Narrow Results

This research discusses the investigation of heat and mass transfer in a magnetohydrodynamic (MHD) Casson fluid (CF) flow over an exponentially porous stretching sheet. The analysis takes into account the existence of thermal radiation, viscous dissipation, chemical reactions, and the influence of velocity and thermal slips. A recognized Casson model is taken into account in order to distinguish the characteristics of Casson fluid from those of Newtonian fluids. Using the geometry under consideration, the current physical problem is modeled. Appropriate similarity conversions are implemented to reduce the resultant set of coupled nonlinear PDEs to a set of nonlinear ODEs. By implementing the Keller-Box technique, numerical solutions to these reduced non-dimensional governing flow field equations are obtained. Tables and diagrams are utilized to illustrate the physical behavior of various control parameters. The temperature profile is enhanced and velocity profile diminished as the CF parameter value increased, according to this study. An increase in the velocity slip factor resulted in a diminution in the velocity field, while a gain in the thermal and concentration contours. With growing amounts of the chemical reaction factor, the concentration profile exhibited a decline. Indeed, the similar outcomes elucidated in this paper exhibit a remarkable correspondence with solutions that have been previously documented in the academic literature. This research may be motivated by a desire to improve the comprehension of fluid flow in different engineering and environmental situations, where these conditions are common, such as geothermal energy extraction, thermal management, chemical processing industries, and environmental control technologies.

This study investigates the impact of variable permeability as well as chemical reactions on the oscillatory free convective flow that passes parallel porous flat plates with fluctuating temperature and concentration in the presence of a magnetic field. A vertical channel is assumed to be rotating at an angular velocity $\mathrm{\Omega }$. Periodic free stream velocity causes oscillations in one plate, while periodic suction velocity causes oscillations in the other plate. Complex variable notations are used to solve the governing equations. The perturbation technique is used to derive analytical expressions for the temperature, concentration, and velocity fields. In this study, various parameters were investigated in relation to mean velocity, mean temperature, mean concentration, amplitude, and phase difference. The study also examines the impact on secondary velocity, primary velocity, temperature, concentration, and heat transfer rate during transients. The outcomes are presented graphically for the physical parameters of the problem. The findings contribute to optimizing systems and improving efficiency in heat transfer, fluid dynamics, and environmental remediation.

The principal aim of the work is to investigate the flow of Williamson fluid on a power law extended lubrication surface with partial sliding under the Magnetohydrodynamic issue on account of alterable thickness lubricated film. Accounting for the influence of microbes, assumption of activation energy, Cattaneo–Christov mass and heat flux took place in the equations of concentration and also in the temperature. Our paper will provide remarkable help to the medical and industrial areas because of the inspection under electro osmosis along with MHD effects in the Williamson fluid flow. With the influence of boundary conditions, built-in equations were studied. The BVP4C technique adopted to solve numerically the transformed ordinary differential equations from nonlinear partial differential equations using innumerable variables. The significant outlines of microorganisms, temperature, concentration and velocity were discussed. Decreases in the velocity distribution were observed with magnetic parameters. Also, an increase in friction coefficient was noticed as 3.1069% for rising magnetic field strength.

Swirling flows are important in rheological devices, spin coatings and lubrication, so we set out to investigate what makes chemically reactive non-Newtonian spinning flows across a disk with a radially applied magnetic field so interesting. Nanofluids are thermally enhanced working fluids with many interesting physical properties. This study takes its inspiration from rotating disk oxidations used in the medical techno industry and builds a mathematical model of a continuous convective von Kármán swirling flow including Jeffrey, magnetic, Joule/ohmic and chemical reactions. The wall anisotropy slips and the concentration-induced blowing effects are included. By using the bvp4c approach, the transformed boundary conditions (BCs) are addressed. Graphical representations of the effects of involved parameters on the density distribution of motile microorganisms, concentration, temperature and dimensionless velocity components are shown. Supporting evidence from prior research is included. Novel bioreactors, membrane oxygenators, bio-chromatography and food processing should take note of the study’s findings. As Jeffrey’s parameter upsurges, there is a decrease in radial velocity. As the Jeffrey parameter increases, there is a decrease in the circumferential velocity. Radial flow is significantly enhanced near the wall as the radial slip parameter ($\delta u)$ increases. As the Eckert number grows, the quantity of temperature increases. Concentration distribution closer to the disk to grow as Le increases. The concentration and diffusivity of microorganisms drop as the number of motile microorganisms thickens.

Industry and space technology have significant issues managing heat energy and controlling mass dispersion. The purpose of this study is to develop motion caused by boundary layer thickness sheets that are increasingly being used in various engineering fields (civil engineering, mechanical, aeronautical, maritime processes and constructions). The activation energy is a critical factor in chemical reactions due to the existence of many applications in gas-cooled reactors, nuclear thermal rockets and liquid-fluoride reactors. This study presents the numerical analysis of activation energy on three dimensional (3D) nanofluid (NFs) motion via Stretching Surface (SS) with nonlinear thermal radiation effect. This is in contrast to the conventional slip condition, convective condition applied at surface. The governing basic equations are translated into nonlinear ODEs by suitable similarity transformations. The relevant boundary value problem was explored for a numerical solution for applying the MATLAB based on Runge–Kutta–Fehlberg (RKF) scheme via shooting technique. The major outcomes of current work have more concentration ($\varphi ({\eta }_1)$) and Mass Transfer Rate (${\mathrm{\text{Sh}}}_x{\mathrm{\text{Re}}}_x^{-1∕2}$) for various numerical values of Activation Energy ($E_A$). The present solutions determine very good correlation with the previously studied ones in a special case as predicted in the tables.

Pseudoplastic fluids are non-Newtonian fluids with intriguing uses in current research and industry. Among many other extant models, the Sutterby fluid model is an essential viscoelastic fluid model that demonstrates shear thinning and shear thickening properties in high polymer aqueous solutions by manifesting viscous and elastic aspects during deformation. The magneto hydrodynamic effects of Sutterby nanofluid on porous elastic surfaces in the presence of chemical processes are examined in this theoretical study. By using similarity transformation, the mathematical model of a governed problem is converted into a collection of differential equations. A shooting strategy is used to solve these nonlinear coupled ordinary differential equations. The velocities, temperatures, and chemical species concentrations of fluids are graphically shown. Physical quantities of importance, such as local heat and mass flow, are visually represented using bar charts. Heat and mass transport, as well as chemical species concentration, decrease with Hartman number in both suction and injection. Chemical concentration of governed fluid rises for homogeneous reactions but drops for heterogeneous reactions. Temperature and concentration of fluid increases for thermophoresis parameter but decreases for Brownian motion parameter, also the effects of injection are much stronger and higher than suction.

This paper addresses the problem of steady two-dimensional magnetohydrodynamics (MHD) migration of dusty fluid across a stretching sheet with the inclusion of Darcy–Forchheimer porosity and Brownian dispersion. The most significant aspect of the ongoing problem is elaborating the entire context with dusty and fluid phases. The governing partial differential equations (PDEs) are transmuted into non-dimensional ordinary differential equations (ODEs) by implementing similarity transformations. The MATLAB script has used the bvp4c strategy to accumulate a visualization of the computational observations. Also, this study illustrates an assessment of the effects of significant parameters on non-Newtonian fluids and fluids with a dusty phase. It is observed that the thermal boundary layer is enhanced with the increasing strength of the mass concentration of dusty granules (${\mathrm{\Gamma }}_{\nu }$) and Eckert number (Ec) for both scenarios, while in the case of fluid interaction temperature parameter (${\beta }_t$) the scenario is opposite for dusty and fluid phases. Moreover, the heat transfer rate intensifies with the increasing effect of thermal radiation, and magnetic field. The significant variations in the various physical quantities are represented through tabular and graphs.

This work objective focuses on studying the combined influences of variable thermal conductivity, chemical reaction, and magnetohydrodynamics (MHD) on the flow of a tangent hyperbolic nanofluid flow over an exponentially stretching surface, considering a first-order velocity slip condition. Additionally, thermophoresis and Brownian motion impacts are taken into account. The phenomena of heat transfer are analyzed considering several factors such as thermal radiation, Joule heating and nonlinear heat source. On the other hand, mass transfer is explored under the effect of chemical reaction. Tangent hyperbolic fluid is an important branch of non-Newtonian fluids known for its ability to describe shear thinning effects. Understanding fluid flow on exponentially stretched surfaces is of great significance due to its applications in various industrial processes. These applications include fluid film condensing methods, plastic production for making plastic covers, fiber manufacturing (where it is used to spin fibers), glass blowing, metallurgical procedures, and the paper industry. The concept of magnetohydrodynamics (MHD) is significant due to its various engineering applications, such as MHD generators, flow meters, heat reservoirs, small components in different devices, and cooling systems for nuclear reactors. To analyze the system, using similarity transformations, the governing equations of continuity, velocity, and concentration are transformed into non-dimensional differential equations. The numerical solution is obtained using the shooting technique. The study presents the physical significance of all the fluid parameters involved, focusing on the velocity, temperature, and concentration profiles. These profiles are presented graphically and discussed in detail. The results show that the fluid velocity profile increases with enhancing values of the $\mathrm{\text{We}}$ and the magnetic number M. The thermal profile increases with higher $\mathrm{\text{Nt}}$ and $\mathrm{\text{Rd}}$ The concentration profile decreases with higher values of $Q_t$ and $\mathrm{\text{Nb}}$.

A fractional technique is used to evaluate the temperature, mass, and velocity flow of single and double wall CNTs over a vertical plate. Slip boundary conditions and applied magnetic force are addressed. Human blood is used to examine how base fluid behaves. Applying the proper dimensionless variables results in the dimensionless formulation of initial and boundary conditions related to the governed dimensional concentration, momentum, and energy equations. The Laplace transform technique is used to resolve the dimensionless governing partial differential equations and get the solutions. The constant proportional Caputo (CPC) time-fractional derivative is a unique class of fractional model used in the simulation technique. The fundamental definitions are used to support the said model first. Using MWCNTs and SWCNTs in comparison to the flow characteristics, a thermal and mass study is given. The heat and mass transfer processes for single-walled carbon nanotubes (SWCNTs) have been shown to typically be progressive. The momentum profile decreases as the fractional variables rise. Multi-walled carbon nanotubes (MWCNTs) show more progressive velocity control as a result of the magnetic parameter. Graphs demonstrate the influence of embedding factors on the velocity, energy, and concentration profiles.

The motivation of this paper is to explore an oscillatory flow of a magnetohydrodynamics (MHD) couple stress fluid with heat and mass transfer analyses through a porous medium. The modified Darcy–Brinkman couple stress fluid model is employed. There could be many potential applications of this study, such as thermal system, solidification of liquid crystals, cooling of metallic plates in a bath, exotic lubricants and colloidal solutions. In the wake of these potential applications, the study of MHD couple stress fluid flow with additional diffusing components has been found to be innovative and very interesting in the analysis of the influence of combined magnetic effects, couple stress fluid and two solutes. A governing coupled nonlinear partial differential equation with boundary constraints represents the modeled flow problem. In addition, these equations are converted into nondimensional form by employing the suitable nondimensionalizing quantities. It is possible to attain analytic solutions to the dimensionless equations that control the liquid flow, and the consequences of the flow’s confines on velocity, temperature, concentration and skin friction profile are examined and also the results are discussed through graphs. It is noteworthy that when the injection level on the heated plate is increased, the skin friction on each channel plate also increases.

The study of nanofluids and hybrid nanofluids is gaining conceivable importance due to their characteristics of being so useful in various daily life applications. This study deals with the motion of an electro conductive, incompressible magneto-hydrodynamic (MHD) hybrid nanofluid across a stretched surface of variable thickness. The objective of this study is motivated by a number of manufacturing and machine-building applications. However, no attempt has been made to establish MHD flow of hybrid nanofluid along a stretching sheet (a sheet with variable thickness) while keeping an eye on the impact of Hall current. In real-life situations, variable-thickness sheets are crucial in the creation of flexible containers and, additionally, in the layout and production of aerospace wings and auto body components. This study extends our fundamental knowledge of fluid dynamics and heat transmission in intricate systems. Recognizing how magnetic effects, nanofluid traits and heat conduction interplay can help researchers make valuable developments and breakthroughs in the areas of fluid mechanics and heat transfer. Hall effects are vital for applications including conductive fluids or plasma as they provide a more precise understanding of the movement of charged nanoparticles in the presence of a magnetic field. For hybrid nanofluid, we mixed the nanoparticles of titanium dioxide and copper (TiO₂–Cu) into the water. Due to the low noxiousness and chemical strength of titanium dioxide-based nanoparticles, they have great uses in research. We also consider the effects of Cattaneo–Christov heat flux to analyze the heat transfer of nanoparticles and Hall current effects, which make the flow three-dimensional. For both fundamental research and real-world applications, it is of the utmost importance to take into account the Hall effects and Cattaneo–Christov heat flux in the MHD flow analysis of hybrid nanofluid over stretched surface. It makes it possible to describe the phenomenon more precisely and can enhance the effectiveness and efficiency of numerous technical procedures. By using appropriate transformations, the equations that govern the flow are transformed into a system of non-dimensional ordinary differential equations. The non-dimensional system of equations has been solved numerically by using the ND Solve command in Mathematica Software, which is based on a multistep predictor-corrector method. For velocity and temperature profiles, the interplay of numerous developing parameters on flow is depicted graphically. The Hall parameter enhances the axial velocity but reduces the transverse velocity, while the magnetic field has the opposite effects. The temperature increases with the volume fraction of nanoparticles but decreases with the thermal relaxation parameter.

This study emphasizes the dual exposition of the impact of convective condition and the significant effects of magnetic field, heat source and velocity slip on hybrid nanofluid flow over the exponentially shrinking sheet. The hybrid nanofluid is regarded as a contemporary variety of nanofluid; consequently, it is utilized to enhance the efficiency of heat transfer. By applying the Tiwari–Das model, the key objective of the current research is to investigate the impact of involving factors Biot number, Eckert number, volume fraction, magnetohydrodynamic (MHD), slip and suction on the temperature and velocity profiles. In addition, the Nusselt number and skin friction variations have been explored against the suction effect on the solid volume fraction of copper ${\varphi }_2\in$ [1–3%] and the Biot number $\mathrm{\text{Bi}}\in$ [1–3%]. The nonlinear partial differential equations are converted to a set of an ordinary differential equations by incorporating exponential similarity vectors. Eventually, ordinary differential equations are rectified utilizing MATLAB bvp4c solver. The results demonstrate the presence of dual exposition for suction with varying values of copper volume fraction and the Biot number. The heat transmission rate is observed to escalate in both cases as the strength of the Biot and Eckert numbers intensifies in the range of 1–3%. In summary, the results show that duality and non-unique solutions occur in the aiding flow situation when the suction $S\ge S_{\mathrm{\text{ci}}}$, while there is no flow and unique solutions of hybrid nanofluid feasible when $S<S_{\mathrm{\text{ci}}}$.

In this research, the combined effect of magnetohydrodynamics (MHD) and slip velocity on squeeze film lubrication of long cylinders and infinite plates with couple stress fluid has been studied. Combining the Stokes couple stress theory and the Lorentz force theory yields analytical solutions for the modified Reynolds equation. Using the Reynolds equation, the mathematical expressions for squeeze film pressure, load-carrying capacity and squeeze film time are obtained. The pressure, load carrying capacity and squeeze film time increase with the combined effect of MHD, couple stress fluids and slip velocity.

The analysis of heat transfer of a hybrid nanofluid $({\mathrm{\text{Al}}}_2{\mathrm{\text{O}}}_3-\mathrm{\text{Cu}}∕{\mathrm{\text{H}}}_2\mathrm{\text{O}})$ in the MHD stagnation point motion toward an exponentially expanding/shrinking sheet with a nonuniform heat source/sink, multiple slip conditions and nonlinear thermal radiation effects are carried out in this study. By applying the similarity transformation, the governing equations are transformed into Ordinary Differential Equations (ODEs). The Runge–Kutta method using bvp4c in MATLAB software is implemented to explain the ODEs. The results show that the local Nusselt number is decreased while the skin friction coefficient remains unaltered by the accumulation of thermal slip, space, and temperature-dependent parameters.

The utilization of a ternary hybrid nanofluid, a recent development in the realm of nanofluids, can result in improved heat transfer. In the ongoing study, a ternary hybrid nanofluid flow is utilized, and it is carried out atop a porous spinning disc, which is exposed to a magnetic field, heat generation, thermal radiation, and Casson fluid. In this study, Blood/Water are taken as base fluids and (Ag–Au–Al₂O₃) are considered as a ternary hybrid nanoparticle. Nanoparticles made of gold and silver are put to use in a vast number of industries and fields, including nanotechnology and medicine. As a result of surface effects and quantum effects, these precious metals exhibit unique features in nanoform that play a vital role in with optical, magnetic, chemical, and mechanical behavior. Al₂O₃ unique optical, physical and biochemical qualities make it worthwhile for numerous uses, including nanophotonic, catalysis and the fabrication of high-energy composites. A set of relevant similarity transformations is used to generate non-dimensional forms of controlling paired nonlinear Partial Differential Equations (PDEs). MATLAB is used to perform a numerical solution with ODE45. In addition, the velocity outline decreases, and the temperature profile increases slightly before decreasing over a revolving disk when the values of magnetic parameters are increased. The distribution and radiant heat components heat up as the level gets higher.

Aim and objectives of the study:

The aim of this study is to analyze the Comprehensive Scrutinization of Ternary hybrid and Casson flow in a conducting porous rotating disk with internal heating. The primary objective of this analysis is to increase awareness of the impending energy crisis among those working in the industrial and technological sectors. Ternary nanoparticles (NPs) have a wide range of applications, which lends credence to the developed model. For example, Al₂O₃ can be used in a variety of ways that benefit society, it is used in water purification to remove water from the gas streams and extend people’s lives.

This paper conducts an extensive comparative analysis of numerical methods employed in modeling blood flow through arteries with Magnetohydrodynamics (MHD) and hybrid nanofluids. The study investigates the effectiveness and precision of distinct numerical approaches: Akbari Ganji’s Method (AGM), Fourth-Order Runge–Kutta (RK4), Finite Volume Method (FVM), and the Finite Element Method (FEM). These methods are essential for comprehending the intricate fluid dynamics that arise in the presence of magnetic fields and hybrid nanofluids a phenomenon relevant in numerous medical applications. Blood flow is subjected to a homogeneous magnetic field in a radial direction while the magneto-hemodynamics effect is taken into account. A variety of medical, physiological, and surgical procedures, as well as the regulation of blood pressure, heat distribution, wound healing, diagnostic imaging, and drug discovery, depend on blood flow through arteries to carry out vital functions such as oxygen and nutrition delivery, organ maintenance, and wound healing. Our findings highlight that while each method has strengths, their applicability varies based on the problem’s characteristics and computational resource constraints. This analysis aids researchers and practitioners in selecting the most suitable method for their modeling requirements, advancing numerical techniques for complex fluid dynamics involving MHD and hybrid nanofluids.

In this work, the Cattaneo–Christov double diffusion model is used to analyze the three-dimensional boundary layer flow of an upper-convected Maxwell fluid flowing over a bi-directional stretching surface. The model assumes that the fluid’s diffusivity is concentration-dependent, whereas dynamic viscosity and thermal conductivity are temperature-dependent; and the model takes into account the influence of the magnetic field of uniform strength. The problem is formulated using the conservation rules of mass, momentum, and energy as well as the boundary layer approximation. The resulting nonlinear system is then numerically solved using the bvp4c procedure in MATLAB. The obtained results are set forth graphically to illustrate the variations of different parameters. It is contemplated that the increase in thermal relaxation parameter results in a drop in fluid temperature while there is an enhancement in the concentration profile. Also, the temperature field has a direct relationship with the temperature-dependent viscosity. This research has proven its utility in industries where there is a need to analyze fluid flow over surfaces that can stretch in two directions.

This novel study unfolds the heat and mass transfer investigation of Williamson nanofluid (WNF) through a porous medium past a stretching plate, along with considering heat generation/absorption. Nanoparticles hold significant significance in thermal engineering, industrial operations, and biomedical advancements, contributing to enhanced heat transfer, cooling mechanisms, thermal extrusion processes, and applications in cancer treatment, particularly in addressing brain tumors. The coupled ordinary differential equations (ODEs) are gained from governing partial differential equations (PDEs) by applying sufficient transformations. Then the ODEs of a nonlinear nature, along with boundary conditions, are solved through bvp4c, a built-in MATLAB program. A good agreement has been found in comparing the present study with already published papers. Numerical values for skin friction, mass transfer, and heat transfer are shown through a table against involved physical parameters, especially for both injection $(S<0)$ and suction $(S>0)$ cases, by varying the values of unsteadiness parameter A and Weissenberg number $W_e$. The effects of the physical parameters on velocity, temperature, and mass profile are graphically depicted and illustrated minutely. It is noted that both the magnetic and Williamson parameters cause the thickness of the boundary layer to be reduced. It can be deduced from these findings that the rate of heat transfer over the surface of the plate decreases as the unsteadiness parameter increases. Furthermore, it is observed that an increase in the parameters of thermophoresis and Brownian motion leads to a higher temperature of the nanofluid.

We explored the three-dimensional flow characteristics of a rotating hybrid nanofluid over a permeable stretching surface, incorporating three prominent hybrid nanofluid models: Yamada–Ota, Xue, and Tiwari–Das. The study covers both multi-wall and single-wall carbon nanotubes, exhibiting properties within water. The fluid motion is induced through standard magnetohydrodynamics (MHD) on the spinning permeable stretched sheet. To establish the theoretical framework for fluid flow, we applied boundary layer approximations to the governing laws, including momentum, energy, and mass, leading to the development of partial differential equations. Applying appropriate transformations to the governing partial differential equations (PDEs) results in coupled ordinary differential equations (ODEs). The BVP4c solver in MATLAB has employed to solve the nonlinear ODEs, along with the specified boundary conditions. A BVP4c technique within the MATLAB program has been utilized for obtaining solutions. The results are presented graphically for various physical variables such as the magnetic parameter, porosity parameter, spinning parameter, inertial coefficient, and Eckert number. Additionally, numerical data are provided in tabular format. It is noteworthy that the Yamada–Ota model demonstrates superior heat transfer performance for hybrid nanofluids compared to both the Xue and Tiwari–Das models.

The steady MHD boundary-layer axis-symmetric flow of a third-grade fluid passing through an exponentially expanded cylinder in the vicinity of a magnetic field is investigated in this study. The problem is mathematically modeled. Suitable similarity transformations are carried out to convert the partial differential equations into nonlinear ordinary differential equations. The Runge–Kutta fourth-order shooting technique is used to solve the transmuted system of nonlinear ODEs. Graphical representations of numerical findings are used to examine the effects of various physical factors on the velocity and temperature profiles. The influence of fluid variables on the velocity curve, such as third-grade parameters, second-grade parameters, and Reynolds number, is illustrated and explored. The skin-friction coefficient expression is computed and given. The widths of the velocity and momentum boundary layers are revealed to be increasing functions of the curvature parameter. It is found that the third-grade fluid has a higher velocity profile than Newtonian and second-grade fluids. Also, the stretched cylinders cause a more progressive shift in heat and mass pattern for flow than flat plates do.