Runge Kutta Fehlberg Technique Research Articles

Gyrotactic bio-magneto-convective combined impacts on magnetorheological hybrid nanoflow: Microbial life on surfaces’ application

Magnetorheological fluid with self-driven ubiquitous motile organisms was affected by various physical factors such as hydrodynamics, dispersion, and distribution of organisms. These fascinating properties attracted us to work on this paper, a Casson hybrid nanoflow model exploring the dynamic behavior of microorganisms along spongy extensible surfaces taking Ohmic heating and chemical reaction into account. Further, the heat transfer phenomena are analyzed in the presence of thermal radiation and space and time-dependent energy sources/sinks. The nanoflow has its base methanol and a mixture of iron and aluminum oxides is suspended. Flow obeying mathematical model is solved numerically using the Runge–Kutta–Fehlberg technique along with the shooting method after imposing self-similarity transformation. Results of various pertinent parameters on the flow characteristic quantities and engineering quantities are portrayed and presented through graphs and tables. Iron oxide methanol nanofluid shows supremacy over hybrid nanoflows as speed is considered, whilst the opposite phenomenon is observed in the case of energy. Hybrid nanoflow makes the surface friction-free. Microorganism concentration difference improvement also helps in the intensification of motile organism growth.

A numerical study on MHD Maxwell fluid with nanoparticles over a stretching surface: Impacts of thermal radiation, convective boundary condition and induced magnetic field

The current study investigates the influence of induced magnetic field and convective boundary condition on the behavior of a magnetohydrodynamic (MHD) Maxwell fluid including nanoparticles flowing through a stretching sheet. Furthermore, the analysis considers the existence of radiation. Numerical solutions to the basic governing equations are achieved using the Runge–Kutta–Fehlberg technique. The impact of various physical parameters on concentration, velocity, and temperature profiles is deployed through graphs. The primary conclusions of this study are that increasing the Magnetic parameter decreases the Induced Magnetic field while increasing the Deborah parameter improves the velocity profile. Temperature distribution is growing due to increased estimates of the Radiation, Biot number, and Deborah number values. When the Lewis number values are raised, the concentration profiles get smaller but there is an opposite scenario deployed as the Thermophoresis parameter upsurges. The findings of this study are consistent with those that have been previously reported. The phenomenon of flow induced by a stretched surface finds several applications in various industrial and technological domains, including copper wire production, polymer extrusion, article manufacturing, and fiber synthesis. Induced magnetic fields are extremely prominent in many areas of science and technology, including electromagnetism, magnetic materials, and magnetic resonance imaging (MRI).

Influence of Soret–Dufour on a magneto‐micropolar fluid over a porous stretching surface with a nonuniform heat source/sink

AbstractThe goal of the current investigation is to ascertain the influence of Soret and Dufour on a magneto‐micropolar fluid past a permeable stretching surface with a nonuniform heat source/sink. Moreover, the investigation considers the influence of melting heat transfer, as well as the implications of velocity, thermal, and solutal slips. Numerical solutions to the basic governing equations are achieved using the Runge–Kutta–Fehlberg technique. The analysis of our results reveals several important findings. Enhancing the material parameter leads to an upsurge in the velocity profile, whereas the opposite trend exists for the boosting estimations of the magnetic parameter. Enhancing the Soret parameter leads to an improvement in the concentration profile while improving the Dufour factor results in an enhancement in the temperature distribution. These findings are critical for understanding the heat transfer over a stretching sheet, subject to different models and they encompass aspects such as temperature distribution, heat transfer rates, boundary layer characteristics, velocity profiles, and the potential for implementing heat transfer enhancement techniques. The findings of this current investigation are consistent with those that have been previously reported. Fluid dynamics, materials processing, cooling systems, and thermal systems are just a few of the real‐life domains that can benefit from understanding magneto‐micropolar fluid flow over a porous stretching surface with a nonuniform heat source–sink and incorporating Soret and Dufour effects.

Thermal radiative and Hall current effects on magneto-natural convective flow of dusty fluid: Numerical Runge–Kutta–Fehlberg technique

This research article intends to examine the consequences of Hall impact on the natural convection of magnetohydrodynamic (MHD) fluid flowing and the thermal characteristics of the fluid flow which containing dusty particles via a vertical stretchable sheet. Slip velocity and convective boundary conditions are taken into consideration in this research article. The flow is mathematically described by partial differential equations. The governing equations are renovated into a group of nonlinear differential equations with the use of similarity conversations, which are then resolved numerically by applying Runge–Kutta–Fehlberg. The behavior of these physical factors on the rapidity and temperature profiles of the dust and fluid phases are exposed graphically in this article. Furthermore, frictional force and heat transfer are also examined and discussed. According to the investigation, the existence of suspended particles in the fluid lead to increment in momentum and thermal boundary layer thickness. The temperature of the fluid seems can be controlled by the significant parameter, such as it seem to be increase due to the magnetic, radiation, and Hall parameters, while it is seen to have decreased due to the slip thermal parameter. While the radiation accelerates up the fluid velocity, the magnetic parameter is excellent at slowing it down. The fluid phase’s temperature and velocity profiles are almost usually higher than those of the dust phase. The outcomes contribute to a better understanding of the two-phase theory’s fluid flow phenomenon.

Modelling heat-mass transport for MHD Eyring-Powell hybrid nanofluid over an expanding surface laden by autocatalytic chemical reaction and nanoparticles diffusion

Prior hybrid nanofluid research is restricted to single or two-phase models, which were dominated by homogeneous nanofluid models. These models adhere a serious flaw with the heat-mass transfer indices. To handle this problem, we develop an entirely novel hybrid nanofluid models based on the fundamental slip mechanisms of nanofluids induced by Brownian and thermophoretic diffusion as well as the chemical reaction of binary species. The laminar flow of MHD rheological model with suspended nanoparticles is striking at the stagnation region laden over a stretching surface. In order to reach the real world scenario, the expression of radiative heat, non-uniform heat source and frictional heating are all included. The coupled Buongiorno's model and Tiwari-Das mixtures models incorporate the effect of particles diffusion and volumetric friction, respectively. It is hypothesized that first order kinetics can describe homogeneous events in the immediate proximity, while isothermal cubic autocatalysis kinetics frequently describe heterogeneous reactions occur on the sheet surface. The restructured mathematical equations are solved with a proficient Runge Kutta Fehlberg technique together with the shooting procedure. According to the outcomes, the presence of higher magnetic field diminished the flow and velocity gradient. The effect of elastic parameters is diminishing frictional coefficient. The effect of first and second Brownian (Nb1, Nb2) and thermophoretic parameters (Nt1, Nt2) are similar for temperature. The mass-heat transfer rate is significant with higher estimations of diffusivity ratio Nbt. The momentum rate of diffusion outweighs the species diffusion rate as the Schmidt number increases, resulting in a decrease in concentration values. Higher intensities of Lorentz's force results in denser streamlines. The streamlines split into an opposite direction because of stagnation point.

Thermal analysis of fully wetted porous longitudinal fin of parabolic profile with variable thermal conductivity and convection–radiation

AbstractThe thermal performance of a fully wetted porous fin with concave parabolic cross section is explored in this study for variable thermal conductivity. We have considered convective and radiative heat transfer with variations of temperature‐dependent surface emissivity, internal heat generation, heat transfer coefficient, and magnetic field effect on the thermal profile of the fin. The heat transfer equation is derived from Darcy's model. Nondimensional governing ordinary differential equation is attempted by Runge–Kutta–Fehlberg technique. Influences of associated parameters on the cooling performance of fin are explored through graphs. From this study, it was found that the rate of heat transfer from the fin whose thermal conductivity varies linearly with temperature is twice than that of the fin with exponential dependency on thermal conductivity in the presence of magnetic effect.

Numerical approach for the modified heat transport process under improved Darcian and nonlinear mixed convective effects

Analysis regarding heat transport subjected to non-Darcian media has increased in science and technological utilization. Typical applications comprise grain storage, nuclear waste discarding, production of crude oil, pollution in groundwater, porous insulation, and several others. This motivates us to demonstrate the heat transport phenomenon in the viscous fluid's flow deformed by a linearly stretchable surface submerged in Darcy-Forchheimer porous medium. Properties of heat transfer are elaborated with nonlinear mixed convection and Cattaneo-Christov heat model. Slip phenomenon is implemented at the wall surface. The governing equations are achieved in dimensionless form through appropriate transformations. Numerical solutions are constructed through shooting scheme along with 4–5 order Runge-Kutta Fehlberg technique. Graphical behaviors of velocity field and liquid temperature are constructed through various physical parameters. Results witnessed that decrement occurs in velocity field for dominant behavior of velocity slip parameter while it enhances for dominant nonlinear mixed convection parameter.

Computational assessment of MHD Carreau tri-hybrid nano-liquid flow along an elongating surface with entropy generation : A comparative study

This research looks at the incompressible, viscous, steady-state laminar stagnation-point-flow of a Carreau ternary-hybrid nano-liquid towards a convectively heated expanding surface, taking into account first-order slippage and Ohmic dissipation. Viscous dissipation and the effects of Lorentz forces are also considered. It is discussed how thermal radiation and heat absorption/generation contribute to the heat transport process. Minimizing entropy during the transport of a Carreau ternary hybrid nano-liquid has also been studied. In addition, convective circumstances are used conceptually in the numerical solution of the current model. tiny-particles of silver (Ag), molybdenum disulfide (MoS2), and multi-wall carbon nanotubes (MWCNT) are analyzed in this work’s flow study. As a base fluid, carboxymethyl cellulose (CMC-water) is used. With the use of appropriate similarity trans-formations, the standard model equations are transformed into dimensionless form. Finding solutions for momentum and heat fields using the Runge–Kutta-Fehlberg technique and a shooting strategy. The figures depict a wide range of characteristics, including fluid flow velocity, temperature, skin friction, the Nusselt number, entropy minimization, and the Bejan number. Key results from the present model include the fact that the velocity curve flattens down as Weissenberg number increases.

Thermal instability in electrically conducting nanoliquid filled in Hele-Shaw cell under 3-types of rotational-speed modulation with impact of through-flow and magnetic-field

Due to the great significance of rotation-speed modulation in various areas of science and technology as well as in day-to-day life, our aim, in this study, is to perform linear/nonlinear studies to examine the impacts of sinusoidal like sine waveform and non-sinusoidal like square and triangular waveforms of rotational-speed modulation in nanoliquid (electrically-conducting) filled within a Hele-Shaw cell with through-flow and magnetic-field. The applied rotational-speed modulation has periodic components that are both constant and time-dependent and changes sinusoidally over time. The linear analysis is done using Galerkin’s approach while the nonlinear analysis is done using truncated Fourier series method. The results of Mathematica’s built-in NDSolve tool are validated using the Runge–Kutta-Fehlberg technique. Hele-Shaw number, Taylor-number, nanoliquid magnetic number and frequency of rotational-speed modulation have a stabilizing impact on the system, whereas through-flow, magnetic Prandtl-number, nanoliquid Prandtl-number and amplitude of rotational-speed modulation have destabilizing impact on the system. Additionally, it has been found that rotational-speed modulation can be utilised to efficiently increase or decrease heat and mass movement. Square-waveform is most efficient for both heat and mass transport in compression of sinusoidal and triangular waveforms.

MHD thermo-radiative heat transfer characteristics of carbon nanotubes based nanofluid over a convective expanding sheet in a porous medium with variable thermal conductivity

ABSTRACT The impacts of thermal radiation (non-linear) and thermal conductivity (variable) on magnetohydrodynamic (MHD) heat transfer over a flat stretchable surface by taking single-walled carbon nanotubes (SWCNTs) and multiple-walled carbon nanotubes (MWCNTs) inside the base liquid (water) embedded in a porous media have been analysed. We have utilized an appropriate set of similarity transformations to rehabilitate the governing partial differential equations into a non-linear ordinary differential equations system. These equations are then solved using a Runge–Kutta–Fehlberg technique with a shooting algorithm. The influences of several critical physical parameters on the velocity (V) and temperature (T) fields, coefficient of skin-friction (C f ), and local Nusselt number (Nu x ) have been examined in detail by plotting several graphs and tables for both SWCNTs and MWCNTs. The acquired outcomes show that SWCNTs in water-based nanofluid has remarkable effects on skin friction and the heat transfer rate when contrasted with MWCNTs. Likewise, the variable thermal conductivity enhancement diminishes the Nu x , whereas the inverse impacts the expanding T ratio parameter.

Impact of Cattaneo–Christov double diffusion theory and Arrhenius pre-exponential factor law on flow of radiative Reiner–Rivlin nanofluid over rotating disk

Reiner–Rivlin nanofluid flow due to rotating disk has significance in manufacturing of computer disks, pumping of liquid metals, spin coating, centrifugal machinery, turbo-machinery, crystal growth and rotational viscometer. In light of such real and relevant industrial applications, this study deals with the numerical investigation of unsteady rotationally symmetric flow of Reiner–Rivlin nanofluid over a stretchable rotating disk. The purpose of this investigation is to explore heat transfer characteristics of Reiner–Rivlin nanofluid subject to radial stretched surface implementable in several thermal systems. For facilitation of heat transport in complex thermal systems, mathematical models, such as Cattaneo–Christov, Buongiorno and nonlinear thermal radiation models, are assumed to be introduced. Runge–Kutta–Fehlberg technique along with shooting method is used for numerical computation of the transformed equations. It is captivating that radial and circumferential velocities decelerate with rise in Reiner–Rivlin and stretching strength parameters, respectively. Amplified thermal relaxation and Reiner–Rivlin parameters led to diminution of wall temperature gradient. Nanoparticle concentration profiles exhibit opposite behavior in response to escalation of activation energy and reaction rate parameters.

Bio-Marangoni convection of Maxwell nanofluid over an inclined plate in a stratified Darcy–Forchheimer porous medium

The work highlights the bioconvective transport phenomena of gyrotactic microorganisms in Maxwell nanofluid passing over an inclined plate enclosed in Darcy–Forchheimer porous medium. The medium is stratified in terms of heat, mass, and motile density gradient and under the influence of an inclined magnetic field. The physical situation is well described by means of governing partial differential equations. Under some capable similarity transformations, the governing equations are remodeled into the system of ordinary differential equations and worked out numerically using Runge–Kutta–Fehlberg technique. The consequences of pertinent flow properties on temperature, nanoparticle volume fraction, and motile microorganisms density distributions are illustrated graphically. Also, the main curiosity is focused to realize the changes in important quantities of engineering interest owing to the alteration in crucial parameters. It is revealed that the dominancy of the Marangoni ratio and Maxwell fluid parameter on motile microorganism gradient accelerates with a higher magnetic effect. Further, the thermophoretic effect plays a remarkable role in the reduction of motile microorganism transport with a smaller Brownian motion effect.

Three Dimensional Modelling of Magnetohydrodynamic Bio-Convective Casson Nanofluid Flow with Buoyancy Effects Over Exponential Stretching Sheet Along with Heat Source & Gyrotactic Micro-Organisms

In current analysis, influence of buoyancy forces in MHD bioconvective non-Newtonian nanofluids over three dimensional exponential sheet has been studied numerically. Additionally, impact of heat source parameter along with convective conditions has been incorporated. Moreover, nanofluid flow with gyrotactic microorganisms has been analyzed in presence of chemical reaction. Initially similarity variables are used for the conversion of partial differential equations into highly non-linear differential equations. Thusly, non-linear behavior of equations makes typical solutions which are not solved analytically. So, computational MATLAB software is used to calculate results and graphs by following shooting algorithm with Runge Kutta Fehlberg technique using ODE45 solver. Present modeling investigates the influence of crucial fluid parameters especially; magnetic M (0.1 ≤ M ≤ 0.9), Casson parameter β (0.09 ≤ β ≤ 0.13), permeability parameter Bθ, Brownian motion Nb (0.5 ≤ Nb ≤ 5.0), thermophoresis Nt (0.2 ≤ Nt ≤ 2.0), thermal Biot number Bt (0.3 ≤ Bt ≤ 0.7), heat generation parameter Q (0.1 ≤ Q ≤ 0.5), Prandtl number Pr (0.1 ≤ Pr ≤ 0.9), concentration Biot number Bc (0.1 ≤ Bc ≤ 0.9), Lewis number Le (1 ≤ Le ≤ 5), chemical reaction parameter Ch (0.1 ≤ Ch ≤ 0.9), bioconvective Lewis number Lb (0.1 ≤ Lb ≤ 2), Peclet number Pe (0.1 ≤ Pe ≤ 5), gyrotactic Biot number Bn (0.1 ≤ Bn ≤ 0.5), stretching ratio parameter c (0.1 ≤ c ≤ 0.3) and microorganism concentration difference parameter Ω (0.1 ≤ Ω ≤ 5.0). Final results are compared for Prandtl number and stretching ratio parameter along with residual errors. It is inferred that motile concentration declines for larger bioconvective Lewis number whereas rises for motile gyrotactic microorganism Biot number.

Chemically radiative and mixed convection solute transfer in boundary‐layer flow of Jeffrey nanofluid along an inclined stretching cylinder with joule heating and double stratification impacts

AbstractAn endeavor is consummate to study the steady mixed convection solute transfer in two‐dimensional viscous fluid. A joule heating incompressible flow along an inclined stretching cylinder and double stratification impact on Jeffrey Nanofluid is scrutinized. Acquired nanomaterial pattern comprises the phenomena of thermophoresis and Brownian motion. By applying rule of approximation transformation, the nonlinear PDEs converted into ODEs. Shooting via Newton Raphson technique is used to solve the ODE'S with boundary conditions into initial conditions, also applying Runge–Kutta–Fehlberg technique of fourth order for numerical purpose. Computation for skin friction, () for Nusselt, and for Sherwood number are fetched in table format by using mathematical software Matlab. The results reveal that M (0, 0.3, 0.6, 0.9), γ (0.1, 0.3, 0.7, 1.3), and α will increase upto 7.51, 8.55, and 10.27%, respectively. The behavior of on and falling off upto 11.20 and 28.38%, respectively with the increase of , but in shows opposite reaction on Sherwood number. It is culminated that for velocity augments with the value of Deborah number of four different sizes (0.8, 1.1, 1.4, and 1.7) upto 14.28, 16.6, and 20% while situation is reversed in case of magnetic factor (0, 0.3, 0.6, 0.9) upto 20, 28.12, and 39.13%. Effect of for temperature enhanced with the value of Brownian motion on (0.0, 0.2, 0.4, and 0.6), about 16.66, 20, and 25%, respectively. The maximum effect percentage with radiation factor on is 39 and minimum percentage is 25.71. Eckert number on ( − 4, 0, 4, 8) and thermophoresis on diminished with pile‐up Prandtl number on (1.2, 1.5, 1.8, and 2.1). Impact of for concentration and local Sherwood number showed reverse situation on Schmidt number.

A numerical study on the flow of water-based ternary hybrid nanomaterials on a stretchable curved sheet.

Nanomaterials are quite promising in electronic cooling systems, heat exchangers, engine lubricants, brake liquids, shock absorbers, radiators, etc. Therefore, the study of heat transfer characteristics on the flow of trihybrid nanofluids on an exponentially stretched curved surface is developed. Purpose: In this study, trihybrid nanofluid is taken into consideration, which is composed of Fe3O4, Ag and Cu as nanoparticles and water as the basefluid. Heat generation and magnetic field impacts are addressed. Based on these assumptions, the governing partial differential equations were reduced to a favorable set of ordinary differential equations using adequate transformations. Formulation: The highly nonlinear coupled system of equations was numerically solved using the shooting method with the Runge-Kutta-Fehlberg technique. Findings: Trihybrid nanofluids improve the thermal performance of fluid when compared with other fluids such as hybrid nanofluids, nanofluids, and basefluids. The trihybrid nanofluid is efficient in heat transfer phenomenon and has a significant impact on the overall performance of a system, including cooling systems, heat exchangers, electronics, and many industrial processes. Graphical representation for the physical variables of the fluid velocity and temperature is discussed. The local Nusselt number and skin friction coefficient are computed and analyzed. A magnetic field decreases the velocity but escalates the temperature. The Nusselt number decreases for larger solid volume fractions. Novelty: The Tiwari and Das model for hybrid nanofluid extended for trihybrid nanoparticles has not been investigated previously. Heat transfer examination on the flow of trihybrid nanomaterials on exponentially curved stretching sheets considering magnetism force and heat generation consequence has not yet been studied.

Numerical investigation of MHD convective free stream nanofluid flow influenced by radiation and chemical reaction over stretching cylinder

AbstractChemically reacting magnetohydrodynamic radiative flow of convective free stream nanofluid through a stretching cylinder using Buongiorno's model is discussed. The behavior of Brownian motion and thermophoresis is also appropriate. By adopting the similarity transformation, the partial differential equation is diminished into a first‐order ordinary differential equation (ODE). Since transformed equations are highly nonlinear these ODEs are solved by using mathematical simulation. The shooting procedure has been adopted to resolve converted equations along the attendant Runge–Kutta–Fehlberg technique. The reason behind the present work is to research the effects of different parameters of fluid, namely, magnetic parameter, free stream velocity, Brownian motion, thermophoresis, chemical reaction, heat radiation, Lewis number on nanoparticle concentration, temperature, and velocity distribution. The impact of significantly participating parameters on velocity, concentration, and temperature distribution is distinguished with appropriate physical significance. The convergence of solutions for temperature, velocity, and concentration profiles is studied carefully. The measured challenges of nanofluids are scale‐up capacity, increase in nanofluid viscosity, nanoparticle dispersion, and nanofluid cost. It is observed that nanoparticle temperature rises for more value of Brownian motion parameter while it declines for higher Lewis number. The current study in the cylindrical region is related to novel free stream flow in the presence of chemical reactions along with convective conditions which find applications in electronic systems like microprocessors and in a wide variety of industries and in the field of biotechnology. The current research helps control the transport phenomena, helping production companies to find the quality of the desired product.

Graphene-gold/PDMS Maxwell hybrid nanofluidic flow in a squeezed channel with linear and irregular radiations

The hydrothermal performance of a hybrid nanofluid made of graphene, gold/polydimethylsiloxane between two squeezing plates is discussed in this study. In the investigation of thermal transport of the flow, both linear and nonlinear thermal radiation's effects are taken into account. Bejan numbers are used to determine the system's energy efficiency. Viscous and thermal radiation effects on Maxwell hybrid nanofluid flow in a squeezing channel are incorporated to explore the outcomes. By applying similarity transformations, the set of ordinary differential equations in this case is transformed into the governing equations. With the use of the Runge–Kutta–Fehlberg technique converted system is solved numerically. Skin friction and heat transfer rates under the influence of certain oriented parameters are numerically analysed and presented in tabular form. The impacts of different factors on the temperature and velocity profiles are illustrated graphically and briefly described. Important findings include that the Bejan number raises as the radiation parameter Rd rises, but that it falls with respect to the Eckert number effect. Additionally, the skin friction values and Nusselt number in the hybrid nanofluid case are larger than in the nanofluid case. Furthermore, it has been found that the Deborah number-described stress relaxation phenomenon causes the flow field and thermal energy transfer to be less efficient when fluids are moving. There is surprisingly little research on the hybrid fluid integrated into their issues.

The performance evolution of hybrid nanofluid flow over a rotating disk using Cattaneo–Christov double diffusion and Yamada–Ota model

The article addresses the problem related to Darcy–Forchheimer flow past a rotating disk with radial stretching and shrinking. The effects of viscous dissipation, convective heating and heating due to porous medium are considered; the heat and mass transfer expressions are modeled according to Cattaneo–Christov double diffusion theory. Additionally, a carbon nanotubes-based hybrid nanofluid is taken as working fluid. The mathematical model for existing problem is quite complex which is reduced to a simpler form (non-dimensional) by utilizing Von-Kármán similarity transformation. Then, these non-dimensional expressions are solved using Runge–Kutta–Fehlberg technique via shooting method. The impact of the sundry variables is illustrated via both graphs and tabular data, and the same is explained in detail. The result elucidates that the thermal boundary layer width of stretching rotating and shrinking rotating disks increase by rising the porosity parameter, Eckert number and thermal relaxation parameter. Moreover, a hike in the value of thermal relaxation parameter from 0.1 to 0.4 causes a reduction of 22.36% in heat transfer rate in stretching rotation, whereas a boost in the value of solutal relaxation parameter from 0.2 to 0.8 is responsible for 4.24% and 0.54% increment in mass transfer rate in stretching rotation and shrinking rotation, respectively.

Carboxymethyl Cellulose Based Second Grade Nanofluid around a Horizontal Circular Cylinder

Modern challenges include improving heat transmission in a range of industries, including electronics, heat exchangers, bio and chemical reactors, etc. Innovative heat transfer fluids like nanofluids have the potential to increase energy transport effectively. This gain is attained as a result of improved effective thermal conductivity and modified fluid flow dynamics. Therefore, the topic of this paper is improving heat transmission using nanofluids. The objective is to deal with the second grade fluid model passing through a horizontal circular cylinder with mixed convection and suspended nanoparticles. The respective nanoparticles and based fluid of Copper (Cu) and carboxymethyl cellulose (CMC-water) are considered. Both non-dimensional and non-similarity transformation variables are utilized to convert the governing equations to a system of partial differential equations (PDEs). Reduction to ordinary differential equations (ODEs) is attained from the resulting PDEs at the lower stagnation area and then tackled via the Runge-Kutta Fehlberg technique (RKF45) in the Maple software. Graphs are used to illustrate the detailed description of the results of dimensionless parameters like the Biot number, mixed convection, and the second grade parameter. Results show that the fluid slows down while the temperature increases as the value of second grade parameter rises.

Homogeneous-heterogeneous chemical reactions and effectiveness of thermo-sloutal time’s relaxation concept in Carreau fluid flow

This study includes Carreau fluid flow at a stagnation point under MHD influence. The Christov-Cattaneo double diffusion formula is to manipulate the heat and mass flux rate with the generation of heat. The flow is manifested by the stretched sheet. The heterogeneous-homogeneous reactions are carried out. The admissible transformations are invoked to alter the flow narrating coupled partial differential framework into a coupled ordinary differential framework. These equations are sorted out in a numerical framework with the aid of the Runge–Kutta Fehlberg technique supported by the shooting scheme. The parameter of thermal and concentration relaxation diminishes fluid temperature and concentration, respectively. The numerical results for the drag force are examined.