1 Introduction

Choi^[1] first used the nanofluid terminology to enhance the thermal conductivity of fluids by dissipating nanosized solid particles in the base liquid. This tremendous model has attained the focus of investigators all over the world. A bundle of articles^[2-10] have been done with the theory of Choi^[1]. The Casson fluid is categorized as a fluid of non-Newtonian type due to its rheological features in corresponding to the shear stress and strain correlation. This fluid functions similarly to a flexible material at a lesser shear strain. It is beyond a significant stress value, and thus acts like a Newtonian fluid. Examples of such fluids include blood, paint, toothpaste, etc. Oyelakin et al.^[11] studied the radiative heat transport of the Casson nanoliquid flow alongside partial slips and convective conditions with the spectral relaxation method, and found that the growing values of the Casson parameter resulted in the reduction of the velocity and temperature profiles.

The Fourier theory of heat diffusion was remodeled by the addition of the heat flux relaxation time. It permits the conveyance of heat over a dissemination of thermal wave with a fixed velocity. Such a heat diffusion model has overwhelming practical applications. Owing to these significant features, Cattaneo^[12] developed a new model of heat diffusion with the consideration of the extra feature of heat flux relaxation. Christov^[13] modified the time derivative in the Maxwell-Cattaneo theory with the formulation of material invariant. Straughan^[14] executed the unique solutions of the Cattaneo-Christov expression. Li et al.^[15] described the nature of the Cattaneo-Christov heat diffusion theory by considering the hydromagnetic flow of viscoelastic liquid. It is heeded from their results that, when the thermal relaxation parameter strengthens the fluid velocity, the temperature will visibly increase. Recently, some monumental pieces of results have been reported^[16-28].

Devi and Prakash^[29] analyzed the flow and heat transfer past a slendering stretching sheet, and investigated the effects of both the temperature dependent conductivity and the viscosity on the magentohydrodynamic (MHD) flow past a slendering sheet. From their results, we can see that the non-dimensional temperature and thermal boundary layer thickness increase when the velocity power index increases. This trend is analyzed by Devi and Prakash^[30] and Babu and Sandeep^[31].

Three-dimensional (3D) fluid flow has been the subject of concern of many researchers due to the fact that every physical flow is usually 3D. Khan and Khan^[32] analyzed the flow and heat transfer features of the 3D flow of the Burgers fluid with the Cattaneo-Christov heat flux model and the homotopy analysis technique. They found that the velocity increased when the Deborah number and the stretching parameter increased. As regards, some recent studies can be found in Refs. [33]-[37]. Hsiao^[38] considered the MHD Carreau fluid in a nanofluid suspension. Ramesh et al.^[39] studied the two-dimensional (2D) steady laminar stagnation point flow of the Casson fluid over a stretching sheet with the thermal radiation and variable thickness. Hsiao^[40] investigated the MHD flow of the Maxwell fluid with radiation and viscous dissipation.

Kumar et al.^[41] studied the partial slip on the MHD dissipative ferro-fluid over a non-linear permeable convectively heated sheet. Ramesh et al.^[42] investigated the hydromagnetic Casson liquid flow through a stretching cylinder. The MHD properties in the flow with various geometries and physical aspects have also been studied^[43-47].

As far as the authors' knowledge with the aid of the above-outlined literature, we find that no existing study has concerned the 3D Casson nanofluid due to the slendering sheet with Cattaneo-Christov heat flux. The present study is concerned with the analysis of the 3D multiple slip flow of the Casson nanofluid due to a slendering sheet with the Cattaneo-Christov heat flux. The governing coupled nonlinear partial equations are transformed into the ordinary differential equations with suitable similarity transformations, and solved numerically with the Runge-Kutta based Newton method. Some previous results are used to ascertain the accuracy of our results. The significance effects of some useful salient parameters are investigated, and both graphs and tables are equally used to manifest the obtained results.

2 Mathematical formulation

In the present investigation, we consider the Brownian motion and thermophoresis on the MHD 3D flow over a slendering sheet filled with multiple slips. For the development of the temperature and concentration fields, we also consider the gyrotactic microorganisms and Cattaneo-Christov heat flux mode. The x-axis is considered as the sheet motion, and the y-axis is perpendicular to it (see Fig. 1). We assume that that the external electric field is negligible, and

Then, the governing equations for continuity, momentum, thermal, and diffusion can be given as follows^[36]:

(1)

(2)

(3)

(4)

(5)

(6)

The corresponding boundary conditions are

(7)

where

(8)

(9)

(10)

(11)

(12)

The irregular heat source/sink parameter is

(13)

In the above equation, A^*>0 and B^*>0 represent the internal heat generation coefficients, while A^* < 0 and B^* < 0 denote the heat absorption coefficients. Now, we transform the partial equations into ordinary differential equations. We introduce the similarity transformations as follows:

(14)

With the help of Eqs. (11), (12), and (13), Eqs. (2)-(4) can be converted to the following differential equations:

(15)

(16)

(17)

(18)

(19)

The corresponding boundary conditions are

(20)

The dimensional parameters M, Pr, Le, N_t, N_B, k_r, h₁, h₂, γ, h₃, h₄, δ, Pe, and β₁ are given by

The skin-friction coefficient C_f, local Nusselt number Nu_x, local Sherwood number Sh_x, and local Sherwood number due to microorganisms NSh_x are defined as follows:

(21)

With Eq. (13), we can rewrite Eq. (20) as follows:

(22)

where

3 Results and discussion

The set of the non-linear ordinary differential equations (15)-(19) have been solved mathematically by the Runge-Kutta based Newton method with the boundary conditions in Eq. (20). The numerical values of our results are demonstrated in Table 1, where the effects of the non-dimensional parameters such as M, m, β, n, A^*, B^*, Pr, , γ, k_r, Le, δ, Pe, h₁, h₂, h₃, and β₁ on the shear stress, the dimensionless heat and mass transfer rate, and the velocity gradient along the y-axis are established. It is noticed that the encouraging variations in the values of the magnetic field M, the Casson fluid β, the power-law index n, the wall thickness δ, and the dimensionless slip velocity h₁ result in the significant hike of the momentum shear stress F"(0). However, the dimensionless heat transfer rate θ'(0) keeps growing with an increase in the values of m, β, A^*, B^*, Pr, γ, and δ while decreases when the values of M, n, h₁, and h₂ increase and the Casson fluid parameter β=3. Similarly, the dimensionless mass transfer rate ϕ'(0) increases when the values of k_r, Le, and δ are strengthened, while decreases when the values of M, β, n, h₁, and h₃ increase. Moreover, when the values of the parameters such as m, β, n, k_r, Pe, h₁, and β₁ increase, the gradient density of the motile microorganisms χ'(0) decreases. When β₁=3 and the physical parameters Le, δ, and h₃ increase, the gradient density of the motile micoorganism χ'(0) increases.

From the velocity profiles along the x-axis of the slendering sheet, we can see that both the velocity gradient and the boundary layer thickness along the x-direction of the flow reduce due to the increase in the values of the magnetic field parameter for both β < 0 and β>0 cases (see Fig. 2). From Fig. 3, one can observe that the growing values of the wall thickness parameter δ result in the significance decreases in the velocity gradient and the momentum boundary layer thickness for both β < 0 and β>0 cases, though a very negligible effect is spotted for a positive Casson fluid parameter. The curves are seen to be increasingly steeper towards the wall. Moreover, the velocity gradients are found to increase for both the negative and positive Casson fluid parameter cases with an increase in the values of the dimensionless velocity slip and power-law index parameters (see Figs. 4 and 5).

Figures 6-8 show the effects of the magnetic field M, the wall thickness δ, and the dimensionless velocity slip h₁ parameters on the velocity gradient along the y-direction of the slendering sheet. From the figures, we can see that the increase in the values of these physical parameters leads to the monotone depreciation of the velocity gradient for both β < 0 and β >0 cases. However, the positive Casson fluid parameter (β>0) case produces no meaningful effect on the velocity gradient (see Fig. 7). When the power-law index parameter n increases, the velocity gradient increases (see Fig. 9).

Figures 10-13 show the effects of the magnetic field M, the power-law index n, the dimensional stretching sheet coefficient A^*, and the dimensionless velocity slip h₁ on the temperature field. From the figures, we can see that the recapitulated increase in these parameters results in the eloquent hike of the temperature distribution. Moreover, the heat transfer rate often goes high when β >0, while the thermal boundary layer thickness stiffens. Besides, it is remarkably found that the temperature and the surface thermal boundary layer thickness are simultaneously reduced due to the monotone increase in the values of m, B^*, h₂, δ, and γ (see Figs. 14-18). It worth noting that both the negative and large values of the Casson fluid parameters (β < 0 and β =∞) share a little common ground on the temperature profiles (see Figs. 15 and 18).

Figures 19-21 show the effects of M, n, and h₁ on the concentration profiles of the slendering sheet. From the figures, we can see that the concentration profiles increase with the increases in M, n, and h₁. It amounts to a small difference in the negative and large values of the Casson fluid parameter (β < 0 and β =∞) cases. Moreover, the concentration profiles obviously decrease with the increases in Le, k_r, h₃, and δ for both β < 0 and β >0 cases (see Figs. 22-25).

Figures 26-30 show the effects of the dimensionless parameters M, n, β₁, Pe, and h₁ on the gyrotactic microorganisms profiles. From the figures, we can see that the gyrotactic microorganisms are greatly encouraged when M, n, β₁, Pe, and h₁ increase, where the negative Casson fluid parameter β motivates the strengthening of the gyrotactic microorganisms faster. However, on the contrary, the gyrotactic microorganisms are significantly reduced due to the increases in the parameters such as Le, h₃, and δ (see Figs. 31-33).

4 Conclusions

In the present study, the analysis of the three-dimensional Casson nanofluid along the slendering sheet with the Cattaneo-Christov heat flux is performed. The effects of various flow parameters on the fluid velocity, temperature, concentration, density of motile organisms, surface shear stress, dimensionless heat and mass transfer rates, and the density of motile organism's gradient are portrayed and illuminated with the help of graphs and tables. The transformed set of differential equations is numerically solved via the Runge-Kutta method with the shooting technique. It is worth noting that the velocity gradient is highly motivated due to the significance increases in the dimensionless velocity slip h₁ and the power-law index n for both the β < 0 and β >0 cases. The gyrotactic microorganisms are greatly encouraged when the dimensionless parameters M, n, β₁, Pe, and h₁ increase, where the negative Casson fluid parameter β motivates the strengthening of the gyrotactic microorganisms faster.