacterial properties could be employed to control bacterial growth. It has been observed that a circulating disc with a stationary cone can achieve the optimum cooling of the cone-disk apparatus while the outer edge temperature remains fixed. The thermal energy profile remarkably upgraded with the magnetic effect, the addition of nanoparticulate in base fluid and Eckert number.

www.nature.com/scientificreports/ using 50% Ethylene glycol as the base fluid. Khan et al. 23 presented about the Marangoni convection of a hybrid nanofluid made up of two nanoparticles and a base fluid. Some related literature may be found in [24][25][26][27] . Shi et al. 28,29 proposed a mathematical framework to evaluate the bio-convection flow of a magneto-cross nanofluid including gyrotactic microorganisms with transient magnetic flow of Cross nanoliquid through a stretched sheet. Yusuf et al. 30 studied the rate of entropy generation in a bio-convective flow of a MHD Williamson nanoliquid across an inclined convectively heated stretchable plate, taking into account the effects of thermal radiation and chemical reaction. The presence of microorganisms has been shown to help in the stabilisation of suspended nanoparticles via a bioconvection mechanism. Khashi'ie et al. 31 and Wahid et al. 32 numerically investigates the influence of gyrotactic microorganisms in the mixed convection stagnation point flow of Cu-Al2O3/water hybrid nanofluid towards an immovable plate.
Highly nonlinear boundary value concerns that cannot be solved are common in the automotive industry. For many problems that are routinely addressed by other numerical methods, convergence is susceptible to relaxation constants and initial framework. The PCM's objective is to uncover the technique's universal applicability as a viable solution to nonlinear issues 33 . The 3D turbulent flow and heat transmission over the surface of an extensible spinning disc is highlighted by Shuaib et al. 34 . The fluid has been studied in the presence of external magnetic strength. Shuaib et al. 35 identified the characteristic of an ionic transitional boundary layer flow across a revolving disc. To find an ionic species, the Poisson's and Nernst-Planck equations were utilized. Wang et al. 36 used a parametric continuation technique to analyze the consistency of complex systems for engineering disciplines. They also investigated the static bifurcation that occurs while solving nonlinear starting value problems with distinct features and developed an algorithm for determining the bifurcation points in detail. Bilal et al. 37 explored the fluctuating Maxwell nanofluid flow around a stretched cylinder guided by suction/injection impact. The resulting system of ODEs was then numerically calculated using the PCM technique, and the results were validated using the Matlab program bvp4c.
The above examinations indicated that no attempt has been made so far to analyze the 3D flow of silver, magnesium oxide and gyrotactic microorganism-based hybrid nanofluid inside the conical space between disc and cone in perspective of thermal energy stabilization. Multiple cases involving the rotation of disc and cone in the same or reverse trajectory have been addressed. The hybrid nanocomposites are formed in the context of silver Ag and magnesium oxide MgO nanomaterials. The magnetic field and viscous dissipation components are included in the simulated equations. To numerically address the posed situation, the computational methodology parametric continuation method is used.

Mathematical formulation
We considered an incompressible flow of silver and magnesium oxide-based hybrid nanofluid between a cone and disk under the consequences of the magnetic field. The motile microorganism has been also considered in the present analysis. Both the devices are supposed to be either stationary or spinning in the r, ϕ, z direction (cylindrical coordinate) with the angular velocity. The and ω elaborate the cone and disk angular velocities. Figure 1 communicates the hybrid nanofluid flow mechanism between the disk and cone. The phenomena. The radial variable surface temperature T w = T ∞ + cr n , where c and n are kept constant. Here, p is the pressure depends on radial r and axial z distance between the conical gaps. Based on the above presumption, the flow mechanism can be stated as 38 :  (7) introduced concentration and motile microorganism profiles respectively. Here, Ñ , D n, Wc, B 0 and p is the motile microorganism's density, microorganism diffusion, floating velocity of cell, magnetic strength and pressure. While k hnf ,ν hnf ,ρ hnf ,µ hnf , ρc p hnf and (u, v, w) is the thermal conductivity, kinematic viscosity, density, dynamic viscosity, electrical conductivity, heat capacitance and velocity terms along r, ϕ, z direction.
The boundary conditions are: Here γ specified the gap angle between the cone and disk.
Similarity conversion. We adopt the following similarity transformation 39 : Using Eqs. (8) in Eqs. (2-5), we get: Thermo-physical properties. The following are the thermal properties of hybrid nanofluid and water are 42 : The dimensionless form of cone and disk are rebound as:
Step 4: Use the superposition approach to each problem and characterize the Cauchy problem For each term, resolve the Cauchy problems below.
Step 5: Solving the Cauchy problems This study employs a numerical implicit methodology, which is detailed below.
we get the iterative form of the solution.

Result and discussion
The goal of this part is to learn about the effects of velocity, temperature, mass, and motile microorganism distributions under the effect of multiple fundamental factors. The flow mechanics of a circulating cone and disc is observed in Fig. 1. Table 1 addressed the numerical properties of silver, magnesium oxide and water. The four different cases are discussed in detail between cone and disk. Case 1 elaborated that the disk is spinning while the cone is fixed. Case 2 revealed that the cone is spinning, while the disk is fixed. Case 3 & 4 highlighted that the disk and cone are co-rotating or counter-rotating respectively. The following observations have been noticed: www.nature.com/scientificreports/ Axial velocity profile. Figure 2a-e revealed the behavior of axial velocity f (η) profile versus magnetic field M, volume friction of silver φ Ag , volume friction of magnesium oxide φ MgO , cone angular velocity Re and disk angular velocity Re ω respectively. The resistive force generated due to the consequences of the magnetic field declines the axial velocity as shown in Fig. 2a. Figure 2b,c manifested that the velocity profile enhances with the increment of both silver Ag and magnesium oxide MgO nanomaterials because the specific heat capacity of MgO and Ag nanoparticles is much less than water, that's why the rising thermal energy inside the hybrid  Radial velocity profile. Figure 3a-e highlighted the behavior of radial velocity profile g(η) versus magnetic field and four different cases of rotation and counter-rotation of both cone and disk respectively. The magnetic effect is also reducing the radial velocity, while keeping disk stationery and cone moving as illustrated in Fig. 3a. Figure 3b,c highlighted the two cases (1 & 2) and pointed that the radial velocity increases in both cases. Physically, the increasing velocity of both apparatuses excited the fluid particles, which become the reason for the enhancement of the radial velocity profile g(η) . Figure 3d,e connived cases 3 & 4 and shows that the velocity distribution reduces in both cases. The counter-rotation of the disk and cone generated resistance to the flow field, which decline the velocity profile.
Tangential velocity profile. Figure 4a-e elaborated the nature of tangential velocity profile h(η) versus magnetic field M, volume friction of silver φ Ag , volume friction of magnesium oxide φ MgO , cone angular velocity www.nature.com/scientificreports/ Re and disk angular velocity Re ω respectively. The tangential velocity of the fluid is significantly reducing with the influence of the magnetic field as shown in Fig. 4a. Figure 4b-e illustrated that the tangential velocity h(η) profile boosts with the rising quantity of nanoparticles (Ag & MgO) and both disk Re ω and cone Re increasing rotation respectively.
Thermal energy profile. Figure 5a-e displayed the characteristics of thermal energy profile �(η) versus magnetic field M, volume friction of silver φ Ag , volume friction of magnesium oxide φ MgO , Eckert number Ec and Prandtl number Pr respectively. Figure 5a-d communicated that the thermal energy profile remarkably upgraded with the magnetic effect, the addition of nanoparticulated in base fluid and Eckert number respectively. As we have discussed in Fig. 2b,c that the growing credit of nanoparticles diminishes the average specific heat capacity of base fluid, that's why such a scenario has been observed in Fig. 5b,c. Similarly, the dissipation energy is added to fluid internal energy and enhances its thermal energy �(η) distribution as exposed in Fig. 5d. The higher Prandtl fluid Pr has always less thermal diffusivity, that's why the Prandtl effect decrease fluid temperature �(η).  Fig. 6a. An opposite scenario has been observed while keeping the disk fixed and spinning cone Fig. 6b. Figure 6c,d revealed that the energy profile reduces with the co-rotation of disk and cone, while enhances with counter-rotation because the opposite direction motion generates resistive force, which encourages fluid temperature �(η) as scrutinized in Fig. 6d. Figure 7a-d manifested the behavior of mass transfer profile �(η) and motile microorganism ¯ (η) versus Schmidt number Sc, volume friction of silver φ Ag , Reynold number Re and Peclet number Pe respectively. The Schmidt number upshot diminished the mass transition rate as shown in Fig. 7a. Physically, the molecular diffusion lowers, and kinetic viscosity rises with Schmidt number, that's why such phenomena have been observed. An opposite scenario is observed against silver nanoparticles φ Ag that the mass transference enhances with the upshot of silver nanoparticles as elaborated through Fig. 7b. Figure 7c Table 6 revealed comparative analysis for the validation of the results between PCM and bvp4c numerical Matlab package.

Conclusion
In the present study, the silver, magnesium oxide and gyrotactic microorganism-based hybrid nanofluid flow inside the conical space between disc and cone is addressed in the perspective of thermal energy stabilization. The hybrid nanofluid has been synthesized in the presence of silver Ag and magnesium oxide MgO nanoparticulate. The viscous dissipation and the magnetic field factors are introduced to the modeled equations. The numerical boundary value solver bvp4c is utilized to numerically handle the modeled problem. The following results have been observed: • The resistive force generated due to the consequences of magnetic field declines the velocity profiles, while enhances with the increment of both silver Ag and magnesium oxide MgO nanomaterials.     Table 3. Numerical outputs of Nusselt number −� ′ (1) at the surface of cone.   Table 4. Numerical outputs of Sherwood number −� ′ (0) at the surface of disc.