Magnetized suspended carbon nanotubes based nanofluid flow with bio-convection and entropy generation past a vertical cone

The captivating attributes of carbon nanotubes (CNT) comprising chemical and mechanical steadiness, outstanding electrical and thermal conductivities, featherweight, and physiochemical consistency make them coveted materials in the manufacturing of electrochemical devices. Keeping in view such exciting features of carbon nanotubes, our objective in the present study is to examine the flow of aqueous based nanofluid comprising single and multi-wall carbon nanotubes (CNTs) past a vertical cone encapsulated in a permeable medium with convective heat and solutal stratification. The impacts of heat generation/absorption, gyrotactic-microorganism, thermal radiation, and Joule heating with chemical reaction are added features towards the novelty of the erected model. The coupled differential equations are attained from the partial differential equations by exercising the local similarity transformation technique. The set of conservation equations supported by the associated boundary conditions are worked out numerically by employing bvp4c MATLAB function. The sway of numerous appearing parameters in the analysis on the allied distributions is scrutinized and the fallouts are portrayed graphically. The physical quantities of interest including Skin friction coefficient, the rate of heat and mass transfers are assessed versus essential parameters and their outcomes are demonstrated in tabulated form. It is witnessed that the velocity of the fluid decreases for boosting values of the magnetic and suction parameters in case of both nanotubes. Moreover, the density of motile microorganism is decreased versus larger estimates of bio-convection constant. A notable highlight of the presented model is the endorsement of the results by matching them to an already published material in the literature. A venerable harmony in this regard is achieved.


Mathematical Modeling
We assume a water based nanofluid flow with CNTs over a vertical cone in a permeable medium. The additional effects accompanied the model are Heat generation/absorption, solutal stratification, Joule heating, chemical reaction with entropy generation. The flow is induced along the x-axis and a magnetic field is applied along the y-axis as shown in Fig. 1. The equations resulting from the above assumptions are modeled as 55 : , and V 0 represent dynamic viscosity, density, thermal and solutal expansion coefficients, cone half-angle, magnetic field of strength, modified thermal diffusivity, thermal radiation coefficient, Dimensional heat generation/absorption parameter, heat capacities, Brownian diffusion coefficient, rate of chemical reaction, maximum cell swimming speed, diffusivity of microorganisms, thermal conductivity, convective parameter, reference temperature and concentration dimensionless constants and suction/injection parameter respectively. Table 1 is erected to depict the characteristics of water and CNTs of both types. Thermal conductivity and effective density of the nanofluid are given by: www.nature.com/scientificreports www.nature.com/scientificreports/ (1 ) 2 ln (7) nf nf The similarity transformations are defined as (1 (0)), (0) 1 , (0) 1, The parameters in non-dimensional form are stated as: Here, Pr, k 1 , M, N r , R b , R d , E c , γ, S c , C r , L b , P e , δ, B 1 , and n characterize Prandtl number, Porous parameter, magnetic parameter, buoyancy ratio parameter, Bio-convection Rayleigh number, Radiation parameter, heat generation/absorption parameter, Schmidt number, Chemical reaction parameter, bio-convection Lewis number, Bio-convection constant, Boit number and solutal stratification respectively.
The physical quantities like drag coefficient, local Nusselt and Sherwood numbers, and local density of motile microorganisms are given by: www.nature.com/scientificreports www.nature.com/scientificreports/ In dimensionless form, the above physical quantities are evaluated as: Table 2 shows the resemblance of the present results with Khan et al. 56 for numerous estimates of φ in limiting case. An exception concurrence between both results is found.

Entropy Generation
The model representing the entropy generation is given by: In Eq. (17), entropy is consisting of three terms viz. i) I (heat transfer irreversibility) ii) II (fluid friction irreversibility) and iii) III (diffusion irreversibility). The entropy generation N G is given by: Here, ‴ S gen is entropy generation rate and ‴ S 0 the characteristic entropy generation rate represented by: f w Here, Ra x , Br, λ, ζ, and α represent Reynold number, Brinkman number, diffusive constant parameter, concentration difference parameter, and temperature difference parameter respectively. Khan et al. 56 Existing Results Khan et al. 56 Existing Results

Results and Discussion
This section is dedicated to discuss and anticipate the impacts of various parameters φ . ≤ ≤ . 0 01 003, and Pr 6 2 = . unless stated separately. Figure 2 demonstrates the action of solid volume fraction φ of nanoparticle on axial velocity. The velocity field enhances for augmenting values of the solid volume fraction φ for both nanoparticles. It is also understood that the velocity distribution upsurge rapidly for SWCNT in comparison to MWCNT. Figure 3 is portrayed to depict the impact of the magnetic parameter M on the velocity field. The velocity of the fluid lessens for boosting values of M. It is because the strong Lorentz force that presents resistance to the fluid's movement that eventually lowers the fluid's movement. In Fig. 4, the consequence of porous parameter k 1 on velocity profile is sketched. It is comprehended that the velocity is a lessening function of k 1 . Further, the momentum boundary layer for both nanoparticles declines while increasing k 1 . The influence of suction parameter v 0 on axial velocity is shown in Fig. 5. It is perceived that the velocity profile diminishes for boosting estimations of the suction parameter v 0 . Also, the momentum boundary layer declines for cumulative values of v 0 for both SWCNTs and MWCNTs. The buoyancy ratio parameter N r and bio-convection Rayleigh number R b effects on axial velocity are examined in Figs 6 and 7. The velocity profile declines with increasing values of N r and R b . It is also found that the velocity profile for MWCNT decreases more rapidly than SWCNT in both cases. Figure 8 is illustrated to depict the impact of Biot number B 1 on temperature field. An upsurge in temperature of the fluid is visualized for larger estimates of B 1 for both types of CNTs. Higher thermal resistance in comparison to the boundary layer inside the cone is witnessed. As a result, temperature of the fluid  www.nature.com/scientificreports www.nature.com/scientificreports/   www.nature.com/scientificreports www.nature.com/scientificreports/ in the vicinity of the boundary layer is seen. The thermal radiation parameter R d effect on the temperature distribution is shown in Fig. 9. More heat is generated with the rise in values of R d . That is why rise in temperature of the fluid is perceived. The outcome of Schmidt number S c on concentration field is visualized in Fig. 10. The weak concentration of the fluid is seen for higher values of S c . As the Schmidt number is the quotient of kinematic viscosity to molecular diffusion coefficient small values of molecular diffusion coefficient results in large estimates of S c that lowers the concentration. Figures 11 and 12 are graphed to comprehend the effect of solutal stratification n and chemical reaction parameter C r on the concentration of the fluid. It is witnessed that concentration of the fluid is on decline for higher estimates of n and C r in case of both types of CNTs. Figures 13, 14 and 15 are portrayed to see the impression of bio-convection constant δ, bio-convection Peclet number P e and bio-convection Lewis number L b on the density of motile microorganism respectively. It is observed that density of motile microorganism is on decline for all three parameters for both types of nanoparticles. Figures 16 to 19 illustrate the impression of different physical parameters versus the entropy generation number N G . From Fig. 16, it is seen that increasing the temperature difference parameter α, the entropy generation number N G decreases for both nanoparticles. Figures 17 and 18 demonstrate the effects of concentration difference ζ and Reynold number Ra x on the entropy generation number. The entropy generation profile enhances with enhancing the value of Ra x and ζ for both nanoparticles. The local entropy generation increases for growing estimates of the diffusive constant parameter λ for both SWCNT and MWCNT which is displayed in Fig. 19. Table 3 is erected for Skin friction coefficient versus numerous estimates of the arising parameters in the defined mathematical model. It is comprehended that Skin friction parameter rises versus suction parameter and solid volume fraction. Likewise, it decreases for the values of the magnetic parameter, bio-convection Rayleigh number, and porous medium. The values of Nusselt number for various values of involved parameters are given in Table 4. It is witnessed that Nusselt number is decreasing function of magnetic parameter and it enhances for  www.nature.com/scientificreports www.nature.com/scientificreports/   www.nature.com/scientificreports www.nature.com/scientificreports/   www.nature.com/scientificreports www.nature.com/scientificreports/ radiation parameter, Biot number, and solid volume fraction. Table 5 is erected to witness the behavior of parameters versus the Sherwood number. For the growing estimates of the buoyancy ratio parameter and concentration stratification, Sherwood number is on the decline and grows versus numerical values of the Schmidt number   www.nature.com/scientificreports www.nature.com/scientificreports/ and chemical reaction parameter. To see the impact of certain parameters on Motile density number, Table 6 is formed. It is witnessed that Motile density number escalates for the values of microorganism concentration difference parameter and Peclet number. Whereas it deteriorates for increasing estimates of Rayleigh number.

Concluding Remarks
The aqueous base nanofluid flow with both types of CNTs (SWCNT and MWCNT) over a vertical cone accompanied by impacts of a gyrotactic microorganism containing motile organisms and solutal stratification in a porous medium is deliberated numerically here. The analysis is performed in the presence of heat generation/absorption, Joule heating, and chemical reaction. An additional effect of Entropy generation is also taken into account. The salient characteristics of the modeled problem are: • With an increase in estimates of Peclet number, Motile density number enhances.
• For both CNTs, the velocity of the fluid escalates versus growing values of suction and magnetic parameters.
• The fluid's temperature is on the rise for the estimates of the buoyancy ratio parameter.
• For both types of CNTs, the concentration of the fluid is on the decrease in the values of solutal stratification.
• For the growth estimates of the buoyancy ratio parameter, Sherwood number is on the decline and grows versus numerical values of chemical reaction parameter. • Skin friction parameter rises for solid volume fraction.
• Nusselt number is decreasing the function of the magnetic parameter.