Heat Source Effect on Three-Dimensional Couple Stress Casson Fluid Flow via Bidirectional Stretching Surface with Chemical Reaction

The couple stress fluid (CSF) has a motivated work by the upcoming research scientists, because it is utilizing in different physical liquid problems like liquid crystals, colloidal liquids, animal and human blood, lubrication. Moreover, such liquids are used in industrial and engineering like aerodynamic heating, electrostatic precipitation, petroleum products, solidification of liquid crystals, exotic lubricants. The CS fluid (“applications in chemical engineering like fluid crystals, polymer thickened oils, polymeric suspensions and physiological fluid mechanics ext.”) was first established by Stokes [1] and is simplest generalization of classical theory of fluids which is polar NNF theories. The theory of couple stress is familiar from theory of elastic shells was presented by Toupin [2]. Srivastava [3] investigated by the CSF model. Mindlin [4] examined the 2D theory of CS model. Srivastava [5] developed under the condition of zero Reynold number in CSF. Yang et al. [6] discussed the classical theory of couple stress. The convection motion of CSF via channel was established [7, 8]. The CSCF motion via bidirectional SS was developed [9-11].

At present modelling of HMT in NN liquid motion via SS have practical and fundamental importance in extensive industrial and production tenders (“bioengineering and polymeric fluids, annealing and thinning of copper wire, nuclear fuel slurries, plasma and mercury,”) involved. Huang [12] explained numerically the HMT of NN power law fluid via vertical cone. Oyelakin et al. [13] presented the HMT on the 3D Casson NFs motion by convective conditions. Some of the scientists [14-17] established heat and mass transfer in NN liquids via SS. Mahabaleshwar et al. [18] established Brinkmann model in NN liquid motion on porous shrinking sheet. Recently, the various characteristics NN NFs motion via different channels was found [19-23]. 

Another most significant work has convective motion regime of temperature, concentration fields an electrically conducting in a SS around a heating or cooling surface. It has more uses in industrial such as cooling of nuclear reactors, thermal insulation, petroleum reservoirs, and geothermal reservoirs ext. The unsteady 3D couple stress liquid via stretching surface with convective condition was discussed by Thammanna et al. [24]. The Casson liquid motion by convective condition via SS was presented [25-29]. The unsteady convective flow of NFs via surface was explained [30-33] numerically. The numerical solutions of convective motion of NFs were established [34, 35]. Marzougui et al. [36] find out that the existence of Hartmann number where the magnetic field dominates via its intrinsic effect.

The primary motivation is to explore realistic flow conditions commonly found in industrial and biomedical processes, such as extrusion, plastic sheet stretching, and tissue engineering, where both surface stretching and microstructural effects are non-negligible. The bidirectional stretching surface generates a three-dimensional velocity field, adding complexity and practical relevance compared to unidirectional cases.

Novelty arises from the combined consideration of Casson fluid rheology, couple stress theory, internal heat generation, and a homogeneous chemical reaction mechanism within a 3D flow regime an area not widely studied in the current literature. The governing partial differential equations are transformed into a system of ordinary differential equations using similarity transformations. These equations are then solved numerically using an efficient Runge-Kutta-based shooting method.

Limitations:

• The governing equations are highly nonlinear and coupled, making analytical solutions nearly impossible and requiring advanced numerical techniques.
• The Casson model applies only to specific non-Newtonian fluids (e.g., blood, ketchup) and may not be valid for all industrial fluids.
• Bidirectional stretching surfaces are idealized assumptions that may not accurately represent real-world surfaces in practical systems.
• Couple stress and Casson parameters are difficult to measure experimentally, limiting the validation of simulation results.
• Most such studies assume laminar flow; however, in real applications, turbulent effects may be significant and must be considered.
• Nanofluid or particulate effects in multiphase systems are often ignored, limiting accuracy in heterogeneous mixtures.
• Assumes a spatially uniform heat source (HS), which may not reflect varying thermal loads in real-world conditions.

The CSCF characteristics of λ^* on velocity motion along x^*, y^*-axis $\left(f^{\prime}\left(\eta_1\right), g^{\prime}\left(\eta_1\right)\right)$ and $\theta\left(\eta_1\right)$ as illustrated respectively in Figure 2 and Figure 3. It is perceived that, the $\left(f^{\prime}\left(\eta_1\right), g^{\prime}\left(\eta_1\right)\right)$ (axial, transverse velocities resp.) smoothly convergence (point at surface area is λ^*=0.8 (not exact value)) monotonically increases along with $x^*, y^*$ axis and associated BL thickness of couple stress NN liquid motion is thinner with ascending numerical values of λ^*. Physically, when stretching ratio increase in NN CS liquid from zero then the lateral surface begins to move in the x^*, y^* axis rises and $\theta\left(\eta_1\right)$ monotonically down for $\left(K_1=\beta^* \rightarrow \infty\right)$ and $\left(K_1=\beta^*=0.5\right)$. For these cases, the results are mentioned that $\theta\left(\eta_1\right)$ is more effected in pure fluid case $\left(K_1=\beta^* \rightarrow \infty\right)$ than NN CS fluid $\left(K_1=\beta^*=0.5\right)$ .

Figure 4 and Figure 5 predict the effect of K₁ on $\left(f^{\prime}\left(\eta_1\right), g^{\prime}\left(\eta_1\right)\right)$ and $\theta\left(\eta_1\right)$. It is analysed that, the speed of NN liquid is enhances along with x^* and y^* directions whereas reverse trend behaviour shows $\theta\left(\eta_1\right)$ for distinct enlarge values of K₁. Physically, the CS parameter is inversely proportional to Kinematic viscosity because it reflects the relative influence of microstructural (rotational or size-dependent) effects in a fluid compared to standard viscous (shear) diffusion.

The most significant characteristic M₁ on $\left(f^{\prime}\left(\eta_1\right), g^{\prime}\left(\eta_1\right)\right)$ respectively presented in Figures 6-9. It is dictated that, the velocity in y^*- axis $g^{\prime}\left(\eta_1\right)$ is high convergence deference than the x^*-axis velocity $f^{\prime}\left(\eta_1\right)$ and also ${Re}_x^{1 / 2} C_{f x}, {Re}_x^{1 / 2} C_{f y}$ (skin friction coefficients) rises in fluid motion along in x^*, y^*-axis while opposite behaviours $\theta\left(\eta_1\right)$ profile with growth numerical values of M₁. Physically, the Lorentz force applied to liquid flow direction which is stronger to high magnetic field in NN couple stress liquid and then, the energy is transformed into heat because of friction forces.

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Figure 2. Effect of λ^* on $f^{\prime}\left(\eta_1\right), \mathrm{g}^{\prime}\left(\eta_1\right)$

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Figure 3. Effect of λ^* on $\theta\left(\eta_1\right)$

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Figure 4. Effect of K₁ on $f^{\prime}\left(\eta_1\right), \mathrm{g}^{\prime}\left(\eta_1\right)$

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Figure 5. Effect of K₁ on $\theta\left(\eta_1\right)$

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Figure 6. Effect of M₁ on $f^{\prime}\left(\eta_1\right), \mathrm{g}^{\prime}\left(\eta_1\right)$

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Figure 7. Effect of M₁ on $\theta\left(\eta_1\right)$

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Figure 8. Effect of M₁ on $\operatorname{Re}_x^{1 / 2} C_{f x}$

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Figure 9. Effect of M₁ on $\operatorname{Re}_x^{1 / 2} C_{f y}$

Figure 10 and Figure 11 predict the significant parameter β^* on velocity profiles $\left(f^{\prime}\left(\eta_1\right), g^{\prime}\left(\eta_1\right)\right)$ and corresponding high BL thickness convergence to η₁=0.8 with enlarge numerical values of β^*. Figure 11 presents that the $R e_x^{-1 / 2} N u_x$ is enlarges in various magnetic field conditions (like M₁=0.5 (hydromagnetic), M₁=0.0 (hydrodynamic) and M₁=-0.5 (terminating field lines)). In CSCF motion involved multiple charges, the magnetic field generate more heat transfer $\left(R e_x^{-1 / 2} N u_x\right)$ in M₁=0.5 positive charges, the field is negative M₁=-0.5 than the heat transfer rate $\left(R e_x^{-1 / 2} N u_x\right)$ is terminating and field should be zero there is no heat transfer $\left(R e_x^{-1 / 2} N u_x\right)$ on stretching surface. Physically, the CF parameter is proportional to dynamic viscosity, it is produced resistance force in the flow is very high and higher values of β^*. Further, this fluid parameter depends on yield stress of fluid and drag the thinner BL thickness of the fluid flow towards a stretching surface.

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Figure 10. Effect of $\beta$ on $f^{\prime}(\eta), \mathrm{g}^{\prime}(\eta)$

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Figure 11. Effect of $\beta$ on $N u_x \operatorname{Re}_x^{-1 / 2}$

Variation of $\theta\left(\eta_1\right)$ for $\left(K_1=\beta^*=0.5\right)$ and $\left(K_1=\beta^* \rightarrow\right.$ $\infty), R e_x^{-1 / 2} N u_x$ for the cases of ( $\beta^*=0.5$ ) (CSCF) and $\left(\beta^* \rightarrow \infty\right)$ (couple stress fluid) enhances with distinct ascending numerical values of $\Gamma_1$ (temperature Biot number) as depicted in Figure 12 and Figure 13, respectively. It is dictates that the stronger intensity of convective heat, the high thermal BL convergence in Newtonian liquid $\left(K_1=\beta^* \rightarrow \infty\right)$ while comparing NN fluid $\left(K_1=\beta^*=0.5\right)$ and moreover, $R e_x^{-1 / 2} N u_x$ is more significant in CS fluid $\left(\beta^* \rightarrow \infty\right)$ is better than CSCF $\left(\beta^*=0.5\right)$ with ascending values of $\Gamma_1$. Physical representation as a measure of the ratio of convective heat transfer at the surface to conductive heat transfer within the fluid. It characterizes how effectively heat is exchanged between the fluid and the stretching surface.

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Figure 12. Effect of $\Gamma_1$ on $\theta\left(\eta_1\right)$

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Figure 13. Effect of $\Gamma_1$ on $N u_x \operatorname{Re}_{\mathrm{x}}^{-1 / 2}$

Figure 14 illustrates that the profile $\phi\left(\eta_1\right)$ enlarges with numerical values of $\Gamma_2$ (concentration Biot number) for the presence $\left(\gamma^*=0.5\right)$ and absence $\left(\gamma^*=0.0\right)$ of CR. It is declined that, the weaker intensity of convective concentration in liquid motion, the ascending $\phi\left(\eta_1\right)$ monotonically convergent to surface of the point $\left(\eta_1=9.5\right)$. The NN couple stress fluid generate heat is more when absence $\left(\gamma^*=0.0\right)$ of chemical reaction while comparing presence $\left(\gamma^*=0.5\right)$ of chemical reaction with ascending values of $\Gamma_2$. Physical reason represents the ratio of MT at the boundary (surface) to mass diffusion within the fluid. It provides a physical measure of how effectively a solute (or chemical species) is transported across the fluid–surface interface compared to its internal diffusion.

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Figure 14. Effect of $\Gamma_2$ on $\phi\left(\eta_1\right)$

The variation of $H_1$ on $\operatorname{Re}_{\mathrm{x}}{ }^{-1 / 2} N u_x$ for Casson couple stress luid ( $K^*=\beta^*=0.5$ ) and pure fluid ( $K^*=\beta^* \rightarrow \infty$ ) as expressed in Figure 15. The heat transfer $\operatorname{Re}_{\mathrm{x}}^{-1 / 2} N u_x$ is high with enlarge values of $H_1$. It is clear view of $\operatorname{Re}_{\mathrm{x}}{ }^{-1 / 2} N u_x$ is more effected in pure fluid $\left(K^*=\beta^* \rightarrow \infty\right)$ while comparison of CSCF (NN CS fluid) $\left(K^*=\beta^*=0.5\right)$. Physically, the HS is inversely proportional to fluid density. Due to this the isothermal temperature and heat transfer rate high in CS liquid motion towards from stretching surface with enhances values of $H_1$.

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Figure 15. Effect of $H_1$ on $N u_x \operatorname{Re}_{\mathrm{x}}^{-1 / 2}$

The most significant physical number is Pr (Prandtl number) on $\left(R e_x^{-1 / 2} N u_x\right)$ with various Thermophysical fluid values of Pr (such as CO₂=4.1, H₂O=6.2, CH₃OH=7.38, C₁₂H₂₆=21) as predict respectively in Figure 16. It is discussed on $\left(R e_x^{-1 / 2} N u_x\right)$ in two cases one is pure fluid $\left(K_1=\beta^* \rightarrow \infty\right)$ and second one is NN CS liquid case $\left(K_1=\beta^*=0.5\right)$. Which is clear that, the couple stress non-Newtonian $\left(K_1=\beta^*=0.5\right)$ fluid has produced more heat transfer when compared to the pure fluid $\left(K_1=\beta^* \rightarrow \infty\right)$. Physically, the Prandtl number is ratio between the viscous diffusion rate and thermal diffusion rate. The thermal diffusion rate is more in CSCF motion then pure fluid, so that the HTR is produce high in surface. 

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Figure 16. Effect of Pr on $N u_x \operatorname{Re}_{\mathrm{x}}^{-1 / 2}$

Figure 17 illustrates that the profile $\phi\left(\eta_1\right)$ enlarges with distinct higher numerical values of y^* (CR parameter) while reverse behaviours displays ShRe_x^-1/2 (MTR) for presence convective Biot numbers $\Gamma_1, \Gamma_2=0.5$ and absence of Biot numbers $\Gamma_1, \Gamma_2=0.0$ as seen in Figure 18. It is demonstrated that, the ShRe_x^-1/2 is produce more in presence of $\Gamma_1, \Gamma_2=0.5$ while comparing absence of Biot number $\left(\Gamma_1, \Gamma_2=0.0\right)$. Physical represents the rate at which a chemical species is generated or consumed within the fluid flow due to a chemical reaction. It directly influences the concentration distribution and indirectly affects mass transfer and possibly thermal or rheological properties, depending on the nature of the reaction.

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Figure 17. Effect of y^* on $\phi\left(\eta_1\right)$

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Figure 18. Effect of y^* on ShRe_x^-1/2

Characterise of Le (Lewis number) with absence and presence of Casson fluid $\left(\beta^*=0.5\right)$ (like CSCF) and absence of Casson fluid $\left(\beta^* \rightarrow \infty\right)$ (like CSCF) on ShRe_x^-1/2 displays in Figure 19. In view of this the ShRe_x^-1/2 profile enlarges with distinct ascending values of Le. It is finally concluded that, the couple stress fluid $\left(\beta^*=0.5\right)$ is produce high MTR while comparing Casson couple stress fluid $\left(\beta^* \rightarrow \infty\right)$. Physically, it describes the relative efficiency of heat diffusion compared to mass (species) diffusion in the fluid.

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Figure 19. Effect of Le on ShRe_x^-1/2

The present effort computes numerical results via R-K-F 4^th order along with shooting technique is matched up to four decimal places and also find extra numerical results. the present results compared to related articles of Sarah et al. [38], Nadeem et al. [39], Gupta and Sharma [40] and Ahmad and Nazar [41] for some specific cases (like K₁=0 and λ=1.0) and getting good agreement results (Table 1 and Table 2) respectively.

Table 1. Comparison of Skin friction coefficient $\left(1+\frac{1}{\beta^*}\right) f^{\prime \prime}(0)$ in the absence of $\left(K_1=0\right)$

Table 2. Comparison of skin friction coefficient in the absence of (K₁=0) and λ^*=1.0

Table 3 explored the numerical results of heat transfer rates with respect to variation of various physical parameters. The Re_x^-1/2Nu_x amplify for ascending values of K₁, Pr, M₁, H₁, $\Gamma_1$, and β^* respectively.

The mass transfer rates of different parameters demonstrated in Table 4. The (MTR) Re_x^-1/2Sh decline with enlarge numerical values of K₁, M₁, γ^* and β^* while opposite trend with enhance values of Pr, Le when presence$\Gamma_1=0.5=\Gamma_2$ and absence $\Gamma_1=0=\Gamma_2$ of temperature and concentration Biot number, respectively.

Table 3. Numerical values of Re_x^-1/2Nu_x with different parameters of K₁, Le, Pr, R_d, M₁, N_t, N_b, θ_w, H₁, $\Gamma_1$, $\Gamma_2$, γ^*, Ec_x, and Ec_y for λ^*=0

Table 4. Numerical values of Re_x^-1/2Sh with different parameters of K₁, Pr, R_d, M₁, θ_w, H₁, γ^*, Ec_x, and Ec_y for λ^*=0