Exploration of Temperature-Dependent Thermal Conductivity andDiffusionCoefficient for Thermal andMassTransportation in Sutterby Nanofluid Model over a Stretching Cylinder

Department of Applied Mathematics and Statistics, Institute of Space Technology, P.O. Box 2750, Islamabad 44000, Pakistan Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia Department of Medical Research, China Medical University, Taichung, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan Renewable Energy Research Centre, Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut’s University of Technology North Bangkok, 1518 Pracharat 1 Road Bangsue, Bangkok 10800, 4ailand


Introduction
In the momentum period, non-Newtonian liquids are progressively significant for analysts as a result of their assorted applications in the mechanical field. For example, the assembling of polymers, gas turbines, power generators, glass fabrics, paper creation, wire drawing, and the sky is the limit from there. Sutterby liquid is one of the significant non-Newtonian fluids under the perceptions of the analysts. Among these, the Sutterby model addresses the high polymer fluid arrangements. It is one of the liquid models which expand the huge conduct of non-Newtonian liquids like pseudoplastic and dilatant liquids and it has the attributes to catch both the shear diminishing and shear thickening nature of the stream. Bilal et al. [1] researched Sutterby's MHD liquid stream in the Darcy medium. ey performed OHAM to appraise the outcomes and found that speed demonstrated the contrary conduct for Darcy's opposition. Stale Sutterby liquid stream disregarding a contracting sheet highlighting the Cattaneo-Christov hypothesis going with heat retention/ age was dissected by Rehman et al. [2]. Nawaz [3] contemplated the warm execution of the mixture Sutterby nanofluid stream and applied the FEM strategy to process results. He noticed that surface warmth transition is higher on account of crossbreed nanofluid than that of the base liquid. Rana and Nawaz [4] delineated the warmth improvement model in Sutterby nanofluid stream. ey likewise considered Koo-Kleinstreuer and Li (KKL) Cattaneo-Christov heat motion models and found that temperature is altogether affected by the Joule warming impact. Sutterby nanofluid electromagnetohydrodynamic limit layer Darcy Forchheimer stream getting over an extending surface with a slip limit was clarified by Bilal et al. [5].
ey applied the shooting technique modernized with Cash and Crap coefficients. ey remarked that, in correlation with a traditional case, the drag on a superficial level for the Sutterby case shows a piddling pattern.
Warmth move is a significant wonder as it is an approach to move energy to obtain the necessary errand through various cycles. Warmth transport is needed in chilling of electrical thing, inner purifier with radiator, heat exchanger, chilling of reactor, and so forth. Transport of warmth and mass exchange assume a vital part in designing and modern fields. Trials show that steady warm conductivity and mass dispersion exist for extremely uncommon cases and results got with consistent properties do not approve for all circumstances. at is the reason there is a need to present a variable warm model. Variable thermophysical properties in the liquid stream are essential to expanding the productivity of the framework. Hayat et al. [6] examined blended, convected Casson liquid stream with temperature-subordinate warm conductivity. ey contemplated heat move within the sight of gooey dispersal and find that warm limit upgrades for developing estimations of little boundary use to fluctuate warm conductivity. Temperature-subordinate warm conductivity in the stagnation guide stream of Jeffrey liquid to nonstraight extended surface having variable divider thickness is examined by Hayat et al. [7]. ey expressed that temperature shows a decrease in the rising warm unwinding boundary. Khan et al. [8] endeavor to delineate the stagnation direct progression of the 2 nd grade to an impermeable extending chamber. ey consider temperature-subordinate warm conductivity and warm definition and infer that Fourier's law model has a higher temperature than the Cattaneo-Christov heat motion model. Hammad et al. [9] examined radiation impact for the gooey liquid stream with temperature-subordinate thickness and warm conductivity disregarding a penetrable plate and pronounce that convective warmth move rots the temperature and speed of the stream.
Nanofluid is standing out enough to be noticed by scientists around the globe to expand the productivity of energy. Nanoparticles can assimilate straightforwardly from the wellspring of the episode radiation and improve the fluids radiative properties. Likewise, warm radiations alongside nanoparticles are primarily centered around the examination. ey help in using sun-oriented energy as radiation heat moves. e sun is the less expensive source of the energy. Ijaz et al. [10] investigated the entropy and system of energy actuation and twofold compound response for the progression of nonlinear radiative Sisko liquid ignoring a radially stretchable rotatory circle. e huge ascribes of this examination are thick dispersal, Joule warming, and blended convection. Stagnation point blended convective MHD stream of cross nanofluid toward the surface which is extending straightly highlighting the idea of initiation energy shown by Khan et al. [11]. Parts of thermophoresis, Brownian movement, and nonlinear radiation are additionally introduced. ey advise that for higher assessment of actuation energy variable types of focus rises. Stagnation point upper convective Maxwell nanofluid streaming over directly extending sheet alongside the credits of sun-oriented radiation and gooey dispersal was broken down by Khan et al. [12]. e shooting strategy is utilized to register results and they found that temperature is emphatically affected for radiative warmth move. Waqas et al. [13] noticed the effects of the attractive dipole and nonlinear warm radiation on liquid whose rheology was described via Carreau nanofluid. ey applied the shooting technique to get an answer and expressed that the bigger ferrohydrodynamic connection variable tumbles down the speed.
Stratification has a critical part in the advancement of businesses. It applies in the dismissal framework, condensers, and nuclear power. Khan et al. [14] analyzed the impacts of variable consistency and warm separation on Williamson nanofluid over a nondirect extending sheet and reasoned that speed and temperature rot for bigger warm delineation boundaries. Axisymmetric Jeffrey liquid stagnation point stream over a slanted porous stretchable chamber was concentrated by Ijaz and Ayub [15]. e ramifications which are thought about are the Cattaneo-Christov hypothesis, warm separation, and homogeneousheterogeneous.
ey uncovered that temperature is diminishing the capacity of warm definition. Rehman et al. [16] outlined the blended convection digression exaggerated stream by extending round and hollow surface which is drenched in twofold defining media with no-slip conditions. ey guaranteed that warm and solutal delineation is the reason for a decrease in temperature and convergence of the stream. Nanofluid extended stream with motile microorganisms with the ascribes of a slanted attractive field, definition, and gooey dissemination being inspected by Alsaedi et al. [17]. For additional investigations and examinations, perusers allude to the investigations announced in [18,19].
Existing writing demonstrates that there is no examination so far to see the highlights of temperature-subordinate warm conductivity and mass dispersion coefficient for the progression of magnetohydrodynamic Sutterby nanofluid past over a stretchable chamber where warm radiation as sunlight based energy is assuming a significant part in the stream model with respect to the fact that its application is concerned. is correspondence fills this gap.

Mathematical Modelling under Boundary
Layer Theory (MMUBLT) e transportation of warmth and mass for the steady and incompressible two-dimensional Sutterby nanofluid stream on a stretch chamber involving variable thermophysical qualities is considered in the current examination in Fig is explained in Appendix. Sutterby's liquid model [1,4,20,21] is characterized as (1) Here, _ c � ������ (1/2)Π, and Π � Tr(A 2 1 ). Under the suspicion of limit layer, the demonstrated incomplete differential condition framework relating to the protection laws of mass, movement, warmth, and species transport is given as e marvel considered with various actual perspectives considered is in Figure 1. Related limit conditions are e numerical models for warm conductivity and mass dissemination coefficient are referenced as [22,23] e similarity transformations mentioned below [1,20] are used to convert equations (3)-(6) into coupled ordinary differential equations:  Complexity η � Exploiting the above-declared renovations, equations (3)-(6) are as the following formation: Transformed conditions for equations (9)-(11) are where P r∞ � P r (1 + εθ), Incorporating equation (13) in equations (10)-(11), we have 4 Complexity e dimensionless parameters which are shown in the above equations are defined as follows: Coefficient of Skin erosion [1] and neighborhood warmth and mass transitions boundaries in nondimensional structures are deciphered as where λ � z 2 U 0 /] f l is a dimensionless parameter.

Solution via OHAM
is plan is broadly used to deal with nonlinear coupled issues that emerge in numerical material science. is technique is utilized by numerous specialists in their investigations. e itemized plans of this technique are treated in [1,[20][21][22][23][24][25][26][27]. e underlying assessments and their comparing direct administrators for speed, temperature, and dimensionless fixation fields are

Graphical Analysis and Physical
Interpretation and Justification

Influence of Involved Parameters on Fluid
Velocity. e emerging nonlinear arrangement of conditions has been handled by means of OHAM. e union of the applied calculation by processing the blunders is introduced in Table 1. In the current segment effect of various boundaries on stream, profiles have been examined as charts and their representation is incorporated.  Figures 6-14 are designed for the impact of several flow parameters on dimensionless temperature θ(η). Figure 6 shows the view of curvature impact on temperature θ(η). As velocity is an upsurging function of curvature and temperature is also defined through average kinetic energy. Higher curvature implies higher velocity which implies higher kinetic energy results in the rising temperature profile. e declining outturns of a small parameter (ε) on temperature are shown in Figure 7. For (ε) � 0, the case reduces to normal temperature-independent thermal conductivity case and shows the maximum temperature on it which implies that variable thermal conductivity cools down the flow with higher temperature. Figure 8 shows the impact of the small parameter (ε 1 ) (used to vary the diffusion coefficient) on temperature θ(η). For higher values of (ε 1 ), the graph shows a rise in temperature. Figure 9 is portraying    Complexity the declining behaviour of temperature curves for growing values of Prandtl number (P r ) since the thickness of the thermal boundary layer reduced by enhancing (P r ). As Prandtl number is inversely related to thermal diffusivity, enhancing it results in cooling the flow. e combined impact of the Brownian motion parameter (N b ) and thermophoresis parameter (N t ) is depicted in Figure 10. Speeding the Brownian motion leads to the faster random motion of nanoparticles in a flow which shows an extension in thermal boundary layer thickness and enhances the temperature of flow more rapidly. A similar pattern is observed for growing values (N t ). As in process of thermophoresis, more heated particles near the surface travel away from heated regions toward the cold region and raise temperature there and collective temperature of the whole system rises. Figure 11 illustrates the impression of radiation parameter (R d ) on temperature θ(η). Radiation parameter tells about the relative contribution of heat transfer through conduction to thermal radiation. So, a clear upsurge in temperature curves is observed for escalating values of (R d ).

Comportment of Contributing Parameters on Temperature Field.
Enhancing thermal radiation means adding energy through radiation in the flow which eventually boosts the temperature distribution of the fluid. Figure 12 shows the impression of the Eckert number (E c ) on dimensionless temperature θ(η). e growth in (E c ) has a positive influence on heating the system due to the fact that it strengthens the kinetic energy of the flow. e graphical view for the behaviour of temperature when magnetic parameter (M) and its inclination angle (α) are taken into account is illustrated in Figure 13. e resistive force which arises due to the presence of a magnetic field is the cause of boosting the temperature distribution of flow. Figure 14 is a pictorial illustration of the stratification parameter (S 1 ) due to temperature difference. e thermal stratification process causes layer formation due to temperature variation, the enlarging stratification parameter falls down the temperature gradient between the wall and reference point, and thus the temperature of the system drops.   Figure 15 is an image of thermophoresis parameter (N t ) influence on concentration. Boosting thermophoresis process boosts the traveling of particles from higher to lower temperature difference which as a result maximizes the concentration of nanoparticles in the flow regime. Figure 16 is plotted to understand the impact of Lewis's number (L e ) on concentration ϕ(η). As (L e ) gives the relation between the thermal boundary layer and the concentration boundary layer, physically, escalation in (L e ) shrinks the thickness of the boundary layer of concentration. Figure 17 is planned to study the behaviour of concentration ϕ(η) on the basis of a small meter (ε 1 ) (used to vary diffusion coefficient). An increasing behaviour is observed for elevating values of (ε 1 ), Figure 18 is sketched to illustrate the impact of the stratification parameter (S 2 ) due to concentration difference. e higher values of (S 2 ) are a reason for the decline in concentration difference between wall and  Figure 19 shows the illustration of the effect of the Sutterby fluid parameter (β) on skin friction coefficient (c f R −1/2 e ). Enhancing (β) has a reverse reaction on flow velocity as it enhances the viscous nature of the fluid due to which opposing force arises which is the cause of boosting the skin friction coefficient. Figure 20 gives a pictorial view for the impressions of radiation parameter (R d ) on the heat flux in the form of local Nusselt number (Nu z R −1/2 e ). Higher radiation parameter lowers the local Nusselt number as it also has a reverse effect on the momentum boundary layer. Physical temperature gradient at the wall surface becomes small which lowers natural convection ability. Figure 21

Complexity 9
shows the footprints of stratification concentration parameter (S 2 ) on local Sherwood number (Sh z R −1/2 e )and the bar chart shows a reverse behaviour for the escalation in (S 2 ). In Table 2, a limiting case of dimensionless stress is computed and a comparison is established. It is found that, due to the factor 2 in the momentum equation, our results are differing by the multiple of 2. If we ignore 2, our results exactly match with the precious finding.

Conclusion and Key Points of Current Research
In the course of the study, a two-dimensional stationary radiation flux of Sutterby nanofluid with an oblique magnetic field and with variable thermophysical properties was taken under observation. To solve the system of transformed ODEs, the OHAM scheme was applied. e main findings of this study are as follows: Larger curvature (c) boosts both velocity and temperature profiles. Small parameter (ε) reduces the temperature and (ε 1 ) shows opposite behaviour. Radiation parameter (R d ) has a positive impact on the temperature profile. Velocity field slows down for higher Reynolds number (R e ), magnetic parameter (M), and Sutterby fluid parameter (β).