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This investigation incorporating the fluctuation in heat and mass transfer associated to the mixed convection magnetized flow of viscous fluid due to inclined surface with porous media. The contribution of Soret effects and viscous dissipation appliances is addressed. Furthermore, the heat transfer improvement is also assessed by thermal radiation, heat source and joule heating effects. The chemical reaction enrollment is also studied for concentration phenomenon. The convection of problem into non-dimensional framework is based on implication of new variables. The perturbation technique is followed to tracking the analytical outcomes. Physical visualization and interpretation of results under the influence of perturbed parameters have been studied. It is observed that heat and mass transfer enhances due to Soret number. Presence of chemical reaction leads to decrement of concentration profile. Claimed results presents applications in heat and mass transfer processes, chemical reaction, manufacturing systems, chemical engineering, extrusion processes etc.

The nanofluids are decomposition of nano-sized metallic particles with base materials maintaining peak thermal properties. Owing to enhanced thermal features, various applications of nanofluids are observed in enhancing energy resources and cooling processes. The objective of current work is exploring the thermal impact of nanofluid associated to the oblique stagnation point flow. The thermal interpretation of nanofluid is subject to active and passive control approach. The heat trnasfer analysis is identified by using convective thermal flow constraints. The Buongiorno nanofluid model is adopted, endorsing the Brownian and thermophoresis consequences. The flow problem is first simplfied intodimensionless form. The numerical computations are performed by using famous shooting scheme with justifiable accuracy. A comparative change in the thermal phenomenon given the passive and active frameworks is presented. It is exmained that heat transfer reduces for stretching ratio parameter. The concentration profile reduces for angle of incidence for both active and passive control approach.

The engine oil signifies key importance in the automobile industry. The thermal capataince of engine oil is quite necessary to retained the peak performances. The aim of current research is to enhance the thermal mechanism of engine oil base fluid with applications of hybrid-nanofluid. The graphene oxide and molybdenum disulfide nanoparticles are used to justify the thermal mechanism of hybrid nanofluid. The contribution of magnetic force and buoyancy forces are consdeired. The equations of velocity and temperature are transformed into non-dimensional forms with dimensionless parameters. The problem is transformed to fractional form by employing the Atangana-Baleanu fractional approach and generalized Fourier’s law. These equations are solved with the Laplace transform. Disparities in momentum and energy are exposed to several fractional and material parameters. It has been observed that temperature profile increases due to nanoparticles volume fraction and fractional parameter. The increasing impact of Grashof number is observed on velocity field.

The thermal improvement in heat transfer investigation using a hybrid nanofluid model is investigated. In a class of hybrid nanofluid, aluminum oxide, and titanium oxide nanoparticles are used. Kerosene oil and water decomposition are used in hybrid nanoparticles. The analysis is carried out for sinusoidal boundary constraints. Moreover, the current phenomenon's gravitational impact and magnetic force features are disclosed. The porous inclined surface, which moves with a sinusoidal pattern, endorsed the flow. The dimensionless system with suitable variables is achieved. The fractional framework with generalized Prabhakar model is used for the solution procedure. The basic concept of the fractional approach is presented. The physical contribution of parameters is observed for analyzing the thermal phenomenon.

In this paper, firstly, the “ Haar wavelet method ” is used to give approximate solutions for coupled systems of linear fractional Fredholm integro-differential equations. Moreover, we consider the fractional derivative to be described in the Caputo sense. The general idea of this technique is simply based on reducing this kinds of coupled systems into systems of algebraic equations which are easily to deal with and solve. Also, Laplace transform operator is included to develop a sophisticated approach which we called “ Laplace Haar wavelet method ” as an adjustment to “ Haar wavelet method ” to reduce the error and computational time. We provide illustrative examples to confirm validity, efficiency, accuracy, and applicability of the proposed methods.

The entropy generation phenomenon is important in thermodynamic systems, chemical engineering, industrial systems, reducing the energy consumptions and optimizing the resource utilization. Owing to such motivated applications, aim of current research is to present the applications entropy generation phenomenon in double diffusion flow Carreau-Yasuda nanofluid due to bidirectional stretched surface. The rheology of non-Newtonian fluid is observed by using the Carreau-Yasuda (CY) nonlinear model. The analysis is further supported with suspension of microorganisms with Carreau-Yasuda nanofluid. The formulated model is extended by entertaining the viscous dissipation, thermal radiation and activation energy consequences. The inertial effects are supported with Darcy Forchheimer approach. Additionally, non-Fourier and Fick theories are implanted to observe the thermal phenomenon. The equations resulted from the given model are analytically entertained via homotopy analysis scheme. The physical insight of thermal model is graphically presented. Moreover, the streamlines are plotted for flow model. An enhancing outcomes for entropy generation phenomenon are noted due to diffusion parameter and magnetic parameter. Moreover, the optimized phenomenon also increases due to Brinkman number.

This paper introduces an efficient numerical method for approximating solutions to geometric boundary value problems. We propose the multiplicative sinc–Galerkin method, tailored specifically for solving multiplicative differential equations. The method utilizes the geometric Whittaker cardinal function to approximate functions and their geometric derivatives. By reducing the geometric differential equation to a system of algebraic equations, we achieve computational efficiency. The method not only proves to be computationally efficient but also showcases a valuable symmetric property, aligning with inherent patterns in geometric structures. This symmetry enhances the method’s compatibility with the often-present symmetries in geometric boundary value problems, offering both computational advantages and a deeper understanding of geometric calculus. To demonstrate the reliability and efficiency of the proposed method, we present several examples with both homogeneous and non-homogeneous boundary conditions. These examples serve to validate the method’s performance in practice.

Hydrodynamics instability is studied for an electrical conducting fluid against small disturbance between the channels by using normal magnetized force. Chebyshev collocation technique is used to determine the stability of three-dimensional Hartmann flow problem. The distinctive case of the perturbations is calculated. It is also considered that perturbations intensity be determined by just on quantified by different parameters. We have considered one of the flow stabilities conditions to analyze our problem. QZ (Qualitat and Zuverlassigkeit) technique is applied to investigate the problem to draw stability curves. αc,βc are critical wave numbers in streamwise and span-wise respectively and for a big range of Hartmann value (Ha), we obtained critical Reynolds number (Rec). It is found that Couette flow is destabilized for a specific value of Magnetize force while with greater or lesser magnitude than the particular one will stabilize the flow. Disturbances with particular oblique angle θ will grow while the others will decay for the three-dimensional disturbance. We observed from over results that for Ha(>2.0) Rec, increases steadily and when Ha more than 3.886, Rec decreases fast to the minimum. It is also established that for drop down from Ha = 3.886 and Rec becomes very large. For distinct Hartmann values, there exist two Rec numbers due to closed contour. The outcomes of current study are utilized in drug-delivery systems and photodynamic therapy.

Abstract The objective of current research is to endorse the thermal aspect of Sutterby nanofluid containing the microorganisms due the stretched cylinder. The features of nonlinear thermal radiation, Darcy resistance and activation energy are also incorporated to inspect the thermal prospective. The problem is further extended with implementation of modified Fourier and Fick’s theories. The results are presented for the stretched cylinder and also for stationary plate. The numerical formulation for the problem is presented by following the shooting technique. The comparative numerical is performed to verify the computed simulations. The results convey that the presence of Darcy resistance parameter enhanced the velocity more effectively for stretched cylinder. A reduction in velocity due to Sutterby fluid parameter and buoyancy ratio parameter has been observed. Moreover, the temperature profile enhanced with larger sponginess parameter more effectively for stretching cylinder.

An unsteady free convective flow of an electrically conducting viscous fluid due to accelerated inestimable inclined perpendicular shield has been presented in presence of heat and mass transfer phenomenon. The applications of thermos-diffusion and heat source are also incorporated. The chemical reaction consequences are considered in the concentration equation. The compelling meadow is considered to be homogeneous and practical perpendicular to the flow direction. Further, the oscillatory suction effects are also taken into observations for porous regime. The closed form expressions are resulted with implementation of perturbation approach. The non-dimensional expression for the proposed governing system is yield out with entertaining appropriate variables. The graphically influence of parameters is studied. Following to obtained observations, it is claimed that declining deviation in velocity is predicted with chemical reactive factor. Further, less thermal transport between container to fluid is noticed for radiative absorption parameter.

The development of innovative heat exchangers with improved heat transfer performance is an important concern for heat transfer applications. The use of passive techniques such as vortex generators, surface roughness, augmentation of heat transfer area, or the combination of passive techniques is effective to improve the performance of heat exchangers. This review briefly summarizes the passive heat transfer enhancement techniques and the most important types of heat exchangers. Many effective design parameters have been briefly discussed to improve the performance of passive heat transfer techniques for heat exchangers.

Owing to the widespread practical significances of nanoparticles in thermal engineering and industrial processes, scientists are devoted to explore the more fundamental thermal aspects of such nano-materials. Recently, the nanoparticles show extensive applications in heat transfer improvement, energy production, cooling and heating processes, refrigeration, thermal extrusion processes, thermoelectric devices, bio-medical applications like cancer treatment, brain tumors and many more. Moreover, the phenomenon of bioconvection utilize interesting applications in bio-technology, bio-fuels, enzymes, fertilizes etc. Keeping such motivations in mind, this research presents the bioconvection flow of nanofluid induced by two stretchable disks separated at confined distance. The Buongiorno nanofluid model is employed to examine the Brownian motion and thermophoresis aspects. The equations govern for the flow of the nanofluid, heat transfer; nanoparticles concentration and density of motile microorganisms’ fields are simplified with the help of similarity transformation. The resultant form of nonlinear equations is solved by using homotopy analysis method. The influence of significantly participating parameters on velocity, nanofluid temperature, concentration and density of motile microorganisms profiles are observed with relevant physical significances. The convergence of solution for velocity, temperature, concentration and motile microorganisms profiles are examined carefully. The simulations are compared with already declared numerical data and found a convincible accuracy. It is noticed that the temperature profile enhances up to specified range and then start decline due to larger values of values of Brownian motion and thermophoresis parameters. A dual nature of velocity profiles for higher the value of stretching parameter has been observed. The concentration profile declines with increasing Brownian motion constant and Lewis number. The motile microorganisms profile decreases due to enhancement of bio-convection Peclet number, Reynolds number and Schmidt number.

Owing to fine thermal prospective of nanomaterials, different applications have been addressed in nanotechnology and thermal management systems. Multidisciplinary applications of such nanomaterials are commonly observed in nuclear reactors, thermal systems, transportation systems, industrial products cooling, power industry, renewable energy etc. The motivated work aims to predicts improved thermal consequences of Maxwell nanofluid with applications of non-uniform heat source and radiated phenomenon. The heat transfer analysis is assessed by using the Cattaneo-Christov theories. The thermal phenomenon is controlled with interference of slip effects. The flow pattern is based on the oblique stagnation point flow confined by stretched cylinder. The Keller Box numerical determination of problem is worked out with fine accuracy. The physical dynamic of flow parameters is visualized graphically. It is claimed that control of thermal transportation phenomenon is resulted when the slip effects are dominant. The exponential heat source with nonlinear radiated phenomenon is important when enhancement in thermal process needed.

The hybrid materials are an impressive class of nanofluids with exciting thermal outcomes and present applications in enhancing the heat transfer procedure, solar energy, extrusion processes, and in different engineering processes. The current contribution aims to reflect the improved mechanism of the heat transfer phenomenon for hybrid nanofluids. Aluminum oxide, copper, and copper oxide at different solid volume fractions are used to report the thermal phenomenon. For the base material, water is used. The mixed convection applications are also encountered. The moving cylinder with a stretched uniform velocity causes the flow. The velocity slip and convective boundary constraints are used to observe the flow phenomenon. The hybrid nanofluid is expressed via different mathematical relations. The shape factors for hybrid nanomaterials are presented. The Keller box numerical method with effective accuracy has been entertained for the simulation process. The applications of parameters for the current model are explained via graphs.

The nanoparticles are frequently used in biomedical science for the treatment of diseases like cancer and these nanoparticles are injected in blood which is transported in the cardiovascular system on the principle of peristalsis. This study elaborates the effects of Lorentz force and joule heating on the peristaltic flow of copper and iron oxide suspended blood based nanofluid in a complex wavy non-uniform curved channel. The Brinkman model is utilized for the temperature dependent viscosity and thermal conductivity. The problem is formulated using the fundamental laws in terms of coupled partial differential equations which are simplified using the creeping flow phenomenon. The graphical results for velocity, temperature, streamlines, and axial pressure are simulated numerically. The concluded observations deduce that the solid volume fraction of nanoparticles reduces the velocity and enhance the pressure gradient and accumulation of trapping bolus in the upper half of the curved channel is noticed for temperature dependent viscosity.

This research communicates the thermal assessment of Sutterby nanofluid containing the gyrotactic microorganisms with solutal and Marangoni boundaries. The applications of melting phenomenon and thermal conductivity are also considered. The flow is confined by a stretched cylinder. The prospective of Brownian motion and thermophoresis diffusions are also taken account via Buongiorno nanofluid model. The problem is formulated with help of governing relations and equations which are altered into dimensionless form via appropriate variables. The numerical scheme based on shooting scheme is employed to access the solution. A comparative analysis is performed to verify the approximated solution. The observations reveal that the velocity profile enhanced with the Marangoni number while a declining velocity profile has been observed with Sutterby nanofluid parameter and Darcy resistance parameter. The nanofluid temperature get rise with thermal conductivity parameter and thermal Biot number. An arising profile of nanofluid concentration is observed for concentration conductivity parameter and buoyancy ratio parameter.

This manuscript is presented to investigate peristaltic phenomenon of a flow driven peristaltically in an asymmetric inclined channel under the influence of externally applied magnetic field without employing the assumption of long wavelength and smaller Reynolds number assumptions which permit the effects of dominant inertial forces and to solve the Navier-Stokes equations in full form. The numerical results are obtained with help of finite element numerical scheme with Galerkin’s residual procedure. A comparative analysis for the obtained numerical data is performed in view of already reported studies and a favorable accuracy of results is noticed. The effects of other involved parameters are also presented in suitable graphs. Trapping and pumping phenomena is also discussed. It is found that trapped bolus shape is increased with increment of Hartmann number which tends to move towards central region of asymmetric channel and is predicted that trapped bolus will disappears at large Reynolds number. The pressure in peristaltic pumping region is enhanced by increasing angle of inclination.

The peristaltic phenomenon has intended the researcher’s attention due to its extensive applications in bio-mechanics, industries and biological sciences. The applications of peristaltic transport problems are observed in the transportation of cilia, food mixing in digestive tract, small blood vessels vasomtion, urine transportation etc. Following to such motivating applications in minds, this theoretical analysis presents the peristaltic flow of magnetized Jeffery fluid through accounted by curved channel in presence of heat transfer. The assumed flow is confined by different wave frames. The heat transfer inspection has been performed by utilizing the non-linear thermal radiation and joule heating features. The curvilinear coordinate system is used to model the two dimensional, laminar and incompressible Jeffrey fluid with peristaltic phenomenon. The partial differential equations are transformed into ordinary differential equations in view of appropriate quantities. The implementation of wave length with long magnitude and the assumptions of smaller Reynolds assumptions, the governing system of ordinary system is solved numerically by using shooting technique. The dynamic of involved quantities and physical variables is observed carefully. A declining observation in the temperature in all forms of waves has been noticed for increasing the viscoelastic parameter. The heat transfer coefficient for all waves enhanced for the radius of curvature. Moreover, the velocity of the fluid declines in the lower half of the channel and rises near upper wall of the curved channel due to magnetic parameter.

The simulations have been performed for the nonlinear radiative flow of Prandtl nanofluid following the peristaltic pumping in a wavy channel. The applications of entropy generation for the electrokinetic pumping phenomenon are also focused as a novelty. The complex wavy channel induced the flow of Prandtl nanofluid. Moreover, the formulated problem is solved by using the convective thermal and concentration boundary conditions. The Keller Box numerical procedure is adopted as a tool for the simulation task. The results are also verified by implementing the built-in numerical technique bvp4c. The comparison tasked against obtained numerical measurement has been done with already reported results with excellent manner. The physical characteristics based on the flow parameters for velocity, heat transfer phenomenon, concentration field, and entropy generation pattern is visualized graphically. It has been observed that the presence of thermal slip and concentration enhanced the heat transfer rate and concentration profile, respectively. The skin friction coefficient declines with electro-osmotic force and slip parameter. The increasing variation in Nusselt number is observed for electro-osmotic parameter for both linear and non-linear radiative phenomenon.

In this study, we have discussed thermophoresis particle deposition effects under the action of both pressure and buoyant forces flow of magneto-Walter’s B nanofluid induced by a stretched surface. The Buongiorno nanofluid model is employed to analyze the dynamic impact of thermophoretic dispersion and Brownian motion. The effects of chemical reaction, Joule heating and non-linear radiation relations are also incorporated. The analysis has been performed in view of solutal and heat convective boundary constraints. The analytical technique namely homotopy analysis scheme followed to solve the resulting non-linear governing equations. The behavior of velocity, temperature and concentration profiles are observed graphically. The physical consequences for all physical parameters are justified. It is noted that heat thermal Biot number, thermophoretic constant and viscoelastic parameter increases the nanofluid temperature and concentration. A decaying concentration profile is noted for Schmidt number.

Abstract On the account of significance of bioconvection in biotechnology and several biological systems, valuable contributions have been performed by scientists in current decade. In current framework, a theoretical bioconvection model is constituted to examine the analyzed the thermally developed magnetized couple stress nanoparticles flow by involving narrative flow characteristics namely activation energy, chemical reaction and radiation features. The accelerated flow is organized on the periodically porous stretched configuration. The heat performances are evaluated via famous Buongiorno’s model which successfully reflects the important features of thermophoretic and Brownian motion. The composed fluid model is based on the governing equations of momentum, energy, nanoparticles concentration and motile microorganisms. The dimensionless problem has been solved analytically via homotopic procedure where the convergence of results is carefully examined. The interesting graphical description for the distribution of velocity, heat transfer of nanoparticles, concentration pattern and gyrotactic microorganism significance are presented with relevant physical significance. The variation in wall shear stress is also graphically underlined which shows an interesting periodic oscillation near the flow domain. The numerical interpretation for examining the heat mass and motile density transfer rate is presented in tubular form.

The mixed convection flow of carbon nanotubes due to porous vertical plate has been presented via fractional simulations. The single and multi-wall carbon nanotubes are used to observe the thermal aspect of hybrid nanofluid model. The blood is treated as base material. The shear thinning and thickening aspect of blood are predicted by using the Casson fluid model. The comparative thermal observations of blood base fluid with multi-wall carbon nanotubes (MWCNTs) and single-wall carbon nanotubes (SWCNTs) are reported. The Prabhakar fractional derivative simulations are performed to analyze the physical pattern of problem. The Laplace transformations are implemented to the integrating task. It is noticed that fractional parameter reduces the velocity profile for both MWCNTs and SWCNTs suspension. With increment in volume fraction and fractional parameters reduces the thermal profile effectively.

The mathematical study is modeled to examine the physical impacts of porosity and thermal slip on flow features of nanofluid through an asymmetric micro-channel. Additionally, the effects of electro-osmosis and magnetic field are also under consideration. Convective boundary conditions are used in the current study by neglecting thermal and buoyancy forces. The obtained rheological equations are transformed into non-dimensional flow systems by using scaling variables. These rheological equations are elucidated by using the creeping approximation and low zeta potential (the Debye–Hückel linearization). The exact solutions of rheological equations are evaluated via Mathematica software 11.0. The dynamic impacts of embedded parameters such as Hartmann number, electro-osmosis parameter, Darcy's number, Brinkman number, Helmholtz–Smoluchowski velocity, slip parameter, thermal radiation, cilia length parameter and Prandtl number on the rheological features are presented graphically via Mathematica software 11.0. The whole analysis is based upon train waves of metachronal propulsion. Three-dimensional graphs are plotted in the current investigation to get more obvious behavior of embedded on the flow features. The cilia length parameter has a remarkable character in enhancing the magnitude of velocity profiles, while, and opposite actions is perceived in the graph of pressure gradient. The thermal slip parameters have an energetic impact in reducing the temperature profile magnitude. The outcomes revealed a good understanding into biomimetic energy frameworks taking advantage of electroosmosis, magnetism and nanotechnology, and, besides, they outfit a valuable benchmark for numerical and experimental multi-physics recreations.

The advances in nanotechnology led to the development of new kinds of engineered fluids called nanofluids. Nanofluids have several industrial and engineering applications, such as solar energy systems, heat conduction processes, nuclear systems, chemical processes, etc. The motivation of the present work is to analyze and explore the thermal and dynamic behaviors of a non-Newtonian fluid flow under time retardation effects. The flow is unsteady and caused by a bidirectional, periodically moving surface. In addition to the convective heat transfer and fluid flow, the radiation and chemical reactions have also been considered. The governing equations are established based on the modified Cattaneo–Christov heat flux formulation. It was found that the bidirectional velocities oscillate periodically, and that the magnitude of the oscillation increases with the retardation time. Higher temperatures occur when the porosity parameter is increased, and lower concentrations are encountered for higher values of the concentration relaxation parameter. The current results can be applied in thermal systems, heat transfer enhancement, chemical synthesis, solar systems, power generation, medical applications, the automotive industry, process industries, refrigeration, etc.

The bioconvective flow of a Jeffrey fluid conveying tiny particles under the effect of an oscillating stretched bidirectional surface is considered in this paper. The effects of thermal radiation and a porous medium are also investigated. The Cattaneo–Christov diffusion theories are used to analyze the heat and mass transfer phenomena. The activation energy effects are included in the concentration equation. The solved dimensionless equations system is established, based on non-dimensional variables. The analytical findings are evaluated using the homotopic analysis technique. The convergence of solutions is ensured. The results are validated by already available published findings and a good concordance is encountered. The fundamental physical aspect of flow parameters is graphically evaluated. The main results reveal that the velocity is reduced by increasing the permeability of the porous medium. An increase in the temperature occurs when the viscosity of the fluid is varied. The obtained results can be useful in thermal systems, energy production, heat transfer devices, solar systems, biofuels, fertilizers, etc.

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained using the Sinc-collocation method. This approach approximates the space dimension of the solution with a cardinal expansion of Sinc functions. First, discretizing the time derivative of the MRLW equation by a classic finite difference formula, while the space derivatives are approximated by a θ—weighted scheme. For comparison purposes, we also find a soliton solution using the Adomian decomposition method (ADM). The Sinc-collocation method was were found to be more accurate and efficient than the ADM schemes. Furthermore, we show that the number of solitons generated can be approximated using the Maxwellian initial condition. The proposed methods’ results, analytical solutions, and numerical methods are compared. Finally, a variety of graphical representations for the obtained solutions makes the dynamics of the MRLW equation visible and provides the mathematical foundation for physical and engineering applications.

In this research, Burgers’ equation, which is well known to be nonlinear partial differential equation, has many applications for studying some physical phenomena in the disciplines we mention, water waves, plasma waves, and ion acoustic plasma waves. The paper demonstrates a comprehensive performance assessment for the use of sinc methodology to solve Burgers’ equation; the method used in this research depends mainly on the use of the sinc function as a basis, where the equation under consideration was converted into an integral equation of Volterra type. Then, the x−derivatives were approximated and replaced by their corresponding sinc matrices and the resulting integral equation was also treated by approximating the integral via the use of definite integral formula for sinc functions. The method used is the sinc method, taking into account the domain on the x−axis and the time axis tt, where transformation functions intervened to make the problem applicable in their domains. The solution has been converted into an algebraic equation that is easy to deal with through any iterative method. As for the proof, the convergence of the resulting solution has been proven using the fixed-point method. It turns out that the approximate solution to the problem under consideration converges to the exact solution with an exponential order. Some examples were presented, and the effectiveness and accuracy of the method were shown by displaying the results in tables and graphs, which showed the efficacy and ease of the sinc method.

This novel investigation suggests interesting solution techniques regarding the magnetized flow of Casson material through a porous space confined by a vertical surface. The heat and mass pattern in non-Newtonian fluid flow is accessed by utilizing the slip features and thermal radiation applications. The solution of fractional dimensionless governing equations is obtained by using Laplace transformation and fractional derivative operators i.e. Atangana-Baleanu (AB) time-fractional derivative and Caputo-Fabrizio (CF) time-fractional derivative. Some limiting cases of velocity distribution, obtained by both AB-time fractional derivative and CF-time fractional derivative, are discussed. To enhance the innovation of this article, recent definitions of non-integer derivative operators are used to explore the numerical results under the conversation problem. The obtained results of velocity distribution are compared graphically and numerically by using different inverse techniques. A comprehensive analysis of employed solution techniques with limiting cases is worked out.

The thermal characteristics of modified hybrid nanoparticles have been studied between two parallel walls with applications of entropy generation and nonlinear thermal radiation. Three types of nanoparticles namely copper oxide (CuO), copper nanoparticles (Cu) and aluminum oxide (Al2O3) are utilized to thermal determination of heat transfer. The contribution of viscous dissipation and Joule heating is also inspected. Moreover, the shape effects for the three types of nanoparticles are addressed as a novelty. The mathematical model is developed in view of associated laws and thermal properties of modified hybrid nanofluid. The problem is first modeled in terms of partial differential equations for which the similarity variables are followed to convert the system into dimensionless form. The Keller box numerical scheme is used to obtain the solution. The thermo-physical properties of nanoparticles and fluctuation in the heat transfer rate are reported for the flow parameters. The results claimed that the temperature profile enhanced solid volume faction of copper oxide nanoparticles. A lower change in the entropy generation is observed for the solid volume fraction of aluminum oxide and copper nanoparticles.

In this investigation, the novel fractional order simulations are performed in order to inspect the characteristics of mass and heat transfer over porous inclined surface subject to the magnetic force impact. The radiative phenomenon and slip wall constraints are utilized to make the model more comprehensive. The convective thermal mechanism of heat transfer is predicted with assumptions of ramped temperature. The recent definitions of fractional derivatives are followed. The famous technique of fractional approach namely Atangana-Baleanu (AB) and Caputo-time fractional derivative are employed for the solution procedure. Various relations expressing the mathematical functions for the fractional derivatives are encountered. The major outcomes are listed for parameters govern for the model. This obtained result reflects many engineering and industrial applications like thermal processes, processing industries, nuclear reactions, solar systems, energy production etc.

The double diffusion thermal mechanism of viscoelastic nano-materials has been suggested under the dynamic consequence of nonlinear thermal radiation and temperature absorption/production phenomenon. The electrically conducting generalized micropolar nanofluid is considered to predict the non-Newtonian behavior. The consideration of modified micropolar fluid successfully retained the viscous features as well as results for the second grade fluid. The uniform flow is induced with the linear movement of a flat surface. Following the convective boundary constraints, the heat and mass characteristics are reported. The equations that governs to the flow are reduced into ordinary ones. The homotopy analysis method (HAM) is imposed to develop the analytical expressions by using computational software MATHEMATICA 8. The evaluation of parameters is inspected by performing graphical analysis for transport process. The computational numerically illustration of the local Nusselt number, Sherwood number and the motile microorganism density number is performed. The results claimed that nanofluid temperature declined with the vortex viscosity parameter. The solutal nanoparticles concentration decreases with the Dufour Lewis number while a lower change in solutal nanoparticles concentration has been observed for regular Lewis number. Moreover, the local Nusselt and Sherwood number increases with vortex viscosity parameter.

The unsteady natural convection flow for the maxwell fluid due to infinite plate is inspected with help off ractional derivatives. The effects of slip conditions, Newtonian heating effect, MHD, and radiation are also considered. The recent definitions of fractional derivatives i.e. Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional derivatives are used to construct the fractional modal of leading partial differential equations. The semi-analytical solution of the governing equations is attained by utilizing the Laplace transformation and some numerical techniques i.e. Stehfest and Tzou's algorithms. Moreover, to analyze the physical significance of the under consideration problem, the graphical analysis is explored. The results concluded that the temperature and velocity profile obtained via CF-fractional derivative shows a declining trend when compared with AB-fractional derivative. Furthermore, the velocity and temperature fields were also decayed by varying the value of fractional parameters in both AB and CF cases.

The thermal mechanism of nanofluids due to advanced features is the most interesting research area owing to the applications in heat transmit devices, cooling procedure, energy production, etc. The nanofluids with motile microorganisms signify their importance in bio-medical engineering and biotechnology. This research communicates the accessed dynamic feature of Reiner–Philippoff nanofluid with applications of the bioconvection phenomenon. The magnetic force impact and activation energy features are also intended to perform the radiative analysis of Reiner–Philippoff nanomaterial. The slip features higher-order relations are incorporated to analyze the flow. The modifications in the energy equation are suggested by using thermal radiation with nonlinear relationship. The flow equations, which in turn to non-dimensionless form, are numerically tackled with a shooting scheme. A comprehensive thermal analysis for the endorsed parameters is presented. The numerical data is achieved to examine the fluctuation in heat, mass, and motile density function. The simulated results show that the velocity profile improves with Philippoff fluid parameter while a declining change is noticed for the slip parameter. With increase of Philippoff fluid parameter, both temperature and concentration profile declined. The consideration of higher order slip is more effective for increasing the temperature, concentration and microorganisms’ profiles.

The bioconvective flow of non-Newtonian fluid induced by a stretched surface under the aspects of combined magnetic and porous medium effects is the main focus of the current investigation. Unlike traditional aspects, here the viscoelastic behavior has been examined by a combination of both micropolar and second grade fluid. Further thermophoresis, Brownian motion and thermodiffusion aspects, along with variable thermal conductivity, have also been utilized for the boundary process. The solution of the nonlinear fundamental flow problem is figured out via convergent approach via Mathematica software. It is noted that this flow model is based on theoretical flow assumptions instead of any experimental data. The efficiency of the simulated solution has been determined by comparing with previously reported results. The engineering parameters’ effects are computationally evaluated for some definite range.

The present study explores the characteristics of 2D MHD melting with reference to mass and heat transportation upon stagnation point Powell–Eyring nanofluid flow on an extensible surface. Melting is an important phenomenon that is involved in many procedures such as permafrost melting, solidification of slag, defrosting frozen ground etc., all of which are examples of soil freezing and melting that involve heat trafficking through a coil in a grounded pump. A mathematical model is developed for the boundary layer flow. The differential equations are solved through a numerical algorithm which makes use of the boundary value problem solver bvp4c, applying MATLAB software. The numerical variations of embedded parameters on velocity lineation, temperature figuration, and concentration delineation are represented graphically, as are the width of the boundary layer value and the delineation rate for the increasing velocity parameter. The velocity function shows a decremental response for M while the opposite behavior is seen against the concentration field.

This paper aims to investigate free convection heat transmission in hybrid nanofluids across an inclined pours plate, which characterizes an asymmetrical hybrid nanofluid flow and heat transfer behavior. With an angled magnetic field applied, sliding on the border of walls is also considered with sinusoidal heat transfer boundary conditions. The non-dimensional leading equations are converted into a fractional model using an effective mathematical fractional approach known as the Prabhakar time fractional derivative. Silver (Ag) and titanium dioxide (TiO2) are both considered nanoparticles, with water (H2O) and sodium alginate (C6H9NaO7) serving as the base fluids. The solution of the momentum, concentration, and energy equation is found by utilizing the Laplace scheme, and different numerical algorithms are considered for the inverse of Laplace, i.e., Stehfest and Tzou’s. The graphical analysis investigates the impact and symmetry of significant physical and fractional parameters. Consequently, we surmise that water-based hybrid nanofluid has a somewhat higher velocity than sodium alginate-based hybrid nanofluid. Furthermore, the Casson parameter has a dual effect on the momentum profile. Furthermore, the memory effect reduces as fractional restriction increases for both the velocity and temperature layers. The results demonstrate that increasing the heat transmission in the solid nanoparticle volume fractions enhanced the heat transmission. In addition, the numerical assessment examined the increase in mass and heat transmission, while shear stress was increased with an increase in the Prabhakar fractional parameter α.

Owing to the multidisciplinary significance and dynamic applications of nanoparticles, the research in areas of thermal and process engineering has been expanding every day. The nanomaterials with extraordinary thermal and physical properties are being studied to pose some astonishing contributions in various fields. The nanomaterials when taken with microorganisms in a solution for collective transport, referred as mix fluids, greatly improve the thermal efficiency during heat transfer phenomena. Usually, the mixed fluids particles have greater advantage of being controlled by some physical stimuli of light, gravity and density along with electrical and magnetic forces. The current study offers some innovative applications of induced magnetic field for the bio-convection pattern third grade nanofluid under the influence of extraordinary activation energy and non-uniform hear source/sink factors. The assumptions of stagnation points are considered under the influence of stretched. The nano-materials have been mixed with microorganism to prepare a mixed fluid in order to obtain more stability and directed transportation. The flow constraints for the thermal transport phenomenon have been explained by using convective boundary approach. The couple and nonlinear equations have been put forward to present the model and the solution has been derived using shooting algorithm technique. Moreover, the numerical and graphical outcomes from the study are presented using tables and figures. The improved profile of velocity is predicted against higher values of velocity ratio, Reynolds number and third grade fluid parameter. The induced magnetic field profile enhanced for reciprocal magnetic Prandtl number and magnetic parameter. The consideration of non-uniform heat source/sink is more effective to improve and control and thermal transportation process.

The main aim of this study is to apply the recent definition of fractional derivative i.e. Atangana-Baleanu time-fractional derivative on second grade fluid flowing on an infinite plate under the MHD (magnetohydrodynamics) effect. The flow is studied under both Darcy's law and heat transfer. To find out the analytical solution, the Laplace transformation technique is utilized on the obtained non-dimensional governing equations momentum and heat equations with initial and boundary conditions. To enhance the innovation of this study, the graphical and numerical comparison of heat and momentum equation is analyzed with different parameters for different values. Main results and conclusions are accomplished at the end of this work. The increasing change in temperature profile is noticed with fractional parameter and Grashof number. The velocity profile increases with the Prandtl number while a reverse trend is noted for the temperature profile.

The thermal applications of nanofluids significantly improved the heat and mass transfer pattern which convey necessary role in many engineering and industrial zones. The consideration of nanofluids contributes many dynamic applications in the solar energy and thermal engineering problems. Moreover, the stability of nanofluids is enhanced perfectively with motile microorganisms which have applications in petroleum sciences, biofuels, bio-engineering, bio-medical, enzymes etc. This research determines the applications of bio-convection in Casson nanoliquid flow subject to the variable thermal conductivity and inertial forces. The Cattaneo–Christov relations are treated to modify the heat and concentration equations. The accelerated surface with sinusoidal type velocity induced the flow. The flow problem is formulated in terms of partial differential equations. The homotopic scheme is followed in order to suggest the analytical relations. After highlighting the convergence region, the graphical simulations with help of MATHEMAITCA are performed. The physical output is addressed in view of all flow parameters. The 3-D behavior of velocity, temperature, concentration and microorganisms is also addressed.

The Sumudu decomposition method was used and developed in this paper to find approximate solutions for a general form of fractional integro-differential equation of Volterra and Fredholm types. The Caputo definition was used to deal with fractional derivatives. As the method under consideration depends mainly on writing non-linear terms, which are often found inside the kernel of the integral equation, writing it in the form of Adomian’s polynomials in the well-known way. After applying the Sumudu transformation to both sides of the integral equation, the solution was written in the form of a convergent infinite series whose terms can be alternately calculated. The method was applied to three examples of non-linear integral equations with fractional derivatives. The results that were presented in the form of tables and graphs showed that the method is accurate, effective and highly efficient.

This investigation explored the significances of heterogeneous and homogeneous reactions on magneto hydrodynamic Maxell fluid flow induced by radially stretching disks under influence of buoyancy driven force. The steady state system is examined for the specific case in which the coefficient of diffusion of the reactants and auto catalytic agent are identical. The mathematical formulation for present flow problem is developed by using flow assumptions. The analytical solution for formulated flow problem is determined analytically by using homotopy analysis method (HAM) by using MATHEMATICA software. It is noticed that presence of Deborah number and thermal Biot number enhances the temperature profile. The homogeneous and heterogeneous parameter and Deborah number decays the concentration profile effectively. Further, an improved concentration distribution is noted for Archimedes number.

Abstract The derivation of this paper is devoted to describing the operational properties of the finite Fourier transform method, with the purpose of acquiring a sufficient theory to enable us to follow the solutions of boundary value problems of partial differential equations, which has some applications on potential and steady-state temperature. Numerical calculations show that the present method gives higher accuracy with less computation time than other, traditional methods, like the finite difference method.

On the account of motivating fabrication of bioconvection phenomenon in various engineering and industrial systems, an attention has been devoted by researchers in current decade. Therefore, this theoretical investigation deals with the utilization of bioconvection phenomenon in flow of tangent hyperbolic nanofluid over an accelerated moving surface. It is assumed that the flow is generated due to periodically motion of the sheet. The energy equation is modified by entertaining the nonlinear thermal radiation features. The chemical reaction effects are elaborated in the concentration equation. Moreover, the significance of present flow problem increases by utilizing the thermophoresis and Brownian motion effects. The governing equations are transmuted into non-dimensional form with utilization of appropriate quantities. The analytical solution is computed by using homotopy analysis method. The implications of promising parameters on velocity profile, temperature profile, nanoparticles volume fraction and microorganisms profile is evaluated graphically. The presence of radiation parameter, thermophoresis and Brownian motion effects are more frequent for enhancement of heat transfer. The reported observations can efficiently use in the improvement of heat transfer devices as well as microbial fuel cells.

Owing to the expensive applications of nanoparticles in engineering sciences, an admirable attention has been intended by researchers on this topic in recent years. The utilization of nanoparticles as asource of energy is intended much attention of investigators in recent decade. This novel attempt investigates the thermal properties of Casson nanofluid containing microorganisms induced by an oscillatory moving surface. The fundamental features of heat and mass phenomenon are inspected by utilizing the temperature-dependent viscosity. Buongiorno’s mathematical model is used to report the famous Brownian motion and thermophoretic diffusion consequences. The flow problem characterizes the partial differential equations for which analytical solution has been computed with a convincible accuracy. The insight physical features are inspected with help of various curves. The physical significances of flow parameters is studied via various graphs.

In this paper, we present a comparative study between Sinc–Galerkin method and a modified version of the variational iteration method (VIM) to solve non-linear Sturm–Liouville eigenvalue problem. In the Sinc method, the problem under consideration was converted from a non-linear differential equation to a non-linear system of equations, that we were able to solve it via the use of some iterative techniques, like Newton’s method. The other method under consideration is the VIM, where the VIM has been modified through the use of the Laplace transform, and another effective modification has also been made to the VIM by replacing the non-linear term in the integral equation resulting from the use of the well-known VIM with the Adomian’s polynomials. In order to explain the advantages of each method over the other, several issues have been studied, including one that has an application in the field of spectral theory. The results in solutions to these problems, which were included in tables, showed that the improved VIM is better than the Sinc method, while the Sinc method addresses some advantages over the VIM when dealing with singular problems.

Ferro liquids derive their magneto–viscous behavior from the suspended magnetic nanomaterial that enables tunable changes in temperature, as well as nano-structured fluid characteristics. A theoretical model that depicts the bioconvection flow of cross nanofluid with a magnetic dipole subjected to a cylindrical surface was developed and numerically solved. The model encountered nonlinear thermal radiation, activation energy, and second order slip. The flow equations were reduced and are presented in dimensionless forms, and they were solved numerically using the shooting technique, which is a built-in feature of MatLab. The model encountered symmetrical constraints for predicting velocity, temperature, concentration, and gyrotactic microorganism distribution and profiles. Moreover, the numerical values were computed for local Nusselt number, local Sherwood number, and motile density number against each physical parameter.

In this paper, we develop an accurate technique via the use of the Adomian decomposition method (ADM) to solve analytically a 2 × 2 systems of partial differential equation that represent balance laws of hyperbolic-elliptic type. We prove that the sequence of iteration obtained by ADM converges strongly to the exact solution by establishing a construction of fixed points. For comparison purposes, we also use the Sinc function methodology to establish a new procedure to solve numerically the same system. It is shown that approximation by Sinc function converges to the exact solution exponentially, also handles changes in type. A numerical example is presented to demonstrate the theoretical results. It is noted that the two methods show the symmetry in the approximate solution. The results obtained by both methods reveal that they are reliable and convenient for solving balance laws where the initial conditions are of the Riemann type.

In this paper, numerical solutions for nonlinear coupled Korteweg-de Vries(abbreviated as KdV) equations are calculated by the Sinc-collocation method. This approach is based on a global collocation method using Sinc basis functions. The first step is to discretize time derivative of the KdV equations by a classic finite difference formula, while the space derivatives are approximated by a weighted scheme. Sinc functions are used to solve these two equations. Soliton solutions are constructed to show the nature of the solution. The numerical results are shown to demonstrate the efficiency of the newly proposed method.

In this paper, the homotopy perturbation method is adopted to find explicit and numerical solutions for systems of non-linear fractional shallow water equations. The fractional derivatives are described in the Caputo sense. We apply both the homotopy perturbation method and the homotopy analysis method, to solve certain shallow water equations with time-fractional derivatives, and explicitly construct convergent power series solutions. The results obtained reveal that these methods are both very effective and simple for finding approximate solutions. Some numerical examples and plots are presented to illustrate the efficiency and reliability of these methods.