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The underlying intention of this study is to oversee the flow, heat, and mass transport of hybrid nanofluid (Casson, micropolar, silica, alumina, and Water), Casson, and micropolar fluid flow across a curved stretching sheet. The influence of radiation, cross-diffusion and the applied magnetic field is also looked into. The total entropy rate of the Casson, micropolar and hybrid fluid is discussed. A curvilinear coordinate system models the flow equation. Appropriate similarity transformations are applied to modify the governing nonlinear PDEqns into ODEqns.The proposed system is numerically solved with R-K 4th order based shooting procedure. Outcomes are explained by preparing graphs and tables—comparative studies of obtained results with previously published results discourse. The main outcomes of this study are (a) Hybrid nanofluid has a higher temperature profile than micropolar and Casson fluid. (b) Entropy generation is maximum for larger values of Curvature, temperature difference, diffusion parameter, Dufour and Brinkman number. (c) The micropolar fluid shows higher entropy generation compared to Casson and hybrid nanofluid. (d) Fluid velocity augments with larger magnetic field and Curvature parameter. (e) Casson fluid shows a higher velocity field compared with micropolar and hybrid nanofluid.

Generally, transformer oil is used in various devices including circuit breakers and (oil-filled) transformers. Functions of the oil are many, for instance, to inhibit corona discharge. Researchers found that the breakdown functioning of transformer oil enhances (to some extent) when nanoparticles (for instance, F e 3 O 4 ) are suspended in it. In the current paper, we aim to examine transformer oil based hybrid fluid flow amongst two rotating (stretchable) disks with activation energy, heat source and cross-diffusion (Dufour-Soret) effects. We have done the irreversibility analysis on the fluid flow. Leading equations (PDEs) are transmuted as nonlinear ODEs with the aid of befitted similarity transmutations. Shooting technique (R-K 4th order based) is executed to resolve the transmuted equations. Outcomes are explicated via graphs in two kind of flows i.e., hybrid fluid (transformer oil with F e 3 O 4 and T i O 2 ) and nanofluid (transformer oil with F e 3 O 4 ). We have done the validation of our results with earlier results and noticed a decorous accord. Some of our ensues are (a) rotation parameter ameliorates the tangential velocity and stretching parameter (due to lower disk) minimizes the same (b) escalation in entropy generation is recognized for the acclivity in Brinkman number (c) concentration boundary layer becomes thinner for reaction rate parameter (d) Dufour number step-down the heat transfer rate near the lower disk but shows opposite behaviour near the upper disk (e) activation energy parameter intensify the mass transfer rate near the upper disk but exhibits reverse conduct near the lower disk.

The consequences of microgravity and micropolarity are assessed on mixed convection flow associated with magnetic and heat sources. The flow fields take place on a stretching vertical sheet. This communication therefore addresses oscillating convective flow driven by g-Jitter forces to facilitate guidelines for upper atmospheric aerodynamic designs through operating thermal fields. The field equations are reframed as ordinary differential types to be resolved numerically by finite element discretization. Matlab code is developed for Glariken formulation to evaluate parametric simulations for reduced skin friction factor, the velocity of fluid, microrotation, heat transfer rate, and thermal characteristics. The oscillating trends of physical outputs are notably influenced by the parameters of micropolarity, the amplitude of modulation, frequency parameter, magnetic and thermal sources. The oscillatory pattern of reduced skin friction factor − f ″ ( τ , 0 ) becomes stronger for progressive inputs of amplitude parameter and magnetic parameter, but the fluid temperature rises up with increasing strength of the magnetic field. The fluid speed f ′ ( τ , η ) responds reciprocal to the magnetic parameter, but it behaves directly with material parameter.

In this paper, we have analysed a binary chemical nanofluid dissipative flow (in two cases i.e., 50% EG $$ + $$ 50% water/silica and 50% EG $$ + $$ 50% water/graphene oxide) due to a curved stretching sheet with activation energy. Appropriate transformations yield the nonlinear ordinary differential system. Shooting procedure (R-K 4th order based) is executed to solve the resultant equations. Graphical illustrations thoroughly demonstrate the features of the involved pertinent parameters. We have deliberated the behaviour of the alike parameters on the rate of transfers (heat and mass) and surface drag force (skin friction coefficient) by means of tables. This investigation reveals that (a) reaction rate parameter and temperature difference parameter are helpful to ameliorate the mass transfer rate (b) concentration enhances for higher estimation of activation energy variable (c) increasing the volume fraction of nanoparticles reflects an escalation in temperature (d) heat transfer rate enhancement is recognized for the influence of heat transfer Biot number. At the end this study, we came to know that the EG-Water $$ + $$ Graphene Oxide mixture has more heat transfer rate compared to EG-Water $$ + $$ Silica mixture. This outcome helps to conclude that, whenever the more heat transport required in manufacturing and industries, we can take the EG-Water $$ + $$ Graphene Oxide mixture.

Compared to (pure) methanol, methanol-based nanofluids are more effective in several aspects like thermophysical properties. For example, specific heat and thermal conductivity are increased at higher temperature. Researchers experimentally discovered that methanol-based hybrid nanofluid ( $${\text{CH}}_{{3}} {\text{OH}} + {\text{SiO}}_{2} + {\text{Al}}_{{2}} {\text{O}}_{3}$$ ) is useful to enhance the absorption rate of $${\text{CO}}_{{2}}$$ in a tray column absorber. Despite numerous reports on hybrid nanofluid via curved stretching sheet, so far, no work is done on the impact of increasing Brinkman number, activation energy on the dynamics of methanol conveying silica and alumina when thermal radiation and cross-diffusion effects are profoundly significant. In the present paper, the examination of methanol-based hybrid fluid flow across a bended (curved) stretching sheet with cross-diffusion effects (Soret–Dufour numbers) and activation energy is made. An analysis of entropy generation on the fluid flow is also visualized. Suitable similarity transformations are applied to modify the leading equations as nonlinear ODEqns. Shooting procedure (R–K 4th order based) is performed to unriddle the consequent equations. Outcomes are explicated via graphs in two cases, i.e., hybrid fluid (methanol + silica + alumina) and nanofluid (methanol + silica). The important outcomes are: curvature parameter ameliorates the velocity but minifies the friction factor and escalation in entropy generation is recognized for the acclivity in Brinkman number and magnetic field parameter. Moreover, we witnessed that activation energy parameter steps up the concentration profile, nanoparticle volume fraction parameters minimizes the friction factor, reaction rate parameter ameliorates concentration, thermal radiation elevates the temperature and Dufour number and Brinkman number are useful to minimize the heat transfer rate. It is interesting to mention that influence of magnetic field on velocity profile depends on the value of nanoparticle volume fraction. Further, our results are validated with previous results for surface drag force under limited conditions and we detected a respectable accord.

An electrically conducting three-dimensional non-Newtonian Carreau fluid induced by bidirectional movement of sheet is demonstrated in this work. The aspects of viscous heating, radiation and first-order chemical reactions are executed in energy transport and mass species expressions. The new variables are defined to restructure the mathematical problem into single independent variable equations (ordinary differential equations, ODE’s). The re-framed ODE’s are exploited numerically through the utilization of MATLAB bvp4c package. The derived results of velocities, temperature and species distributions are executed graphically and interpreted physically for both shear-thinning and shear-thickening cases. Also, the numerical results for coefficients of skin friction, rate of mass transfer in terms of Sherwood number and rate of heat transfer in terms of Nusselt number are presented in tables. The influence of local Weissenberg numbers on Sherwood and Nusselt numbers for shear-thinning fluids is reversed to shear-thickening fluids. The thermal radiation parameter enhances the fluid temperature, and chemical reaction parameter decelerates the fluid concentration.

In the production of ethelene glycol, graphene nanoparticles is inevitable and even suggested due to monomolecular layer of carbon atoms which are bounded like honey comb structure is known as graphene due to this structure, graphene has several types of exceptional and unique structural, optical and electronic properties. However, little is known on the enhancement of the transport phenomenon when Joule dissipation, inclined magnetic field, thermal jump and partial slip are apparent. With emphasis to the inherent aforementioned concepts together with heat source/sink and thermal radiation, this paper presents insight into the dynamics of unsteady Ethelene glycol conveying graphene nanoparticles through porous medium. The dimensional governing equation was non-dimenzionalized using fitting similarity variables and solved the dimensionless equations using Runge-Kutta Fehlberg algorithms along with the shooting technique. Also, a statistical method was implemented for multiple quadratic regression estimation analysis on the numerical figures of wall velocity gradient and local Nusselt number to establish the connection among heat transfer rate and physical parameters. Our numerical findings reveal that the magnetic field and porosity parameters boost the graphene Maxwell nanofluid velocity while Maxwell parameter has a reversal impact on it. The regression analysis confers that Nusselt number is more prone to heat absorption parameter as compared to Eckert number. The rate of heat transfer is higher in case of with slip compare to without slip flow in the presence of thermal radiation, viscous dissipation and unsteady parameter. The fluid velocity and temperature distribution is higher in without slip compare to with slip flow.

This study aims to unfold the significance of numerous physical parameters such as magnetic field, heat absorption, thermal radiation, viscous and Joule dissipations, etc. on the flow of graphene Maxwell nanofluid over a linearly stretched sheet with considerations of momentum and thermal slip conditions. The prevailing mathematical equations are reformed into extremely nonlinear coupled ordinary differential equations (ODE) utilizing similarity variables and then the equations are solved numerically by the scheme of Runge-Kutta Fehlberg method along with the shooting technique. The variations in graphene Maxwell nanofluid velocity and temperature owing to different physical parameters are shown via numerous graphs whereas numerical values of skin friction coefficients and Nusselt numbers are illustrated and reported in different tables. In addition, statistical approach is followed for the multiple regression estimation analysis on the numerical findings of wall velocity gradient and local Nusselt number and are reported in tabular form to demonstrate the relationship among the heat transfer rate and physical parameters. Our results reveal that the graphene Maxwell nanofluid velocity gets reduced owing to enhancement in magnetic field, angle of inclination of magnetic field, porosity and unsteadiness parameters whereas behavior of nanofluid velocity is reversed due to Maxwell parameter. Further, it is noticed that the heat transfer rate of nanofluid is augmented owing to heat absorption, radiation and thermal slip parameters while it is reduced due to increase in viscous dissipation and unsteadiness parameters. The numerical results of the paper are validated by making comparisons with the earlier published paper under the restricted conditions and we found an excellent agreement with those results. A careful review of research papers reported in literature reveals that none of the authors has attempted this problem earlier although the thoughts and methodology explained in this paper can be anticipated to lead to enormously prolific connections across disciplines.

Adding variety of nanoparticles to the base fluid is current technique in order to boost the thermal performance of conventional fluids and mononanofluids. The forthright intention of the present investigation is to analyze numerically the up-to-date progress in flow and heat transport nature of magnetohydrodynamic, radiative Newtonian fluid, water-based Al2O3 nanofluid, water-based graphene nanofluid and water based Al2O3 + graphene hybrid nanofluid due to convectively heated stretching sheet. The flow equations are transformed by applying appropriate transformations into a pair of self-similarity equations. Further similarity equivalences are numerically solved through Runge–Kutta based shooting method. Graphs and tables are structured to analyze the behavior of sundry influential variables. From this study it is found that rate of heat transfer for Graphene + water is 2.921934, Al2O3 + H2O + Graphene is 2.250658 and Al2O3 + H2O is 3.260554. From this we conclude that water based Al2O3 + graphene hybrid nanofluid can be opted for cooling performance. Water based Al2O3 nanofluid significantly enhance convection heat transfer performance over a stretching sheet. Friction at the wall for Graphene + water is (− 1.719525), Al2O3 + H2O + Graphene is (− 2.256614) and Al2O3 + H2O is (− 1.959539). From this we conclude that water based Al2O3 + graphene hybrid nanofluid shows lower wall friction rate compared to other two mixture compositions.

The forthright intention of the present investigation is to analyze the up-to-date progress in Jeffrey nanofluid flow past an electromagnetic sheet by utilizing the properties of nonlinear convection, radiation, convective boundary condition, zero mass flux condition and Arrhenius activation energy. The flow equations are transformed by applying appropriate transformations into a pair of self-similarity equations. Further similarity equivalences are numerically solved through Runge–Kutta-based shooting method. Graphs and tables are structured to analyze the behavior of sundry influential variables. The results acquired showed good agreement with the previous notable works. Through this study we observed that improvement in Lorentz force in the positive x-direction strengthens the momentum, which intensifies the transfer of heat energy from the boundary, resulting in reduced fluid temperature.

Nonlinear convective flow and heat transfer characteristics are analyzed between stationary nonporous and porous rotating disks utilizing graphene nanoparticles in a water and ethylene glycol base fluid. Heat transfer characteristics are analyzed via incorporating thermal radiation and heat absorption/generation. The governing fluid equations are computed numerically using Runge–Kutta based shooting technique after employing appropriate transformations. Characteristics of sundry variables are elaborated graphically as well as through the construction of Table for water base and ethylene glycol based graphene nanoparticles. It is observed that improvements in nonlinear convection variable owing to temperature and heat generation variable improve wall friction in radial direction. Improvement in Hartman number decreased wall friction in radial and tangential directions along with Nusselt number in graphene/ethylene glycol and graphene/water nanofluid. Ethylene glycol based graphene nanofluid takes less time for execution as compared to water based nanofluid.

In this paper, stagnation point region (suspended with nanofluid and microorganisms) is examined subjected to velocity and thermal slips. The flow is assumed to be flowing across an exponentially stretching flat surface under magnetic field environment. Governing transport equations (framed under Buongiorno's model) are converted into self-similar form through suitable unsteady exponential similarity transformations which are then solved by employing implicit finite difference scheme (IFDS) known as Keller-box method (KBM). A parametric analysis is performed along with limited details of KBM. Influence of pertinent constraints on local skin friction, local Nusselt number, local Sherwood number and local density number of micro-organisms are also uncovered through graphs. A significant decrease in skin-friction coefficients has been detected with respect to thermal slip in comparison to velocity slip. Keywords: Unsteady flow, Stagnation point region, Mixed convection, Microorganisms, Nanoparticles, Slip regime, Keller-box scheme

Magnetohydrodynamics (MHD) boundary layer slip Casson fluid flow over a dissipated moving cylinder is explored. Casson fluid model is employed as a non-Newtonian material that demonstrates the phenomenon of yield stress. Blood material is considered to be an example of Casson liquid. The non-linear partial differential quantities are transformed into expressions of ordinary derivatives through transformation of similarity variables. These equations are computed for numeric solutions by using Runge-Kutta method along with shooting scheme. The impact of pertinent constraints on the fluid velocity and temperature are examined through graphs. The coefficient of the skin friction and the rate of heat transfer are found numerically. Comparing of the present study with the earlier results is also presented. We observed that the coefficient of skin friction increases for higher values of Hartmann number.

In this paper, we propose a mathematical model to analyze the effect of a non-uniform heat source or sink on chemically reacting magnetohydrodynamic Casson fluid flowing across a slendering sheet. A chemically reacting fluid with multiple slips has various applications in controlling heating and mass transport phenomena such as biological reactants being converted to products, biomedical signal processing and image processing to establish a dynamic area of specialization in research aspects of biomedical engineering. For these reasons multiple slips (diffusion, thermal and momentum slips) are applied in modeling the heat and mass transfer process. The non-linear ordinary differential equations (ODEs) are solved using the Runge-Kutta-Fehlberg integration method. The characteristics of the velocity, temperature and concentration boundary layers are presented for different physical parameters such as space and temperature-dependent heat source or sink parameter, porosity parameter, chemical reaction parameter, velocity slip parameter, and temperature and concentration jump parameters. Moreover, the skin friction coefficient, and the local Nusselt and Sherwood numbers have been estimated for different parameters are discussed from an engineering point of view. We found a significant increase in thickness of the thermal and concentration boundary layer when the strength of the thermophoresis parameter is increased. In contrast, the thermal boundary layer increases with increasing the Brownian motion parameter while the reverse trend holds true for the concentration field.

In this investigation, we compared the effect of unsteady magneto hydrodynamic Carreau fluid in the interruption of graphene nanoparticles with heat generation over stretching and shrinking surface. The simulation is performed by combination of graphene nanoparticles into base water. Dispersion of graphene nanoparticles into base fluid finds several applications, such as biosensors, detection and purification of cancer cells, biocompatibility, and bio-imaging. The arising set of dimensional partial differential equations (PDEs) is converted into a set of non-dimensional ordinary differential equations (ODEs) by suitable similarity transformations, and then solved by the help of Runge-Kutta with shooting technique. The numerical results for non-dimensional velocity and temperature are displayed with graphs. We also analyzed the physical quantities (friction factor and local Nusselt number) for various values of non-dimensional physical parameters mathematically, and they are presented through tables. We also compared the present results with existing literature under some limited case. At the end of our analysis, we noted that radiation amplifies temperature profiles of both stretching and shrinking sheet cases. But, compared to shrinking sheet stretching sheet gains higher values. Heat generation parameter enriches temperature profiles in both stretching and shrinking sheets; compared to stretching sheet, shrinking sheet gains higher values.

A theoretical investigation is executed for exploring the flow, heat and mass transfer behavior of magnetohydrodynamic radiated ferrofluid flow caused by a cone in the existence of non-uniform heat source/sink. Dispersion of ferrous nanoparticles finds applications in heat exchanger system, recharge able batteries, chemical catalysts, metallurgy, conducting paints, magnetic recording media, drug delivery, nanofibers, textiles etc. in this study we have used the two types of temperatures namely water (At 10 °C and 50 °C), having less thermal conductivity as compared with magnetic nanoparticles (CoFe2O4). A simulation is performed by mixing of cobalt ferrous particles in base fluid water with different temperatures. Also, we presented dual solutions for flow over a cone with CoFe2O4+water at 10 °C and CoFe2O4+water at 50 °C. Governing equations are solved using Runge-Kutta with shooting method and compared with published literature. From the formulated model we found that the friction factor coefficient and rate of heat transfer is more in mixture of CoFe2O4+water at 10 °C case compare to CoFe2O4 +water at 50 °C case. Similarly the mass transfer rate is more in CoFe2O4 +water at 50 °C when compared to CoFe2O4 +water at 10 °C case. These results help us to conclude that depending on the industrial appliances we can use heating or cooling processes CoFe2O4 +water at 10 °C and CoFe2O4 +water at 10 °C fluids respectively.

Time-dependent, two-dimensional Sakiadis flow of quiescent fluid is considered. The flow is induced by stationary flat plate via uniform free-stream (Blasius flow). The variable conductivity and viscosity ratio parameters and non-linear chemical reaction are employed in the mathematical equations. Similarity variables are employed in the governing transport expressions to convert into the ordinary differential system. The transformed system is computed numerically by employing Runge-Kutta scheme via shooting criteria. Results of concentration, velocity and temperature distributions are studied through plots. Moreover, mass and heat transfer rates and friction factor have been discussed in detail. The constraint of chemical reaction slow down the friction-factors and heat transportation rates for the Sakiadis-Blasius flow situations and enhances the mass transportation rate in both cases. Rate of mass transportation is smaller in Sakiadis flow as comparative to Blasius flow. The present results of the heat transfer rate are matched with the literature and excellent agreement is noticed.

The performance of copper selenide and effectiveness of chemical catalytic reactors are dependent on an inclined magnetic field, the nature of the chemical reaction, introduction of space heat source, changes in both distributions of temperature and concentration of nanofluids. This report presents the significance of increasing radius of nanoparticles, energy flux due to the concentration gradient, and mass flux due to the temperature gradient in the dynamics of the fluid subject to inclined magnetic strength is presented. The non-dimensionalization and parameterization of the dimensional governing equation were obtained by introducing suitable similarity variables. Thereafter, the numerical solutions were obtained through shooting techniques together with 4th order Runge–Kutta Scheme and MATLAB in-built package. It was concluded that at all the levels of energy flux due to concentration gradient, reduction in the viscosity of water-based nanofluid due to a higher radius of copper nanoparticles causes an enhancement of the velocity. The emergence of both energy flux and mass flux due to gradients in concentration and temperature affect the distribution of temperature and concentration at the free stream.

The present paper is on 2D Casson magnetohydrodynamic flow through exponentially shrinking surface. The set of nonlinear PDEs are established into ODEs by using R-K scheme connected by shooting methodology. Some of the physical quantities are implemented to demonstrate the effects of exponential, Eckert number, Casson and radiation parameters on temperature and velocity distributions. The constraint of Eckert numbers decelerates the rate of heat transfer in both \( \Pr = 2 \) and \( \Pr = 9 \) cases.

The analysis of thin film flow of non-Newtonian fluid with MHD heat transfer in an unsteady stretching sheet in the suspension of variable thickness and thermal conductivity. The governing PDEs are reduced to nonlinear coupled ODEs by applying the appropriate similarity transformations. These coupled nonlinear ODEs subject to the suitable margin conditions and then solved numerically via RK with shooting method. The influences of various parameters on the stream and heat transfer behaviour of the problem are studied through tables and graphs. The friction factor and reduced Nusselt number are obtainable in tables.

A theoretical study on the effect of magnetohydrodynamic field on the classical Blasius and Sakiadis flows of heat transfer characteristics with variable conditions and variable properties are studied in this paper. Hydromagnetic flows and heat transfer in porous media have been considered extensively in recent years due to their occurrence in several engineering processes such as compact heat exchangers, metallurgy, casting, filtration of liquid metals, cooling of nuclear reactors and fusion control. The governing partial differential equations are converted into ordinary differential equations using the suitable similarity transformations. The transformed boundary layer equations like momentum and energy equations are solved numerically by using Runge–Kutta method. The effect of governing physical parameters over the velocity and temperature distributions is demonstrated graphically. Moreover, the local friction factor coefficient and rate of heat transfer in terms of Nusselt number are computed and discussed through tables. We validated the present solutions with already existing studies under limited cases. It is found that the thermal buoyancy parameter enhances the velocity depreciates the temperature field for both the Sakiadis and Blasius flow cases. The influence of Biot number increases skin friction coefficient and local Nusselt number for both Sakiadis and Blasius flow cases.The rate of heat transfer for temperature ratio parameter is high in Blasius flow case when compared with Sakiadis flow case.

In this present analysis we investigated the steady-state magnetohydrodynamic boundary layer flow of tangent hyperbolic fluid over an exponentially stretching surface in the presence of heat source and chemical reaction. The chemical reaction with combination of exponential surface has significance in many industrial and manufacturing systems. The partial nonlinear differential equations are transformed into ordinary differential equations by using the similarity conversion and the accomplished boundary layer ordinary differential equations are elucidated numerically by using Shooting technique. The effects of numerous non-dimensional governing factors on velocity, temperature and concentration profiles were depicted graphically and analyzed in detail. The numerically computed results of Skin friction factor, Nusselt and Sherwood numbers are presented in tabular form for suction and injection cases separately.Heat transfer rate at the surface increases with increasing values of power law of index and whereas it declines with the magnetic field, heat source and chemical reaction parameters. It observed that Biot number enhances the skin friction, Nusselt number and decrease the Sherwood number.Heat transfer rate and mass transfer rate increases and skin friction decreases with increasing Eckert number.

This study gives unsteady radiative magneto hydrodynamic Carreau-Casson fluids in suspension of graphene particle with Cattaneo-Christov model. A simulation is performed by mixing of graphene nanoparticles into the base water. The arising set of governing partial differential equations (PDEs) are transformed into set of ordinary differential equations (ODEs) using similarity transformations and then solved numerically using shooting technique with Runge-Kutta (RK) method. The computational results for non-dimensional temperature and velocity profiles are presented through graphs and tables. We also presented the numerical values of physical quantities (friction factor and local numbers) for various physical parameters. We compared the present results with existing literature under some limited case. At the end of this analysis we concluded that, the temperature profiles are higher in Casson fluid when compared to Carreau fluid. Similarly, the friction between the particles is more in Casson fluid compare to Carreau fluid, and heat transfer rate is high in Carreau fluid compared to Casson fluid. This help us to conclude that the cooling treatment by using Casson fluid is useful compared to Carreau fluid over unsteady sheet.

The impact of magnetic field, nanoparticles on the two-dimensional flow of a generalized non-Newtonian Carreau fluid flow over a permeable cylindrical surface with no slip condition has been investigated in the existence of heat generation/absorption, thermal stratification and thermal radiation. The mentioned sets of nonlinear governing differential equations are converted into nonlinear ordinary differential equations by using appropriate similarity transformations and solved numerically and fiend with shooting technique performed in this concern. It is initiate that the two various values of magnetic parameter $$(M)$$ , heat generation/absorption parameter $$(H)$$ and power law index $$(n)$$ influences velocity, temperature and nanoparticle volume fraction profiles. Finally, in this article the impact of various parameters on skin friction coefficient, Nusselt number and Sherwood numbers are examined with the support of the tabular form.

In this paper, we analyzed the mathematical model of the convective conditions on Casson fluid across a Paraboloid of Revolution (PR) in the presence of pertinent parameters like magnetic field parameter, Soret, and Dufour. We used R-K-Felhberg-integration scheme based shooting technique to solve the modified governing equations and discussed the appearance of curious parameters on the profiles velocity, temperature and concentration profiles in three cases n = 0, n = 0.5 and n = 1 with the aid of plots. We found that the profiles of velocity, temperature, and concentration are non-uniform for all these three cases. Also, we have taken the help of tables to explore the skin friction coefficient, local Nusselt and Sherwood numbers, which are useful for the purpose of engineering interest. It is noticed that the cross diffusion helps to control the thermal, diffusion and momentum boundary layers for all the three cases. The mixed convective conditions are useful in improving the heat and mass transport phenomena. We also validated the present methodology with already existing methodologies under some limited cases. Keywords: Convective conditions, Paraboloid of revolution, Soret and Dufour effects, Magnetic field parameter, Internal heat source parameter

This is a speculative investigation of the magnetohydrodynamic flow of Eyring-Powell liquid under suspension of nano-particles and dust. A simulation is executed by fraternization of Ferrous oxide (Fe3O4) and aluminum oxide (Al2O3) nanoparticles in Eyring-Powell dusty fluid. Dispersion of ferrous oxide (Fe3O4) and aluminum oxide (Al2O3) nano-particles in dusty fluid have applications in heat exchanger system, biocompatibility, biosensors, nuclear reactor heating process, detection and cancer treatment, in monitoring stem cells differentiation etc. Ferrous oxide (Fe3O4) and aluminum oxide (Al2O3) mixtures are substantially useful in optimizing the heat transport occurrences. Implementation of similarity variables leads to the systems of ordinary differential expressions. These nonlinear systems of ODEs are tackled with the use of Runge-Kutta-Fehlberg Scheme (RKFS). The analysis of dimensionless temperature and velocity fields is given via plots. The numerical benchmarks of friction-factors and heat transport rate are for different constraints are given and examined. Obtained results are matched with previously published material and noted to be satisfactory. This model expresses that the rate of heat transportation is more in aluminum oxide nanofluid compared to ferrous oxide nanofluid with existing of viscous variation parameter. The presence of thermal and momentum slips correspond the enhancement in local Nusselt number in case of ferrous oxide nanoparticles when compared to aluminum oxide nanoparticles.

The present framework addresses the buoyancy assisting heat and mass transfer of a slender body of revolution, positioned in a saturated porous medium considering thermal radiation and Buongiorno's model. In this analysis, the results are presented for the flow behavior over a paraboloid, a cone, and a cylinder. The governing flow equations are solved and validated numerically by adopting Runge-Kutta-Fehlberg method. The problem of this kind may encounter in automatic industries related to geophysical and geothermal applications where surface heat and mass transport takes place. Effects of involved controlling Physical quantities of the governing systems on heat and mass transfer rates as well as temperature and concentration profiles are reported with the help of tables and graphs. The obtained result indicates that temperature profiles are lesser in the revolution of flow over cone compared to paraboloid and cylinder type of body revolution. This result helps us to conclude the revolution of flow over cone is useful for cooling treatments.

In the present paper, the flow of an incompressible, electrically conducting dusty fluid over a stretching sheet is considered. The Cattaneo- Christov heat flux theory is employed to control the thermal boundary layer. The flow equations are transformed into nonlinear ordinary differential equations (NODEs) and which are solved with help of Runge-Kutta 4th order method. Flow equations are examined with respect to boundary conditions namely prescribed wall temperature (PWT) and prescribed heat flux (PHF) cases. In general PWT and PHF boundary conditions are very useful in the industrial as well as manufacturing up and down processes. Impact of the emerging parameters on the dimensionless velocity and temperature as well as friction coefficient and local Nusselt number are examined. We also validated my results with already available literature. It is found that the heat transfer rate of the flow in PWT case is higher than that of PHF case. These results can help us to conclude that for higher heating processes (Heating industries) PWT case and lesser heating processes (Cooling industries) PHF boundary condition is useful.

The present framework addresses Darcy-Forchheimer steady incompressible magneto hydrodynamic hyperbolic tangent fluid with deferment of dust particles over a stretching surface along with exponentially decaying heat source. To control the thermal boundary layer Convective conditions are considered. Appropriate transformations were utilized to convert partial differential equations (PDEs) into nonlinear ordinary differential equations (NODEs). To present numerical approximations Runge-Kutta Fehlberg integration is implemented. Computational results of the flow and energy transport are interpreted for both fluid and dust phase with the support of graph and table illustrations. It is found that non-uniform inertia coefficient of porous medium decreases velocity boundary layer thickness and enhances thermal boundary layer. Improvement in Weissenberg number improves the velocity boundary layer and declines the thermal boundary layer.

In this study, we numerically investigate the hydromagnetic three dimensional flow of a radiating Maxwell fluid over a stretching sheet embedded in a porous medium with heat source/sink, first ordered chemical reaction and Soret effect. The corresponding boundary layer equations are reduced into set of non-linear ordinary differential equations by means of similarity transformations. The resulting coupled non-linear equations are solved numerically by employing boundary value problem default solver in MATLAB bvp4c package. The obtained results are presented and discussed through graphs and tables. It is noticed that the Deborah number reduces the velocity fields and improves the temperature and concentration fields. Nomenclature

Numerical study of three dimensional Carreau liquid flow with heat and mass transport features over a variable thickness sheet filled with microorganisms is analyzed. We considered the non-uniform heat sink or source and multiple slip effects. The governing non-linear partially differential expressions are developed into ordinary differential systems by using variable transformations. These expressions are solved numerically by using Runge-Kutta fourth order method connected with shooting methodology. A Parametric study is implemented to demonstrate the effects of Hartmann number, Prandtl number, Weissenberg number, Peclet number, chemical reaction and heat sink/source parameters on liquid velocity, temperature and concentration profiles. The quantities of physical interest are described within the boundary layer. From this analysis, we found that the magnetic parameter decrease the local Sherwood and local Nusselt numbers for both and cases. The constraint of chemical reaction enhances the mass transfer rate and decelerates the density of motile mass transfer rate. The space dependent and temperature dependent heat source/sink suppress the local Nusselt number.

This research contemplates the flow and heat transport of MHD rheological Eyring–Powell fluid embedded with dust and graphene nanoparticles (GP) in an ethylene–glycol (EG) mixture in the presence of nonlinear convection, Cattaneo–Christov heat flux, and thermal radiation. Primarily existing PDEs (fluid and dust phase) are transferred to non-dimensional form by invoking similarity transformations then solved numerically through RKF-45 method. The graphene particles are significantly used in energy transmission in aerospace, power and propulsion generation etc. Through graphical illustrations, velocity and temperature profiles (fluid and dust phases) converse for various prominent parameters. The results of friction factor and heat transfer rate are presented and analyzed. Validation of the present result is made with the existing data. Results demonstrate that increasing nonlinear convection parameter has an inverse relationship with the Nusselt number and the velocity in the dust and fluid phases. This may happen due to the domination of unsteadiness in the flow.

The present work explores the analytical study of nanofluid flow above a stretching medium with the heat source and viscous dissipation. Additional radiative effects are also incorporated. The main physical problem is offered and changed into an arrangement of combined nonlinear differential equations with appropriate transformations. Optimal homotopy analysis method is used to attain the analytical solutions of the set of nonlinear differential equations. Important predictions of the flow phenomena are explored and deliberated by means of graphs and numerical tables. Moreover, the accurateness of the existing findings is verified by equating them with the previously available work.

This is a theoretical study of unsteady nonlinear convection on magnetohydrodynamic fluid in a suspension of dust and graphene nanoparticles. For boosting the heat transport phenomena we consider the Cattaneo-Christov heat flux and thermal radiation. Dispersal of graphene nanoparticles in dusty fluids finds applications in biocompatibility, bio-imaging, biosensors, detection and cancer treatment, in monitoring stem cells differentiation etc. Initially the simulation is performed by amalgamation of dust (micron size) and nanoparticles into base fluid. Primarily existing partial differential system (PDEs) is changed to ordinary differential system (ODEs) with the support of usual similarity transformations. Consequently, the highly nonlinear ODEs are solved numerically through Runge-Kutta and Shooting method. The computational results for Non-dimensional temperature and velocity profiles are offered through graphs (ϕ=0 and ϕ=0.05) cases. Additionally, the numerical values of friction factor and heat transfer rate are tabulated numerically for various physical parameters obtained. We also validated the current outcomes with previously available study and found to be extremely acceptable. From this study we conclude that in the presence of nanofluid heat transfer rate and temperature distribution is higher compared to micro fluid. Keywords: Nonlinear convection, Thermal radiation, Graphene nanoparticles, Dust particles, Cattaneo-Christov heat flux

The present study deals with steady incompressible magneto hydrodynamic hyperbolic tangent fluid flow induced by a convectively heated stretching surface with the suspension of dust particles, Darcy-Forchheimer, thermal radiation, viscous dissipation and Ohmic heating. Similarity transformations were used to convert partial differential equations (PDEs) to a system of nonlinear ordinary differential equations (NODEs) which are solved numerically by Runge-Kutta Fehlberg method. The effect of pertinent parameters on velocity and temperature profiles of both fluid and dust phase within the boundary layer has been studied by considering various values of controlling parameters. Additionally, the skin friction coefficient and reduced heat transfer coefficient have been examined for various values of the governing parameters. It is found that Hartmann number and Forchheimer parameter reduce friction factor and heat transfer rates.

In this study, we addressed the impact of magnetic field on fluid flow and heat transfer of an in compressible Carreau fluid over exponentially stretching sheet in addition with fluid and dust particle suspension. Thermal radiation and non-uniform heat source/sink were included to develop heat transport phenomena. Dusty fluids have various applications such as processing of material, nuclear heat treatment, cooling process, treatment of waste water etc. The relevant governing equations are converted into ordinary differential equation using similarity transformation the transformed ordinary differential equations are then solved numerically by shooting technique along with Runge-Kutta method The effect of certain parameters on the dimensionless velocity and temperature are presented graphically. The physical quantities of the flow such as the friction factor and Local Nusselt number were calculated. It was found from the study that the velocity slip parameter increases the temperature profiles.