Implications of Multiple Numerical Aspects for Carreau Nanouids With Heat Generation/absorption via Nonuniform Channels

: Nanomaterials are unique work fluids with preeminent thermal performance for improving heat dissipation. We present theoretical and mathematical insights into nanofluid heat transfer and flow dynamics in nonuniform channels utilizing a non-Newtonian fluid. Therefore, the impacts of heat absorption/generation and Joule heating in a magneto hydrodynamic flow of a Carreau nanofluid into a convergent channel with viscous dissipation are addressed in this mathematical approach. Brownian and thermophoresis diffusion are considered to investigate the behavior of temperature and concentration. The magnetic effects on the flow performance are measured. The leading nonlinear equations are solved numerically using the BVP4c solver and RK-4 (Runge – Kutta) along with the shooting algorithm using the computer software MATLAB. The obtained dual solutions are presented graphically. The consequences of the variable magnetic field, heat absorption/generation and numerous physical parameters on the temperature and concentration field are surveyed. The outcomes show that increasing the rates of the heat absorption/generation parameter and Eckert number enhances the thickness of the thermal profile of the convergent channels, while increasing the value of the Prandtl number expands the thickness of the momentum boundary layer of the convergent channels. The key findings related to the study models are presented and discussed. An assessment of solutions achieved in this article is made with existing data in the literature.


Introduction
The idea of magneto hydrodynamics plays a vital role in the flow of Newtonian and non-Newtonian fluids. There are many promising applications of MHD, which appear in cooling systems with liquid metal, flow meters, sensors, blood flow measurement pumps, magnetic drugs, nuclear reactors, the formation of new stars, atmospheric heating, etc. Flow through nonuniform channels is a significant part of research due to many practical and industrial engineering and many physical applications, such as flow through rivers and canals, blood flow through arteries and capillaries, supersonic jets and nozzles. The two-dimensional flow of viscous fluid through a converging-diverging channel was introduced by Jeffrey [1] and Hamel [2]. They done pioneering work in this area. The Jeffrey-Hamel flow models are interesting and important to analyze the boundary layer separation in divergent channels. Jeffrey and Hamel provide a solution to Navier-Stokes equations by introducing the similarity transformation concept that depends on two nondimensional parameters, the Reynolds number and the angle of the channel width. This model was further analyzed by Axfold [3] by considering the effects of an external magnetic field. He predicts that the magnetic field, the Reynold number, and angle of inclination act as control parameters. Imani et al. [4] extend the study by converting Maxwell's equations and Navier-Stokes equations to nonlinear ordinary differential equations for the modeled problem of Jaffrey-Hamel flow in the presence of a high magnetic field and nanoparticles. The flow region in the divergent channel was analyzed for different values of the Hartmann number and angle of the channel. The obtained results were matched with the exact solution obtained by ADM. Makinde [5] studied the effects of MHD on classical Jaffrey-Hamel flow in a convergent-divergent channel using Pade approximations. They interpret various numbers of Reynolds numbers on the velocity field of the flow and constant shear rate in both convergent-divergent channels. Using an analytical (DRA) approach, Dogonchi et al. [6] explored a two-dimensional steady and incompressible viscous water-based MHD nanofluid from the origin between two stretchy/shrinkable walls. They found that the fluid velocity and temperature distribution increased with an increase in the stretching parameters. Reza et al. evaluated the copper-kerosine nanofluid in a channel across a stretched sheet under the action of a magnetic field. [7]. They used a three-stage Lobato IIIA formula to solve the nonlinear ODEs. Their outcomes show that solid volume friction decreases the velocity of nanoparticles close to the wall of the channels and escalates the thickness of the thermal boundary layer of the channels. The study of magnetohydrodynamic flow of nanofluids over a rotating stretchable plate in the presence of a magnetic field was captured by Ram et al. [8].
The unsteady magnetic nanofluid over a stretchable rotating disk was analyzed.
A homogeneous concentration of nanosized magnetic particles (1-100 nm) in a base liquid with an exterior coating is known as a nanofluid. The most important feature of nanofluids is their high thermal conductivity compared to pure liquids. Choi et al. [9]. introduced a model and, for the first time, used the term nanofluid. After the groundbreaking idea, many researchers proposed different models to incorporate Brownian motion and thermophore effects. In the presence of stretched shrink walls, Mohyud-Din et al. [10] addressed the MHD flow of an incompressible fluid comprising nanosized particles between nonparallel walls. MHD nanofluid flow and heat transportation over two horizontal slabs in a rotating device were addressed by Sheikholeslami et al. [11]. The KKL (Koo-Kleinstreuer-Li) correlation was used to analyze the heat conduction and stiffness of the nanofluid. They noticed that with the increase in magnetic parameters, rotational parameters, and Reynolds number, the skin friction coefficient increased but decreased as the nanoparticle volume friction increased. Mebarek-Oudina [12] examined the fluid flow structure of Titania nanofluids in a cylinder annulus containing a variety of base fluids. He used the finite volume strategy to address the Maxwell fluid model for connective energy transfer. In a few other articles, Mebarek-Oudina [12] analyzed the natural convection and heat transport stability of nanofluids in the presence of an applied magnetic field. The impact of molybdenum disulfide (MoS2) nanoparticle shapes on the rotating flow of nanofluids along an elastic stretched sheet was examined by Usman et al. [13]. They discovered that as radiative heat parameters and Prandtl numbers improve, the local Nusselt number diminishes. Once the thermophysical parameters are improved, they improve. Alam et al. [14] explored the role of a magnetic field on the entropy rate of production of conductivity fluid flow in a converging-diverging channel. Khan et al. [15] provide a new track of such flows by takiing velocity and temperature slip on flows in a convergent-divergent channel.
Viscous dissipation effects are often negligible in different flows; however, their involvement becomes significant when fluid viscosity is higher. It affects the heat field by playing an energy source role, which leads to a change in the heat transfer rate. Kayalvizhi and Ganga [16] analyzed the viscous and Joule dissipations on MHD flow over a stretchable porous region immersed in porous medium for common fluids. Pal and Mandal [17] used the RKF algorithm to estimate a numerical solution for convective nanofluids over a stretchable/shrinkable surface underneath the impact of viscous dissipations. For large values of current shrinkable sheet parameters, they execute a dual solution for the temperature profile. In the presence of ohmic dissipations, Hayat et al. [18] address the boundary layer problem for MHD Williamson fluid past a porous stretchy sheet. They observed that the magnetic field expanded the surface drag force.
The effect of viscus and ohmic dissipations of MHD nanofluids over an upright plate with energy generation/absorption was explored by Ganga et al. [19]. They observed that due to an increase in solid volume friction, the temperature spreading declines. Singh et al. [20] explored the impact of MHD and velocity slip on a vertical plate in an alumina-water nanofluid. They observed that the energy transport ratio rises as the volume friction of dense particles increases. Mishra et al. [21] conducted a numerical study for the current equations of MHD (Ag-2 O) nanofluid fluid through an upright cone subject to viscus dissipations. They conclude that the mass transference rate decreases as velocity slip limitations grow. Alamri et al. [22] where is the radial velocity, is the fluid density, describes the kinematic viscosity and is the fluid pressure. Equations (2) and (3) are coupled into equation (4) by eliminating the pressure gradient term the associated conditions at the boundaries are: (3) From equation (1), the radial velocity is a function of r and , as provided in Eq. (5). Hence, Eq.
(5) suggest the following form of radial velocity: Jaffrey-Hamel flow is the mechanism of the dual channels where at one end the fluid transports inside called convergent channel and at the other end the fluid removesward called divergent channel. Furthermore, it is well known that the peak fluid velocity is certainly achieved at = 0.
In fact, we find the following: Incorporating Eq. (5) into Eq. (4), we subsequently obtain the following nondimensional ODE: With the aid of dimensionless variables: At the center of the channel (± ) = 0, = 1,

Heat transfer analysis
This section presents the energy transport throughout the fluid drift through narrow and opening channels. In the presence of viscous dissipation due to nanoparticles, a heat generating/absorbing source is present within the system, and the heat Joule effects energy equation can be proposed as: With the subsequent boundary conditions: , at = 0, = , at = ± , Here, and ( ) indicate the thermal conductivity and specific heat at constant pressure.
Furthermore 0 indicate the heat generating/absorbing coefficient. 0 < 0, denotes heat absorption, while 0 > 0 represents heat generation. In addition, = ( ) ( ) , is the ratio of the thermal capacity ratio. and are Brownian and thermophoresis diffusion, respectively. , signify the density of the nanoparticles. and ∞ demonstrate the wall and ambient temperature.
The following dimensionless transformations are considered: In indicate temperature at the wall. The partial differential equation in (14) is transmuted into ODE using (16), and we have (14) (13)

Concentration analysis
The corresponding concentration equation for the flow model:

Engineering parameters
The skin friction coefficient, Nusselt number, and Sherwood number are several physical quantities of interest.

Results declaration and discussion
The nonlinear equations (10), (17) and (22)  Fehlberg method with the Nachtsherim-Swigert Shooting approach is used, which is a rapid and adaptable numerical technique. The governing dimensionless boundary value problem (10), (17) and (22)     layer is wider than the thickness of the momentum boundary layer, magnified Prandtl numbers correlate to high thermal conductivity, i.e., heat spreads out more efficiently. The temperature evolution with expanding Brownian diffusion parameters is shown in Fig. 9. Improved Brownian diffusion parameters raise the thermal field. The nanoparticle collides with a fluid molecule because of Brownian drift, and a part of the kinetic energy is turned to thermal energy, leading the fluid temperature to rise. In Fig. 10, the significance of the thermophoresis diffusion parameters is prescribed. Thermophoresis transmits fluid molecules with a high thermal energy transfer from a hotter to a colder region, improving the fluid temperature. An increase in the Eckert number provides an upsurge in temperature, as expected, as shown in Fig. 11. This enhancement occurs due to , which directly affects the heat dissipation process and hence the thermal profile boosts. The viscous dissipation effect, which is always positive and indicates a source of heat due to frictional forces within the fluid molecules, is illustrated by the fourth term on the right-hand side of Eq. (14). In addition, as the Eckert number grows, the thickness of the thermal boundary layer decreases. Figure 12 indicates the impact of the Hartmann number on temperature. In a convergent channel, the temperature is a decreasing function of the Hartmann number, as per this graphic. This is because a perpendicular magnetic field creates a resistive force in an electrically conducting fluid known as the Lorentz force. This force causes a resistance within the fluid particles by creating frictional force between its layers, dropping its temperature. Fig. 13 concludes that the temperature distribution grows for convergent channels against distinct values of the heat generation/absorption parameter g. Furthermore, with g, the corresponding boundary layer expands. Physically, large heat generation/absorption rates transfer more heat to the working fluid, driving the thermal profile to expand. Heat generation (heat source) is indicated by positive values of g, whereas heat absorption is indicated by a negative sign (heat sink). The term "heat source" describes the process of producing heat from the surroundings, which heats up in the flow field. As previously stated, the thermal phase of the fluid increases due to the presence of a heat source or a heat generating mechanism, causing the thermal boundary layer to increase.   Brownian motion promotes the random movement that disperses the nanoparticles, resulting in a rise in concentration. Dropping of concentration against growing thermophoresis parameters is witnessed in Fig. 16. It is obvious that as the thermophoresis parameters are improved, the concentration of nanofluid drops, which may be interpreted by the increasing temperature of the flow field and the temperature of the boundaries. A decrease in concentration due to an enhancement in the Eckert number is described in Fig. 17. Even at the boundary value, the concentration of nanoparticles diminishes for high Eckert number values. This is because as rises, the rate of heat transmission at the surface drops. Dropping of the concentration in convergent channels due to a gradual increase in the heat generation absorption parameter is depicted in Fig. 18. Since the concentration of the fluid does not vary with the fluctuation of heat in the fluid, the heat generation/absorption coefficient, , has little influence on the concentration distribution. The heat flux coefficient is responsible for increasing the fluid flow heat gradient, although it has no consequence on the fluid particle concentration levels.           3. A strong magnetic field decreases the fluid flow rate and reduces the temperature of the fluid and can also be used to diminish the concentration of the fluid in both channels.
4. The rise in temperature is recorded due to growing values of Reynolds and Prandtl in both channels.

Brownian and thermophoresis diffusion parameters
and display similar effects in both channels. Decreases in temperature and concentration are noticed upon increasing these parameters. 6. Growing values of Eckert number , temperature and concentration boosts.
7. The concentration and temperature are maximal in the middle of the channel and decrease rapidly near the wall of the channels.
8. The contribution of the heat generation parameter to the velocity and concentration profiles is quite minor, although the temperature of the nanofluid is rapidly growing.
9. The role of the Reynold number in the surface drag force and Nusselt number is contrary in narrow and deviating channels.