Inclined hydromagnetic impact on tangent hyperbolic fluid flow over a vertical stretched sheet

The current research aims to examine the impact of a tangent hyperbolic fluid flow confined by a stretching sheet with the existence of variable thermal conductivity, mixed convection, and magneto hydrodynamics. A mathematical model is developed in the form of partial differential equations (PDEs) and then converted into ordinary differential equations by using self-felicitous transformations. The technique of BVP4C (MATLAB package) has been used to simplify these ordinary differential equations. The numerical solution of skin friction, mixed convection, Nusselt number, and velocity and temperature profiles for different values of the involved parameters is indicated through tables and graphs. It can be noticed that the velocity profile decreases when the Hartmann number increases. The effect of Weissenberg number, inclined angle, and power law index for velocity profiles is also identical to the Hartmann number. The temperature profile decays due to an increment in the Prandtl number. Skin friction and the Nusselt number have also been explained. The physical reasoning for growth or decay of these parameters has been discussed in detail.


INTRODUCTION
The tangent hyperbolic fluid model is one of the most important fluid models in the group of non-Newtonian fluids. From laboratory experiments, it is found that this model predicts the shear thinning phenomenon very precisely. Tangent hyperbolic fluids are being used mostly in laboratory experiments and industries. Whipped cream, blood, solutions, melts, paint, polymers, and ketchup are the main examples of the tangent hyperbolic model in the fields of industry and biology. Kumar et al. 1 probed heat transfer in dusty hyperbolic tangent fluid with respect to magnetic field and thermal radiation toward a deformable sheet. The main focus of this study was to analyze the MHD flow and radiative heat transfer of the tangent hyperbolic fluid with fluid particle suspension. A Runge-Kutta-Fehlberg 4th and 5th order method, with the help of a shooting technique, is utilized to solve mathematical equations. Akbar et al. 2 investigated the numerical solutions of the tangent hyperbolic model toward a deformable surface in the existence of MHD and discussed the behavior of parameters that occurred in the modeled equations. The Runge-Kutta method is used to solve the flow equations. They found that the Weissenberg number is counter positive for fluid momentum. Hayat et al. 3 examined the tangent hyperbolic fluid for the progression of thermal and momentum boundary layers. They concluded that with large quantities of the Weissenberg number and power-law index, the profile of momentum was shortened. Salahuddin et al. 4 examined the tangent hyperbolic fluid model with a flow of stagnation point toward a deformable cylinder. Kumar et al. 5 examined the tangent hyperbolic squeezed flow with a sensor surface along variable thermal conductivity. Rehman et al. 6 discussed the tangent hyperbolic fluid flow toward the inclined cylindrical surfaces and the surfaces that are deformable.
The transformation of heat developing nanofluids is among the hot fields of analysis due to their encouraging heat transfer characteristics. In this field, the most recent published work can be studied in Refs. 7-10. Nasir et al. 11 scrutinized the Darcy Forchheimer nanofluid thin film flow of single-walled carbon nanotubes and heat transfer analysis over an unsteady stretching sheet. Shah et al. 12 studied radiative heat and mass transfer analysis of the micropolar nanofluid flow of a Casson fluid ARTICLE scitation.org/journal/adv between two rotating parallel plates with effects of the Hall current. Sheikholeslami et al. 13 scrutinized the applications of electric field for the augmentation of ferrofluid heat transfer in an enclosure including double moving walls. Chaim 14 has performed a noteworthy work on variable thermal conductivity toward a deformable surface. In his analysis, it was investigated that the fluid's temperature increases as the variable thermal conductivity rises, but, at the same time, the wall gradient declines. Reddy C et al. 15 illustrated MHD and the heat transfer flow along variable thermal conductivity and variable thickness toward a deformable sheet. Shokouhmand et al. 16 explored variable thermal conductivity with two-dimensional porous fins. Sreenivasulu et al. 17 carried out the study of variable thermal conductivity on the MHD flow for a deformable surface with a thermally stratified medium. Magneto hydrodynamics is the analysis of highly electrically conducted fluids with magnetic properties. It performs a vital role in different fields such as geophysics, agriculture, meteorology, solar physics, petroleum industries, and astrophysics. Plasmas, electrolytes, metals, liquids, and salt water are some examples of magneto fluids. Rashidi et al. 18 investigated the mixed convection of heat transfer of the nanofluid flow in a vertical channel with sinusoidal walls under the MHD effect. Ahmad et al. 19 analyzed the Darcy-Forchheimer MHD couple stress 3D nanofluid over an exponentially stretching sheet through Cattaneo-Christov convective heat flux. Shah et al. 20 investigated the transient process in a finned triplex tube during phase changing of aluminum oxide enhanced pulsed-code modulation (PCM). Again, Shah et al. 21 investigated the Darcy-Forchheimer 3D micropolar rotational nanofluid flow of single wall and multiwall carbon nanotubes based on fluids (water, engine oil, ethylene glycol, and kerosene oil). A uniform MHD effect on water based nanofluid thermal behavior in a porous enclosure with an ellipse shaped obstacle has been studied by Sheikholeslami et al. 22 Shah et al. 23 analyzed the electrical MHD and Hall current impact on the micropolar nanofluid flow between rotating parallel plates. They examined the combined effect of magnetic and electric fields on micropolar nanofluids between two parallel plates in a rotating system. The heat transfer and MHD flow toward an exponentially deformable sheet with radiation and viscous dissipation have been examined by Sungu. 24 Ruslan and Yaroslav 25 studied the MHD numerical direct simulation of heat transfer with the combined influences of the thermo-gravitational and longitudinal magnetic field.
In very-high-power output devices, forced convection alone is not enough to dissipate all the heat. In such cases, combining natural convection with forced convection (mixed convection) will often give required results. Mixed convection mainly occurs in many technical and industrial applications. A heat exchanger placed in a lowvelocity environment, solar collectors, electronic devices cooled by fans, and cooling of nuclear reactors during an emergency shutdown are some of the examples of the mixed convection phenomenon. Yang and Wu 26 carried out the effect of the aspect ratio and assisted buoyancy on flow reversal for mixed convection with an imposed flow rate in a vertical 3D rectangular duct. They obtained the results for the mixed convection flow with an imposed inlet flow rate in a heated duct with a uniform wall temperature. Thermal patterns are presented and investigated for different buoyancy parameters and aspect ratios. Khan et al. 27 examined mixed convective heat transfer to the Sisko fluid over a stretching surface with convective boundary conditions. Ahmad et al. 28 discussed an MHD mixed convection Jeffrey fluid and heat transfer toward an exponentially stretching surface. The mixed convection flow of the Eyring-Powell nanofluid toward a plate and cone was examined by Khan et al. 29 Izadi et al. 30 presented the mixed convection heat transfer and entropy generation of a nanofluid containing carbon nanotubes, flowing in a three dimensional rectangular channel. They investigated that with an increase in the opposed buoyancy parameter, the nanofluid velocity near the channel wall reduces and, therefore, causes a reduction in the Nusselt number.
In the glance of the aforesaid literature survey, it has to be noticed that no work has been done to investigate the output of mixed convection of the tangent hyperbolic fluid flow in the existence of MHD and variable thermal conductivity. Therefore, the present endeavor is concentrated on this direction. The exclusive intention of this paper is to examine the effect of mixed convection of the hyperbolic tangent fluid flow with MHD and variable thermal conductivity toward a deformable sheet. The configuration of the present article is derived in such a way that partial differential equations (PDEs) can be converted into ordinary differential equations (ODEs) and then solved by BVP4C (MATLAB package). The behavior of different parameters, i.e., mixed convection, power law index, Hartmann number, aligned angle, Prandtl number, and Weissenberg number, has been examined for velocity and temperature profiles. The obtained results are expressed through graphs and tables in detail.

MATHEMATICAL FORMULATION
For a steady, two dimensional, incompressible, and electrically conducted flow of the tangent hyperbolic fluid, consider a deformable sheet coexisting along the plane y = 0, and the flow is being limited to y > 0, as shown in Fig. 1.
Here, the y-axis is the direction of the flow along the sheet, and the x-axis is perpendicular to the y-axis. Variable thermal conductivity mixed convection and MHD effects are also taken into account. A uniform magnetic force is applied in an inclined direction. The tensor of the tangent hyperbolic fluid model is 1,2 where μ 0 and μ∞ represent the zero and infinite shear rate viscosities, respectively,τ is the stress tensor, n represents the power law index, Γ symbolizes the time dependent material constant, and A 1 is the first Rivilin-Erickson tensor.γ is defined aṡ where Π = 1 2 tr(grad V + (grad V) T ) 2 . It is not possible to consider the problem by taking infinite shear rate viscosity, so we take μ∞ = 0, and since a hyperbolic tangent fluid model has shear thinning behavior (Γγ < 1), then, from Eq. (1), After applying the technique of the boundary layer, the governing equations of temperature and momentum are 2 where u and v are the velocity components along the x and y directions, respectively, ν represents the kinematic viscosity, ρ is the density of the fluid, ϕ symbolizes the aligned angle, λ is the mixed convection parameter, σ denotes thermal diffusivity, β is the magnetic field, and T represents temperature. The boundary conditions are u = uw(x) = ax, v = 0, T → Tw at y = 0, u → 0, T → T∞ as y → ∞.
where uw is the velocity of the fluid surface along the wall.
The following similarity transformations have been used: where ε denotes the parameter of variable thermal conductivity, α * is the parameter of thermal diffusivity, T represents fluids temperature, Tw is the wall temperature, and T∞ is the surrounding fluid temperature. Equations (4)-(7) take the form by using the following equation: In these expressions, We = √ 2xΓa 3 2 √ ν is the Weissenberg number, ρa is the Hartmann number, λ = λ a 2 x (Tw − T∞) is the mixed convection parameter, and Pr = ν α is the Prandtl number. Then, the boundary conditions become The coefficient of skin friction can be defined as Using the values ofτw and μw in Eq. (12), we get Also, the local Nusselt number is Nux (Rex)

RESULTS AND DISCUSSION
Graphical interpretation is used to epitomize the repercussions of different expedient parameters for velocity and temperature profiles, as shown in Figs. 2-7. Figure 2 describes the demeanor of the Weissenberg number We for a velocity distribution f ′ (η). It can be noticed that the velocity distribution f ′ (η) dwindles owing to the   hindrance to the fluid and, consequently, the velocity distribution dwindles. Figure 3 shows the behavior of the Hartmann number M for a velocity profile f ′ (η). It is an apparent fact that the velocity profile f ′ (η) decreases for greater values of the Hartmann number M.
The physical reasoning behind this phenomenon is that the Lorentz force is strengthened due to greater values of the Hartmann number, as a result of which it creates resistance in the fluid flow. For incrementing values of the inclined angle ϕ, there is a decline in the velocity distribution f ′ (η), as shown in Fig. 4. It is just because of the fact that when there is escalation in the aligned angle ϕ, the magnetic field enhances. The demeanor of the power-law index n for velocity distribution has been shown in Fig. 5. It is depicted that for every incrementing value of the power law index n, the velocity distribution declines. Figure 6 elucidates the demeanor of the Prandtl number Pr on the temperature profile θ(η). It is interpreted that the Prandtl number is elucidated as the relation between momentum diffusivity and thermal diffusivity. From this figure, it is palpable certitude that due to exceeding quantities of the Prandtl number Pr, there is growth in temperature distribution. The logic of this phenomenon is that due to exceeding quantities of the Prandtl number Pr, the thermal diffusivity of the fluid declines. The demeanor of the variable thermal conductivity ε for temperature distribution is depicted in Fig. 7. It is seen that augmentation in the variable thermal conductivity ε leads to escalation in the temperature distribution θ(η). As far as skin friction is concerned, there is a decline due to the escalation of the power law index n. The demeanor of skin friction is   Fig. 8. Figures 9-11 represent the behaviour of the Nusselt number for assorted values of ε, power law-index, and Prandtl number. Figure 9 depicts the behavior of the Nusselt number for different values of Pr. It has to be noticed that due to greater values of Pr, the Nusselt number decreases. Subsequently, from Fig. 10, it is evident that due to magnifying values of ε, there is a decline in Nusselt number distribution. Illustration of the power law index parameter n is expressed in Fig. 11. Owing to incrementing values of the power law index n, the Nusselt number dwindles. Figure 12 has been plotted to depict the behavior of the mixed convection parameter λ. It can be observed that velocity distribution increases for greater values of the mixed convection parameter λ. Numerical values for the coefficient of skin friction and its comparison with Ref. 33 are presented in Table I, while the Nusselt number and its comparison with Ref. 34 has been also described in Table II.

CONCLUDING REMARKS
The current analysis describes the numerical study of mixed convection of a hyperbolic tangent fluid flow over a stretching sheet with MHD and variable thermal conductivity. The numerical results are fetched and also compared with existing published data, depicting excellent similarity. 31,32 The major outcomes are cataloged below: • The velocity distribution f ′ (η) dwindles for the Hartmann number (M), Weissenberg number (We), power law index (n), and inclined angle (ϕ). It is perceived that these parameters resist the fluid flow, whereas velocity distribution increases for greater values of the mixed convection parameter λ. • Temperature distribution increases against the small parameter (ε), whereas for greater values of the Prandtl number (Pr), the temperature profile decreases. • The coefficient of skin friction declines due to growth in the power law index (n). • The Nusselt number increases with the Prandtl number, whereas for incrementing values of the small parameter (ε) and power law index (n), it declines.