Dual solution for double-diffusive mixed convection opposing flow through a vertical cylinder saturated in a Darcy porous media containing gyrotactic microorganisms

The steady mixed convection flow towards an isothermal permeable vertical cylinder nested in a fluid-saturated porous medium is studied. The Darcy model is applied to observe bioconvection through porous media. The suspension of gyrotactic microorganisms is considered for various applications in bioconvection. Appropriate similarity variables are opted to attain the dimensionless form of governing equations. The resulting momentum, energy, concentration, and motile microorganism density equations are then solved numerically. The resulting dual solutions are graphically visualized and physically analyzed. The results indicate that depending on the systems' parameters, dual solutions exist in opposing flow beyond a critical point where both solutions are connected. Our results were also compared with existing literature.


Mathematical formulation
We consider the steady mixed convection boundary layer flow over a vertical cylinder with a radius r 0 implanted in a saturated permeable medium that contains gyrotactic microorganisms, as shown in Fig. 1. In our work, we assume that the mainstream velocity is U(x) and the cylinder surface is maintained at a constant temperature of T w . We denote the concentration of fluid by C w and motile microorganism concentration by n w . The velocity, temperature, and concentrations are u ∞ , T ∞ , C ∞ and n ∞ . When it is far from the cylinder's surface, the axial and radial coordinates are x and r; in contrast, the x-axis is measured vertically upward along the cylinder's axis, and the r-axis is measured normal to the x-axis. The gravitational acceleration g acts in the downward direction in opposition to the x-direction.
We use the Darcy model in this research, and it assumes less velocity and porosity. It is worth mentioning that water has been chosen as the base fluid for the survival of microorganisms. The buoyancy term is used in the momentum (Darcy) equation due to up swimming microorganisms. Based on the model proposed by Sudnagar et al. 54 , under the assumptions along with the physical phenomena and Boussinesq approximations, the governing equations are (1) to (5) above, T , C , and n are the temperature, concentration, and volume fraction of motile microorganisms. k is the permeability of the porous medium, µ is the fluid viscosity, ρ is the density of the fluid, g is the acceleration due to gravity, β is the thermal expansion coefficient, α is the effective thermal diffusivity of the porous medium, D m is the solute diffusivity, D n is the diffusivity of the microorganism, b is the chemotaxis constant, and finally, W c is the maximum cell swimming speed. The product b · W c is assumed to be a constant.
The boundary conditions take the following form: Following Mahmood and Merkin 34 , we also assume in this paper the following: We now introduce the following dimensionless quantities: where L is the characteristic length, and Pe is the Peclet number. The continuity equation is satisfied by a stream function ψ such that: Substituting Eqs. (9) and (10) in Eqs. (1) to (7) leads to the following coupled differential equations:   In the coupled differential equations, the mixed convection parameter is = Ra Pe , Raleigh number is (1−C∞)β∇T , Lewis number is Le = α D m , bioconvection Lewis number is Lb = α D n , bioconvection Peclet number is Pb = bW c D n , and the microorganism concentration difference parameter is A = n ∞ n w −n ∞ .

Heat, mass, and motile microorganism transfer coefficient
The heat transfer rate, the Sherwood number, and the density parameter for the motile microorganisms are defined as: where q w , q m , and q n represent the constant wall heat, mass, and microorganisms' fluxes, respectively, and they are written as: By using Eqs. (9), (10), (17), and (18), we obtain the dimensionless Nusselt number, Sherwood number, and the local density number of the motile microorganisms at the surface of the cylinder, respectively:

Method of solution
Using similarity transformations, the governing partial differential equations were converted into ordinary differential equations, which are then solved numerically using Matlab bvp4c solver. Matlab bvp4c solver is a finite difference method with fourth-degree accuracy that is applied on a general two-point boundary value problem with an initial solution guess. It does this by integrating a system of ordinary differential equations on the interval [a, b]. From this method, and using a diversity of initial guess f , f ′ , θ, θ ′ , φ, φ ′ , χ, and χ ′ we were able to find the first and second solutions. In the context of the bvp4c function described earlier, we need to transform the governing equations into a system of first order differential equations as follows: First, we arrange Eqs. (11) through (14) as: Next, we transform the above equations into a system of first order differential equations, and for this, we let η = x , and this gives us www.nature.com/scientificreports/ Therefore, the corresponding system of first order differential equations become: For the boundary conditions, we consider that ya is the left boundary, and yb be the right boundary such that: To validate our results, the differential equations are solved numerically using Maple 14.0 dsolve command. The asymptotic boundary conditions in Eqs. (15) and (16) are replaced by using a value of 8 for the similarity variable η max = 8 . The results for both cases are displayed in Table 1, and they indicate that there are good agreement and preciseness of the numerical calculations. To further validate our results, in Table 2, we compare our present results for the special case against the results of investigations by Chamkha and Khaled 66 and Nima et al. 67 .  Table 2.   Fig. 2, the velocity profile increases with the augmented values of mixed convection parameter for the dominance of buoyancy force for the up-swimming microorganisms. Figure 3 shows the velocity profile f ′ (η) against η for random values of Nr when = −3 . The velocity profiles provide the existence of the dual solution when < c with a diversity of Nr . From the figure, we see that the first solutions are stable as the velocity profile went into the positive range. We also see that the second solutions  www.nature.com/scientificreports/ are unstable as the velocity profile became negative. Figure 3 illustrates the influence of Buoyancy parameter Nr over the dual solution. Figure 3 shows a decrease in Buoyancy parameter Nr , where velocity profile decreases for the first solution but increases in the second solution. Although the second solutions have negative values, there are no physical significances that can be made.
The velocity profile f ′ (η) against η for several values of Rb is visualized in Fig. 4 when = −3 . The velocity profile provides the existence of the dual solution with = −3 with a certain change of bioconvection Rayleigh number Rb . Figure 4 shows the effect of Rb over the dual solution when the curvature parameter γ = 5 , Peclet number Pb = 0.5 , bioconvection Lewis number Lb = 0.5 , Lewis number Le = 0.5 , buoyancy parameter Nr = 0.5 , and microorganism concentration difference parameter A = 0.2 . The greater values of Rb increase the buoyancy force because of the bio-convection process. It is observed in Fig. 4 that when the parameter Rbisdecreased , the first solutions of the dual velocity profile decrease, and the second solutions increase, which implies that the first solution is the stable one.
The velocity profile f ′ (η) against η for several values of γ is shown in Fig. 5 for = −4 . The velocity profiles provide the existence of the dual solution with = −4 with a certain change of curvature parameter γ . Figure 5 shows the effect of γ over the dual solution when the bioconvection Rayleigh number Rb = 0.6 , Peclet number  Also, less contact within the surface area will produce less resistance towards the fluid particles. As a result, the velocity profile shows stimulant values. Variation of Nusselt number with for different values of γ is shown in Fig. 6. It is seen that dual solutions exist for the temperature profile > c where c = −4.80, −4.81, −4.92, and γ = 3, 4, 5, respectively. The critical value c is where both the upper and lower branch solutions connect, and at this exact point, a unique solution exists. From these critical values, the boundary layer separates, and the solution becomes invalid. It is found from the heat transfer rate −θ ′ (0) that it increases strongly with the parameter and decreases relatively weakly with the curvature parameter γ. Figure 7 shows the temperature profile θ(η) against η for different values of γ when = −4 . The temperature profiles provide the existence of the dual solution when > c for different values of γ . An increase of the curvature parameter γ causes a decrease in curvature radius because the fluid velocity particle enhances. As a result, the average kinetic energy increases, which causes an increment in the temperature profile. It is seen in Fig. 7 that when the curvature parameter γ decreases, the temperature profiles also decrease for both solutions.
Variation of Sherwood number with for different values of γ is shown in Fig. 8.   www.nature.com/scientificreports/ become invalid. It is also observed that the Sherwood number increases with the increasing values of and γ for the first solution, while it decreases for the second solution. Figure 9 shows the concentration profile φ(η) against η for random values of γ when = −4 . The concentration profile provides the existence of the dual solution when > c with different values of γ . We see that both solutions are stable as the velocity profile went into the positive range. Figure 9 illustrates the influence of curvature parameter γ over the dual solution. We also see that a decrease in curvature parameter γ makes the concentration profile decrease for both the first and second solutions.
The concentration profile φ(η) against η for several values of Le in Fig. 10 when = −4 . The concentration profiles provide the existence of the dual solution with = −4 ( > c ) with a certain change of Lewis number Le . Both solutions are shown to be stable as the concentration profile went into the positive range. Figure 10 shows the effect of Le over the dual solution when the curvature parameter γ = 5 , Peclet number Pb = 0.5 , bioconvection Lewis number Lb = 0.5 , bioconvection Rayleigh number Rb = 0.6 , buoyancy parameter Nr = 0.5 , and the microorganism concentration difference parameter A = 0.2 . The Lewis number Le is defined as the ratio of thermal diffusivity and mass diffusivity, which is the prominent factor in studying heat and mass transfer. As Lewis number Le reduces the mass diffusivity, this in turn decreases the penetration depth of the concentration boundary layer. We observe in Fig. 10 that as the parameter Le decreases, the first solutions of concentration profile increase, and the second solutions decrease.
In Fig. 11, the density of motile microorganism transfer rates is also increased with the mixed convection parameter and curvature parameter. It is known that the motile microorganism density is higher than liquid, and they usually swim in an upward direction of the exterior of the cylinder wall. Therefore, the curvature parameter

Conclusion
The steady mixed convection boundary layer flow with gyrotactic microorganisms past a vertical cylinder is analyzed. Dual solutions are found to exist in case of opposing flow when the mixed convection parameter is negative (Cylinder is cooled T w < T ∞ ) . The consequences of various flow influencing parameters have been thoroughly discussed in detail. The critical reviews are summarized as follows: • The variation of Nusselt number indicates that dual solutions exist for temperature profile > c , where the critical value c = −4.80, −4.81, −4.92 for the curvature parameter γ = 3, 4, 5 . The curvature parameter γ increases heat transfer rate and temperature profile for the first solution, which is physically stable.   www.nature.com/scientificreports/ Several studies were performed on dual solutions for mixed convection along a vertical cylinder for different engineering applications. Moreover, there are many engineering and practical bio-microsystems where mixed convection flow over a vertical cylinder in porous media with Gyrotactic Microorganism occurs. However, very few works have been done on dual solutions for mixed convection with gyrotactic microorganisms. Analyzing the existence of a dual solution in heat, mass, motile microorganism transfer rate, temperature, and concentration microorganism profile beyond a critical point along a vertical cylinder is a novel concept. The obtained results are also unique. Our study shows mutual relations between different parameters, which can affect the performance of those systems.
In this paper, dual solution phenomena in the presence of gyrotactic microorganisms are observed only in the case of mixed convective opposing flow. For further extensions of this paper, we can consider non-Newtonian fluid with the effect of an aligned magnetic field to observe dual solution phenomena for both assisting flow and opposing flow.  License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.