This paper discusses the model of the boundary layer(BL)flow and the heat transfer characteristics of hybrid nanofluid(HNF)over shrinking/stretching disks.In addition,the thermal radiation and the impact of velocity a...This paper discusses the model of the boundary layer(BL)flow and the heat transfer characteristics of hybrid nanofluid(HNF)over shrinking/stretching disks.In addition,the thermal radiation and the impact of velocity and thermal slip boundary conditions are also examined.The considered hybrid nano-fluid contains silver(Ag)and iron oxide(Fe_(3)O_(4))nanoparticles dispersed in the water to prepare the Ag-Fe_(3)O_(4)/water-based hybrid nanofluid.The requisite posited partial differential equations model is converted to ordinary differential equations using similarity transformations.For a numerical solution,the shooting method in Maple is employed.Moreover,the duality in solutions is achieved for both cases of the disk(stretching(λ>0)and shrinking(λ<0)).At the same time,a unique solution is observed for λ=0.No solution is found for them at λ<λ_(c),whereas the solutions are split at the λ=λ_(c).Besides,the value of the λ_(c) is dependent on the φ_(hnf).Meanwhile,the values of f″(0)and -θ′(0)intensified with increasing φ_(hnf).Stability analysis has been applied using bvp4c in MATLAB software due to a dual solution.Furthermore,analysis shows that the first solution is stable and feasible physically.For the slip parameters,an increase in the velocity slip parameter increases the velocity and shear stress profiles while increasing the temperature profile in the first solutions.While the rise in thermal slip parameter reduces the temperature profile nanoparticle volume fractions increase it.展开更多
The present work is concerned with the effects of viscous dissipation and heat source/sink on a three-dimensional magnetohydrodynamic boundary layer axisymmetric stagnation flow, and the heat transfer of an electrical...The present work is concerned with the effects of viscous dissipation and heat source/sink on a three-dimensional magnetohydrodynamic boundary layer axisymmetric stagnation flow, and the heat transfer of an electrically conducting fluid over a sheet, which shrinks or stretches axisymmetrically in its own plane where the line of the symmetry of the stagnation flow and that of the shrinking （stretching） sheet are, in general, not aligned. The governing equations are transformed into ordinary differential equations by using suitable similarity transformations and then solved numerically by a shooting technique. This investigation explores the conditions of the non-existence, existence and uniqueness of the solutions of the similar equations numerically. It is noted that the range of the velocity ratio parameter, where the similarity solution exists, is increased with the increase of the value of the magnetic parameter. Furthermore, the study reveals that the non-alignment function affects the shrinking sheet more than the stretching sheet. In addition, the numerical results of the velocity profile, temperature profile, skin-friction coefficient, and rate of heat transfer at the sheet are discussed in detail with different parameters.展开更多
In this paper, the boundary layer stagnation-point slip flow and heat transfer towards a shrinking/stretching cylinder over a permeable surface is considered. The governing equations are first transformed into a syste...In this paper, the boundary layer stagnation-point slip flow and heat transfer towards a shrinking/stretching cylinder over a permeable surface is considered. The governing equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into self-similar ordinary differential equations before they are solved numerically using the shooting method. Numerical results are obtained for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles for some values of the governing parameters, namely the velocity slip parameter (α), the thermal slip parameter (β), the curvature parameter (γ) and the velocity ratio parameter (c/a). The physical quantities of interest are the skin friction coefficient and the local Nusselt number measured by f’’(0) and –θ’(0), respectively. The numerical results show that the velocity slip parameter α increases the heat transfer rate at the surface, while the thermal slip parameter β decreases it. On the other hand, increasing the velocity slip parameter α causes the decrease in the flow velocity. Further, it is found that the solutions for a shrinking cylinder (c/ac/a>0) case. Finally, it is also found that the values of f’’(0) and –θ’(0) increase as the curvature parameter γ increases.展开更多
Time-dependent,two-dimensional(2 D)magnetohydrodynamic(MHD)micropolar nanomaterial flow over a shrinking/stretching surface near the stagnant point is considered.Mass and heat transfer characteristics are incorporated...Time-dependent,two-dimensional(2 D)magnetohydrodynamic(MHD)micropolar nanomaterial flow over a shrinking/stretching surface near the stagnant point is considered.Mass and heat transfer characteristics are incorporated in the problem.A model of the partial differential expressions is altered into the forms of the ordinary differential equations via similarity transformations.The obtained equations are numerically solved by a shooting scheme in the MAPLE software.Dual solutions are observed at different values of the specified physical parameters.The stability of first and second solutions is examined through the stability analysis process.This analysis interprets that the first solution is stabilized and physically feasible while the second one is un-stable and not feasible.Furthermore,the natures of various physical factors on the drag force,skin-friction factor,and rate of mass and heat transfer are determined and interpreted.The micropolar nanofluid velocity declines with a rise in the suction and magnetic parameters,whereas it increases by increasing the unsteadiness parameter.The temperature of the micropolar nanofluid rises with increase in the Brownian motion,radiation,thermophoresis,unsteady and magnetic parameters,but it decreases against an increment in the thermal slip constraint and Prandtl number.The concentration of nanoparticles reduces against the augmented Schmidt number and Brownian movement values but rises for incremented thermophoresis parameter values.展开更多
We report on the magnetohydrodynamic impact on the axisymmetric flow of Al_(2)O_(3)/Cu nanoparticles suspended in H_(2)O past a stretched/shrinked sheet.With the use of partial differential equations and the correspon...We report on the magnetohydrodynamic impact on the axisymmetric flow of Al_(2)O_(3)/Cu nanoparticles suspended in H_(2)O past a stretched/shrinked sheet.With the use of partial differential equations and the corresponding thermophysical characteristics of nanoparticles,the physical flow process is illustrated.The resultant nonlinear system of partial differential equations is converted into a system of ordinary differential equations using the suitable similarity transformations.The transformed differential equations are solved analytically.Impacts of the magnetic parameter,solid volume fraction and stretching/shrinking parameter on momentum and temperature distribution have been analyzed and interpreted graphically.The skin friction and Nusselt number were also evaluated.In addition,existence of dual solution was deduced for the shrinking sheet and unique solution for the stretching one.Further,Al_(2)O_(3)/H_(2)O nanofluid flow has better thermal conductivity on comparing with Cu/H_(2)O nanofluid.Furthermore,it was found that the first solutions of the stream are stable and physically realizable,whereas those of the second ones are unstable.展开更多
The viscous fluid flow and heat transfer over a stretching(shrinking)and porous sheets of nonuniform thickness are investigated in this paper.The modeled problem is presented by utilizing the stretching(shrinking)and ...The viscous fluid flow and heat transfer over a stretching(shrinking)and porous sheets of nonuniform thickness are investigated in this paper.The modeled problem is presented by utilizing the stretching(shrinking)and porous velocities and variable thickness of the sheet and they are combined in a relation.Consequently,the new problem reproduces the different available forms of flow motion and heat transfer maintained over a stretching(shrinking)and porous sheet of variable thickness in one go.As a result,the governing equations are embedded in several parameters which can be transformed into classical cases of stretched(shrunk)flows over porous sheets.A set of general,unusual and new variables is formed to simplify the governing partial differential equations and boundary conditions.The final equations are compared with the classical models to get the validity of the current simulations and they are exactly matched with each other for different choices of parameters of the current problem when their values are properly adjusted and manipulated.Moreover,we have recovered the classical results for special and appropriate values of the parameters(δ_(1),δ_(2),δ_(3),c,and B).The individual and combined effects of all inputs from the boundary are seen on flow and heat transfer properties with the help of a numerical method and the results are compared with classical solutions in special cases.It is noteworthy that the problem describes and enhances the behavior of all field quantities in view of the governing parameters.Numerical result shows that the dual solutions can be found for different possible values of the shrinking parameter.A stability analysis is accomplished and apprehended in order to establish a criterion for the determinations of linearly stable and physically compatible solutions.The significant features and diversity of the modeled equations are scrutinized by recovering the previous problems of fluid flow and heat transfer from a uniformly heated sheet of variable(uniform)thickness with variable(uniform)stretching/shrinking and injection/suction velocities.展开更多
This study examines the behavior of a micropolar nanofluidflowing over a sheet in the presence of a transverse magneticfield and thermal effects.In addition,chemical(first-order homogeneous)reactions are taken into accoun...This study examines the behavior of a micropolar nanofluidflowing over a sheet in the presence of a transverse magneticfield and thermal effects.In addition,chemical(first-order homogeneous)reactions are taken into account.A similarity transformation is used to reduce the system of governing coupled non-linear partial differ-ential equations(PDEs),which account for the transport of mass,momentum,angular momentum,energy and species,to a set of non-linear ordinary differential equations(ODEs).The Runge-Kutta method along with shoot-ing method is used to solve them.The impact of several parameters is evaluated.It is shown that the micro-rota-tional velocity of thefluid rises with the micropolar factor.Moreover,the radiation parameter can have a remarkable influence on theflow and temperature profiles and on the angular momentum distribution.展开更多
Hybrid nanofluids are remarkable functioning liquids that are intended to reduce the energy loss while maximizing the heat transmission.In the involvement of suction and nonlinear thermal radiation effects,this study ...Hybrid nanofluids are remarkable functioning liquids that are intended to reduce the energy loss while maximizing the heat transmission.In the involvement of suction and nonlinear thermal radiation effects,this study attempted to explore the energy transmission features of the inclined magnetohydrodynamic(MHD)stagnation flow of CNTs-hybrid nanofluid across the nonlinear permeable stretching or shrinking sheet.This work also included some noteworthy features like chemical reactions,variable molecular diffusivity,quadratic convection,viscous dissipation,velocity slip and heat omission assessment.Employing appropriate similarity components,the model equations were modified to ODEs and computed by using the HAM technique.The impact of various relevant flow characteristics on movement,heat and concentration profiles was investigated and plotted on a graph.Considering various model factors,the significance of drag friction,heat and mass transfer rate were also computed in tabular and graphical form.This leads to the conclusion that such factors have a considerable impact on the dynamics of fluid as well as other engineering measurements of interest.Furthermore,viscous forces are dominated by increasing the values ofλ_(p),δ_(m)andδ_(q),and as a result,F(ξ)accelerates while the opposite trend is observed for M andφ.The drag friction is boosted by the augmentation M,λ_(p)andφ,but the rate of heat transfer declined.According to our findings,hybrid nanoliquid effects dominate that of ordinary nanofluid in terms of F(ξ),Θ(ξ)andφ(ξ)profiles.The HAM and the numerical technique(shooting method)were found to be in good agreement.展开更多
In the present study,we concentrate on finding the dual solutions of biomagnetic fluid namely blood flow and heat transfer along with magnetic particles over a two dimensional shrinking cylinder in the presence of a m...In the present study,we concentrate on finding the dual solutions of biomagnetic fluid namely blood flow and heat transfer along with magnetic particles over a two dimensional shrinking cylinder in the presence of a magnetic dipole.To make the results physically realistic,stability analysis is also carried out in this study so that we realized which solution is stable and which is not.The governing partial equations are converted into ordinary differential equations by using similarity transformations and the numerical solution is calculated by applying bvp4c function technique in MATLAB software.The effects of different physical parameters are plotted graphically and discussed according to the outcomes of results.From the present study we observe that ferromagnetic interaction parameter had a great influenced on fluid velocity and temperature distributions.It is also found from the current analysis that the first and second solutions of shrinking cylinder obtained only when we applied particular ranges values of suction parameter.The most important characteristics part of study is to analyze the skin friction coefficient and rate of heat transfer which also covered in this analysis.It reveals that both skin friction coefficient and rate of heat transfer are reduced with rising values of ferromagnetic number.A comparison has also been made to make the solution feasible.展开更多
In this work,a morphology transition mode is revealed in ultra-high molecular weight polyethylene(UHMWPE)when stretching at 120℃:moving from the slightly deformed region to the necked region,the morphology transfers ...In this work,a morphology transition mode is revealed in ultra-high molecular weight polyethylene(UHMWPE)when stretching at 120℃:moving from the slightly deformed region to the necked region,the morphology transfers from small spherulites to a mixture of transcrystalline and enlarged spherulites,and finally to pure transcrystalline;meanwhile,the lamellae making up the transcrystalline or spherulite were fragmented into smaller ones;spatial scan by wide-angle X-ray scattering(WAXS)and small angle X-ray scattering(SAXS)revealed that the crystallinity is increased from 25.3%to 30.1%and the crystal orientation was enhanced greatly,but the lamellae orientation was quite weak.The rise of enlarged spherulites or a mixture of transcrystalline and spherulites can also be found in UHMWPE stretched at 140 and 148℃,whereas absent in UHMWPE stretched at 30℃.In situ WAXS/SAXS measurements suggest that during stretching at 30℃,the crystallinity is reduced drastically,and a few voids are formed as the size increases from 50 nm to 210 nm;during stretching at 120℃,the crystallinity is reduced only slightly,and the kinking of lamellae occurs at large Hencky strain;during stretching at 140 and 148℃,an increase in crystallinity with stretching strain can be found,and the lamellae are also kinked.Taking the microstructure and morphology transition into consideration,a mesoscale morphology transition mode is proposed,in the stretching-induced crystallization the fragmented lamellae can be rearranged into new supra-structures such as spherulite or transcrystalline during hot stretching.展开更多
Background:Stretching has wide appeal,but there seems to exist some mismatch between its purported applications and what the evidence shows.There is compelling evidence for some stretching applications,but for others,...Background:Stretching has wide appeal,but there seems to exist some mismatch between its purported applications and what the evidence shows.There is compelling evidence for some stretching applications,but for others,the evidence seems heterogeneous or unsupportive.The discrepancies even affect some systematic reviews,possibly due to heterogeneous eligibility criteria and search strategies.This consensus paper seeks to unify the divergent findings on stretching and its implications for both athletic performance and clinical practices by delivering evidence-based recommendations.Methods:A panel of 20 experts with a blend of practical experience and scholarly knowledge was assembled.The panel meticulously reviewed existing systematic reviews,defined key terminologies(e.g.,consensus definitions for different stretching modes),and crafted guidelines using a Delphi consensus approach(minimum required agreement:80%).The analysis focused on 8 topics,including stretching's acute and chronic(long-term)effects on range of motion,strength performance,muscle hypertrophy,stiffness,injury prevention,muscle recovery,posture correction,and cardiovascular health.Results:There was consensus that chronic and acute stretching(a)improves range of motion(although alternatives exist)and(b)reduces muscle stiffness(which may not always be desirable);the panel also agreed that chronic stretching(c)may promote vascular health,but more research is warranted.In contrast,consensus was found that stretch training does not(a)contribute substantively to muscle growth,(b)serve as an allencompassing injury prevention strategy,(c)improve posture,or(d)acutely enhance post-exercise recovery.Conclusion:These recommendations provide guidance for athletes and practitioners,highlighting research gaps that should be addressed to more comprehensively understand the full scope of stretching effects.展开更多
Maximizing the efficiency of thermal engineering equipment involves minimizing entropy generation,which arises from irreversible processes.This study examines thermal transport and entropy generation in viscous flow o...Maximizing the efficiency of thermal engineering equipment involves minimizing entropy generation,which arises from irreversible processes.This study examines thermal transport and entropy generation in viscous flow over a radially stretching disk,incorporating the effects of magnetohydrodynamics(MHD),viscous dissipation,Joule heating,and radiation.Similarity transformations are used to obtain dimensionless nonlinear ordinary differential equations(ODEs)from the governing coupled partial differential equations(PDEs).The converted equations are then solved by using the BVP4C solver in MATLAB.To validate the findings,the results are compared with previously published studies under fixed parameter conditions,demonstrating strong agreement.Various key parameters are analyzed graphically to assess their impact on velocity and temperature distributions.Additionally,Bejan number and entropy generation variations are presented for different physical parameters.The injection parameter(S<0)increases the heat transfer rate,while the suction parameter(S>0)reduces it,exhibiting similar effects on fluid velocity.The magnetic parameter(M)effectively decreases entropy generation within the range of approximately 0≤η≤0.6.Beyond this interval,its influence diminishes as entropy generation values converge,with similar trends observed for the Bejan number.Furthermore,increased thermal radiation intensity is identified as a critical factor in enhancing entropy generation and the Bejan number.展开更多
Highly oriented poly(vinylidene fluoride)(PVDF)ultrathin films with differentβ-phase contents were prepared using the melt-draw method.The effect ofβ-phase content onα-βphase transition of highly oriented PVDF ult...Highly oriented poly(vinylidene fluoride)(PVDF)ultrathin films with differentβ-phase contents were prepared using the melt-draw method.The effect ofβ-phase content onα-βphase transition of highly oriented PVDF ultrathin films induced by stretching was investigated using transmission electron microscopy(TEM)and Fourier transform infrared(FTIR)spectroscopy.The results show that stretching can enhance the crystallinity and increase the average thickness of the lamellae.A fullα-βphase transition can be achieved for PVDF ultrathin films of 20.6%βphase stretched to aλ(stretching ratio)of 1.5,while fewαphases still exist for ultrathin films of 35.0%βphase,together with bent and tilted lamellae.Compared to thicker PVDF films undergoing stretching-inducedα-βphase transition,the higherα-βphase transition efficiency of the PVDF ultrathin films can be attributed to the parallel aligned lamellar structure.Moreover,a higherβ-phase content can suppressα-βphase transition because of the stress concentration effect ofβ-phase.Ultimately,these results provide valuable insights into the stretching-inducedα-βphase transition of PVDF ultrathin films.展开更多
The steady flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge with magnetic field and radiation effects are studied. The governing equations of the hybrid nanofluid are converted ...The steady flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge with magnetic field and radiation effects are studied. The governing equations of the hybrid nanofluid are converted to the similarity equations by techniques of the similarity transformation. The bvp4c function that is available in MATLAB software is utilized for solving the similarity equations numerically. The numerical results are obtained for selected different values of parameters. The results discover that two solutions exist, up to a certain value of the stretching/shrinking and suction strengths. The critical value in which the solution is in existence decreases as nanoparticle volume fractions for copper and wedge angle parameter increase. It is also found that the hybrid nanofluid enhances the heat transfer rate compared with the regular nanofluid. The reduction of the heat transfer rate is observed with the increase in radiation parameter. The temporal stability analysis is performed to analyze the stability of the dual solutions, and it is revealed that only one of them is stable and physically reliable.展开更多
The boundary layer flow of a nanofluid past a stretching/shrinking sheet with hydrodynamic and thermal slip boundary conditions is studied. Numerical solutions to the governing equations are obtained using a shooting ...The boundary layer flow of a nanofluid past a stretching/shrinking sheet with hydrodynamic and thermal slip boundary conditions is studied. Numerical solutions to the governing equations are obtained using a shooting method. The results are found for the skin friction coefficient, the local Nusselt number, and the local Sherwood number as well as the velocity, temperature, and concentration profiles for some values of the velocity slip parameter, thermal slip parameter, stretching/shrinking parameter, thermophoresis parameter, and Brownian motion parameter. The results show that the local Nusselt number, which represents the heat transfer rate, is lower for higher values of thermal slip parameter, thermophoresis parameter, and Brownian motion parameter.展开更多
In this study, a reliable algorithm to develop approximate solutions for the problem of fluid flow over a stretching or shrinking sheet is proposed. It is depicted that the differential transform method (DTM) solution...In this study, a reliable algorithm to develop approximate solutions for the problem of fluid flow over a stretching or shrinking sheet is proposed. It is depicted that the differential transform method (DTM) solutions are only valid for small values of the independent variable. The DTM solutions diverge for some differential equations that extremely have nonlinear behaviors or have boundary-conditions at infinity. For this reason the governing boundary-layer equations are solved by the Multi-step Differential Transform Method (MDTM). The main advantage of this method is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization, or perturbation. It is a semi analytical-numerical technique that formulizes Taylor series in a very different manner. By applying the MDTM the interval of convergence for the series solution is increased. The MDTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. It is predicted that the MDTM can be applied to a wide range of engineering applications.展开更多
In this analysis,the magnetohydrodynamic boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet in the presence of wall mass transfer is studied.Using similarity transformations,the governing ...In this analysis,the magnetohydrodynamic boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet in the presence of wall mass transfer is studied.Using similarity transformations,the governing equations are converted to an ordinary differential equation and then solved analytically.The introduction of a magnetic field changes the behavior of the entire flow dynamics in the shrinking sheet case and also has a major impact in the stretching sheet case.The similarity solution is always unique in the stretching case,and in the shrinking case the solution shows dual nature for certain values of the parameters.For stronger magnetic field,the similarity solution for the shrinking sheet case becomes unique.展开更多
The present study considers the magnetohydrodynamic(MHD)stagnation point flow with chemical reaction effect over a permeable stretching/shrinking sheet.The partial differential equations are reduced to a set of ordina...The present study considers the magnetohydrodynamic(MHD)stagnation point flow with chemical reaction effect over a permeable stretching/shrinking sheet.The partial differential equations are reduced to a set of ordinary differential equations using a similarity transformation.The transformed equations are then solved numerically by employing the bvp4c function available in the MATLAB software.The numerical results illustrate the effects of several parameters on the skin friction coefficient,local Nusselt number and the local Sherwood number.Dual solutions are obtained for a certain range of parameters.The temporal stability analysis is carried out to determine which one of these solutions is stable and thus physically reliable in a long run.展开更多
We investigate the dual solutions for the MHD flow of micropolar fluid over a stretching/shrinking sheet with heat transfer. Suitable relations transform the partial differential equations into the ordinary differenti...We investigate the dual solutions for the MHD flow of micropolar fluid over a stretching/shrinking sheet with heat transfer. Suitable relations transform the partial differential equations into the ordinary differential equations.Closed forms solutions are also obtained in terms of confluent hypergeometric function. This is the first attempt to determine the exact solutions for the non-linear equations of MHD micropolar fluid model. It is demonstrated that the microrotation parameter helps in increasing Nusselt number and the dual solutions exist for all fluid flow parameters under consideration. The dual behavior of dimensionless velocity, temperature, microrotation, skin-friction coefficient,local Nusselt number is displayed on graphs and examined.展开更多
The non-Newtonian fluid model reflects the behavior of the fluid flow in global manufacturing progress and increases product performance.Therefore,the present work strives to analyze the unsteady Maxwell hybrid nanofl...The non-Newtonian fluid model reflects the behavior of the fluid flow in global manufacturing progress and increases product performance.Therefore,the present work strives to analyze the unsteady Maxwell hybrid nanofluid toward a stretching/shrinking surface with thermal radiation effect and heat transfer.The partial derivatives of the multivariable differential equations are transformed into ordinary differential equations in a specified form by applying appropriate transformations.The resulting mathematical model is clarified by utilizing the bvp4c technique.Different control parameters are investigated to see how they affect the outcomes.The results reveal that the skin friction coefficient increases by adding nanoparticles and suction parameters.The inclusion of the Maxwell parameter and thermal radiation effect both show a declining tendency in the local Nusselt number,and as a result,the thermal flow efficacy is reduced.The reduction of the unsteadiness characteristic,on the other hand,considerably promotes the improvement of heat transfer performance.The existence of more than one solution is proven,and this invariably leads to an analysis of solution stability,which validates the first solution viability.展开更多
基金the Researchers Supporting Project number(RSPD2025R997),King Saud University,Riyadh,Saudi Arabia.
文摘This paper discusses the model of the boundary layer(BL)flow and the heat transfer characteristics of hybrid nanofluid(HNF)over shrinking/stretching disks.In addition,the thermal radiation and the impact of velocity and thermal slip boundary conditions are also examined.The considered hybrid nano-fluid contains silver(Ag)and iron oxide(Fe_(3)O_(4))nanoparticles dispersed in the water to prepare the Ag-Fe_(3)O_(4)/water-based hybrid nanofluid.The requisite posited partial differential equations model is converted to ordinary differential equations using similarity transformations.For a numerical solution,the shooting method in Maple is employed.Moreover,the duality in solutions is achieved for both cases of the disk(stretching(λ>0)and shrinking(λ<0)).At the same time,a unique solution is observed for λ=0.No solution is found for them at λ<λ_(c),whereas the solutions are split at the λ=λ_(c).Besides,the value of the λ_(c) is dependent on the φ_(hnf).Meanwhile,the values of f″(0)and -θ′(0)intensified with increasing φ_(hnf).Stability analysis has been applied using bvp4c in MATLAB software due to a dual solution.Furthermore,analysis shows that the first solution is stable and feasible physically.For the slip parameters,an increase in the velocity slip parameter increases the velocity and shear stress profiles while increasing the temperature profile in the first solutions.While the rise in thermal slip parameter reduces the temperature profile nanoparticle volume fractions increase it.
基金supported by the C.S.I.R.,India in the form of Junior Research Fellowship(JRF)(Grant No.09/149(0593)/2011-EMR-I)
文摘The present work is concerned with the effects of viscous dissipation and heat source/sink on a three-dimensional magnetohydrodynamic boundary layer axisymmetric stagnation flow, and the heat transfer of an electrically conducting fluid over a sheet, which shrinks or stretches axisymmetrically in its own plane where the line of the symmetry of the stagnation flow and that of the shrinking （stretching） sheet are, in general, not aligned. The governing equations are transformed into ordinary differential equations by using suitable similarity transformations and then solved numerically by a shooting technique. This investigation explores the conditions of the non-existence, existence and uniqueness of the solutions of the similar equations numerically. It is noted that the range of the velocity ratio parameter, where the similarity solution exists, is increased with the increase of the value of the magnetic parameter. Furthermore, the study reveals that the non-alignment function affects the shrinking sheet more than the stretching sheet. In addition, the numerical results of the velocity profile, temperature profile, skin-friction coefficient, and rate of heat transfer at the sheet are discussed in detail with different parameters.
文摘In this paper, the boundary layer stagnation-point slip flow and heat transfer towards a shrinking/stretching cylinder over a permeable surface is considered. The governing equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into self-similar ordinary differential equations before they are solved numerically using the shooting method. Numerical results are obtained for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles for some values of the governing parameters, namely the velocity slip parameter (α), the thermal slip parameter (β), the curvature parameter (γ) and the velocity ratio parameter (c/a). The physical quantities of interest are the skin friction coefficient and the local Nusselt number measured by f’’(0) and –θ’(0), respectively. The numerical results show that the velocity slip parameter α increases the heat transfer rate at the surface, while the thermal slip parameter β decreases it. On the other hand, increasing the velocity slip parameter α causes the decrease in the flow velocity. Further, it is found that the solutions for a shrinking cylinder (c/ac/a>0) case. Finally, it is also found that the values of f’’(0) and –θ’(0) increase as the curvature parameter γ increases.
文摘Time-dependent,two-dimensional(2 D)magnetohydrodynamic(MHD)micropolar nanomaterial flow over a shrinking/stretching surface near the stagnant point is considered.Mass and heat transfer characteristics are incorporated in the problem.A model of the partial differential expressions is altered into the forms of the ordinary differential equations via similarity transformations.The obtained equations are numerically solved by a shooting scheme in the MAPLE software.Dual solutions are observed at different values of the specified physical parameters.The stability of first and second solutions is examined through the stability analysis process.This analysis interprets that the first solution is stabilized and physically feasible while the second one is un-stable and not feasible.Furthermore,the natures of various physical factors on the drag force,skin-friction factor,and rate of mass and heat transfer are determined and interpreted.The micropolar nanofluid velocity declines with a rise in the suction and magnetic parameters,whereas it increases by increasing the unsteadiness parameter.The temperature of the micropolar nanofluid rises with increase in the Brownian motion,radiation,thermophoresis,unsteady and magnetic parameters,but it decreases against an increment in the thermal slip constraint and Prandtl number.The concentration of nanoparticles reduces against the augmented Schmidt number and Brownian movement values but rises for incremented thermophoresis parameter values.
基金LMP acknowledges financial support from ANID through Convocatoria Nacional Subvención a Instalación en la Academia Convocatoria Año 2021,Grant SA77210040。
文摘We report on the magnetohydrodynamic impact on the axisymmetric flow of Al_(2)O_(3)/Cu nanoparticles suspended in H_(2)O past a stretched/shrinked sheet.With the use of partial differential equations and the corresponding thermophysical characteristics of nanoparticles,the physical flow process is illustrated.The resultant nonlinear system of partial differential equations is converted into a system of ordinary differential equations using the suitable similarity transformations.The transformed differential equations are solved analytically.Impacts of the magnetic parameter,solid volume fraction and stretching/shrinking parameter on momentum and temperature distribution have been analyzed and interpreted graphically.The skin friction and Nusselt number were also evaluated.In addition,existence of dual solution was deduced for the shrinking sheet and unique solution for the stretching one.Further,Al_(2)O_(3)/H_(2)O nanofluid flow has better thermal conductivity on comparing with Cu/H_(2)O nanofluid.Furthermore,it was found that the first solutions of the stream are stable and physically realizable,whereas those of the second ones are unstable.
文摘The viscous fluid flow and heat transfer over a stretching(shrinking)and porous sheets of nonuniform thickness are investigated in this paper.The modeled problem is presented by utilizing the stretching(shrinking)and porous velocities and variable thickness of the sheet and they are combined in a relation.Consequently,the new problem reproduces the different available forms of flow motion and heat transfer maintained over a stretching(shrinking)and porous sheet of variable thickness in one go.As a result,the governing equations are embedded in several parameters which can be transformed into classical cases of stretched(shrunk)flows over porous sheets.A set of general,unusual and new variables is formed to simplify the governing partial differential equations and boundary conditions.The final equations are compared with the classical models to get the validity of the current simulations and they are exactly matched with each other for different choices of parameters of the current problem when their values are properly adjusted and manipulated.Moreover,we have recovered the classical results for special and appropriate values of the parameters(δ_(1),δ_(2),δ_(3),c,and B).The individual and combined effects of all inputs from the boundary are seen on flow and heat transfer properties with the help of a numerical method and the results are compared with classical solutions in special cases.It is noteworthy that the problem describes and enhances the behavior of all field quantities in view of the governing parameters.Numerical result shows that the dual solutions can be found for different possible values of the shrinking parameter.A stability analysis is accomplished and apprehended in order to establish a criterion for the determinations of linearly stable and physically compatible solutions.The significant features and diversity of the modeled equations are scrutinized by recovering the previous problems of fluid flow and heat transfer from a uniformly heated sheet of variable(uniform)thickness with variable(uniform)stretching/shrinking and injection/suction velocities.
文摘This study examines the behavior of a micropolar nanofluidflowing over a sheet in the presence of a transverse magneticfield and thermal effects.In addition,chemical(first-order homogeneous)reactions are taken into account.A similarity transformation is used to reduce the system of governing coupled non-linear partial differ-ential equations(PDEs),which account for the transport of mass,momentum,angular momentum,energy and species,to a set of non-linear ordinary differential equations(ODEs).The Runge-Kutta method along with shoot-ing method is used to solve them.The impact of several parameters is evaluated.It is shown that the micro-rota-tional velocity of thefluid rises with the micropolar factor.Moreover,the radiation parameter can have a remarkable influence on theflow and temperature profiles and on the angular momentum distribution.
基金funded by King Mongkut’s University of Technology North Bangkok with Contract no.KMUTNB-Post-65-07。
文摘Hybrid nanofluids are remarkable functioning liquids that are intended to reduce the energy loss while maximizing the heat transmission.In the involvement of suction and nonlinear thermal radiation effects,this study attempted to explore the energy transmission features of the inclined magnetohydrodynamic(MHD)stagnation flow of CNTs-hybrid nanofluid across the nonlinear permeable stretching or shrinking sheet.This work also included some noteworthy features like chemical reactions,variable molecular diffusivity,quadratic convection,viscous dissipation,velocity slip and heat omission assessment.Employing appropriate similarity components,the model equations were modified to ODEs and computed by using the HAM technique.The impact of various relevant flow characteristics on movement,heat and concentration profiles was investigated and plotted on a graph.Considering various model factors,the significance of drag friction,heat and mass transfer rate were also computed in tabular and graphical form.This leads to the conclusion that such factors have a considerable impact on the dynamics of fluid as well as other engineering measurements of interest.Furthermore,viscous forces are dominated by increasing the values ofλ_(p),δ_(m)andδ_(q),and as a result,F(ξ)accelerates while the opposite trend is observed for M andφ.The drag friction is boosted by the augmentation M,λ_(p)andφ,but the rate of heat transfer declined.According to our findings,hybrid nanoliquid effects dominate that of ordinary nanofluid in terms of F(ξ),Θ(ξ)andφ(ξ)profiles.The HAM and the numerical technique(shooting method)were found to be in good agreement.
文摘In the present study,we concentrate on finding the dual solutions of biomagnetic fluid namely blood flow and heat transfer along with magnetic particles over a two dimensional shrinking cylinder in the presence of a magnetic dipole.To make the results physically realistic,stability analysis is also carried out in this study so that we realized which solution is stable and which is not.The governing partial equations are converted into ordinary differential equations by using similarity transformations and the numerical solution is calculated by applying bvp4c function technique in MATLAB software.The effects of different physical parameters are plotted graphically and discussed according to the outcomes of results.From the present study we observe that ferromagnetic interaction parameter had a great influenced on fluid velocity and temperature distributions.It is also found from the current analysis that the first and second solutions of shrinking cylinder obtained only when we applied particular ranges values of suction parameter.The most important characteristics part of study is to analyze the skin friction coefficient and rate of heat transfer which also covered in this analysis.It reveals that both skin friction coefficient and rate of heat transfer are reduced with rising values of ferromagnetic number.A comparison has also been made to make the solution feasible.
基金supported by the National Natural Science Foundation of China(Nos.52003249,12072325 and 52273027)the Natural Science Foundation of Henan(No.242300421236).
文摘In this work,a morphology transition mode is revealed in ultra-high molecular weight polyethylene(UHMWPE)when stretching at 120℃:moving from the slightly deformed region to the necked region,the morphology transfers from small spherulites to a mixture of transcrystalline and enlarged spherulites,and finally to pure transcrystalline;meanwhile,the lamellae making up the transcrystalline or spherulite were fragmented into smaller ones;spatial scan by wide-angle X-ray scattering(WAXS)and small angle X-ray scattering(SAXS)revealed that the crystallinity is increased from 25.3%to 30.1%and the crystal orientation was enhanced greatly,but the lamellae orientation was quite weak.The rise of enlarged spherulites or a mixture of transcrystalline and spherulites can also be found in UHMWPE stretched at 140 and 148℃,whereas absent in UHMWPE stretched at 30℃.In situ WAXS/SAXS measurements suggest that during stretching at 30℃,the crystallinity is reduced drastically,and a few voids are formed as the size increases from 50 nm to 210 nm;during stretching at 120℃,the crystallinity is reduced only slightly,and the kinking of lamellae occurs at large Hencky strain;during stretching at 140 and 148℃,an increase in crystallinity with stretching strain can be found,and the lamellae are also kinked.Taking the microstructure and morphology transition into consideration,a mesoscale morphology transition mode is proposed,in the stretching-induced crystallization the fragmented lamellae can be rearranged into new supra-structures such as spherulite or transcrystalline during hot stretching.
文摘Background:Stretching has wide appeal,but there seems to exist some mismatch between its purported applications and what the evidence shows.There is compelling evidence for some stretching applications,but for others,the evidence seems heterogeneous or unsupportive.The discrepancies even affect some systematic reviews,possibly due to heterogeneous eligibility criteria and search strategies.This consensus paper seeks to unify the divergent findings on stretching and its implications for both athletic performance and clinical practices by delivering evidence-based recommendations.Methods:A panel of 20 experts with a blend of practical experience and scholarly knowledge was assembled.The panel meticulously reviewed existing systematic reviews,defined key terminologies(e.g.,consensus definitions for different stretching modes),and crafted guidelines using a Delphi consensus approach(minimum required agreement:80%).The analysis focused on 8 topics,including stretching's acute and chronic(long-term)effects on range of motion,strength performance,muscle hypertrophy,stiffness,injury prevention,muscle recovery,posture correction,and cardiovascular health.Results:There was consensus that chronic and acute stretching(a)improves range of motion(although alternatives exist)and(b)reduces muscle stiffness(which may not always be desirable);the panel also agreed that chronic stretching(c)may promote vascular health,but more research is warranted.In contrast,consensus was found that stretch training does not(a)contribute substantively to muscle growth,(b)serve as an allencompassing injury prevention strategy,(c)improve posture,or(d)acutely enhance post-exercise recovery.Conclusion:These recommendations provide guidance for athletes and practitioners,highlighting research gaps that should be addressed to more comprehensively understand the full scope of stretching effects.
文摘Maximizing the efficiency of thermal engineering equipment involves minimizing entropy generation,which arises from irreversible processes.This study examines thermal transport and entropy generation in viscous flow over a radially stretching disk,incorporating the effects of magnetohydrodynamics(MHD),viscous dissipation,Joule heating,and radiation.Similarity transformations are used to obtain dimensionless nonlinear ordinary differential equations(ODEs)from the governing coupled partial differential equations(PDEs).The converted equations are then solved by using the BVP4C solver in MATLAB.To validate the findings,the results are compared with previously published studies under fixed parameter conditions,demonstrating strong agreement.Various key parameters are analyzed graphically to assess their impact on velocity and temperature distributions.Additionally,Bejan number and entropy generation variations are presented for different physical parameters.The injection parameter(S<0)increases the heat transfer rate,while the suction parameter(S>0)reduces it,exhibiting similar effects on fluid velocity.The magnetic parameter(M)effectively decreases entropy generation within the range of approximately 0≤η≤0.6.Beyond this interval,its influence diminishes as entropy generation values converge,with similar trends observed for the Bejan number.Furthermore,increased thermal radiation intensity is identified as a critical factor in enhancing entropy generation and the Bejan number.
基金financially supported from the National Natural Science Foundation of China(Nos.52203026 and 52027804)。
文摘Highly oriented poly(vinylidene fluoride)(PVDF)ultrathin films with differentβ-phase contents were prepared using the melt-draw method.The effect ofβ-phase content onα-βphase transition of highly oriented PVDF ultrathin films induced by stretching was investigated using transmission electron microscopy(TEM)and Fourier transform infrared(FTIR)spectroscopy.The results show that stretching can enhance the crystallinity and increase the average thickness of the lamellae.A fullα-βphase transition can be achieved for PVDF ultrathin films of 20.6%βphase stretched to aλ(stretching ratio)of 1.5,while fewαphases still exist for ultrathin films of 35.0%βphase,together with bent and tilted lamellae.Compared to thicker PVDF films undergoing stretching-inducedα-βphase transition,the higherα-βphase transition efficiency of the PVDF ultrathin films can be attributed to the parallel aligned lamellar structure.Moreover,a higherβ-phase content can suppressα-βphase transition because of the stress concentration effect ofβ-phase.Ultimately,these results provide valuable insights into the stretching-inducedα-βphase transition of PVDF ultrathin films.
基金the Ministry of Education of Malaysia(No.FRGS/1/2019/STG06/UKM/01/4)Ministry of Science of Romania(No.PN-III-P4-ID-PCE-2016-0036)。
文摘The steady flow and heat transfer of a hybrid nanofluid past a permeable stretching/shrinking wedge with magnetic field and radiation effects are studied. The governing equations of the hybrid nanofluid are converted to the similarity equations by techniques of the similarity transformation. The bvp4c function that is available in MATLAB software is utilized for solving the similarity equations numerically. The numerical results are obtained for selected different values of parameters. The results discover that two solutions exist, up to a certain value of the stretching/shrinking and suction strengths. The critical value in which the solution is in existence decreases as nanoparticle volume fractions for copper and wedge angle parameter increase. It is also found that the hybrid nanofluid enhances the heat transfer rate compared with the regular nanofluid. The reduction of the heat transfer rate is observed with the increase in radiation parameter. The temporal stability analysis is performed to analyze the stability of the dual solutions, and it is revealed that only one of them is stable and physically reliable.
基金Project supported by the Ministry of Higher Education in Malaysia(No.FRGS/1/2012/SG04/UKM/2001/1)the Universiti Kebangsaan Malaysia(No.DIP-2012-31)
文摘The boundary layer flow of a nanofluid past a stretching/shrinking sheet with hydrodynamic and thermal slip boundary conditions is studied. Numerical solutions to the governing equations are obtained using a shooting method. The results are found for the skin friction coefficient, the local Nusselt number, and the local Sherwood number as well as the velocity, temperature, and concentration profiles for some values of the velocity slip parameter, thermal slip parameter, stretching/shrinking parameter, thermophoresis parameter, and Brownian motion parameter. The results show that the local Nusselt number, which represents the heat transfer rate, is lower for higher values of thermal slip parameter, thermophoresis parameter, and Brownian motion parameter.
文摘In this study, a reliable algorithm to develop approximate solutions for the problem of fluid flow over a stretching or shrinking sheet is proposed. It is depicted that the differential transform method (DTM) solutions are only valid for small values of the independent variable. The DTM solutions diverge for some differential equations that extremely have nonlinear behaviors or have boundary-conditions at infinity. For this reason the governing boundary-layer equations are solved by the Multi-step Differential Transform Method (MDTM). The main advantage of this method is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization, or perturbation. It is a semi analytical-numerical technique that formulizes Taylor series in a very different manner. By applying the MDTM the interval of convergence for the series solution is increased. The MDTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. It is predicted that the MDTM can be applied to a wide range of engineering applications.
基金the financial support of National Board forHigher Mathematics (NBHM),DAE,Mumbai,India for pursuing this workThe research of A. Alsaedi is partially supported by the Deanship of Scientific Research (DSR),King Abdulaziz University,Jeddah,Saudi Arabia
文摘In this analysis,the magnetohydrodynamic boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet in the presence of wall mass transfer is studied.Using similarity transformations,the governing equations are converted to an ordinary differential equation and then solved analytically.The introduction of a magnetic field changes the behavior of the entire flow dynamics in the shrinking sheet case and also has a major impact in the stretching sheet case.The similarity solution is always unique in the stretching case,and in the shrinking case the solution shows dual nature for certain values of the parameters.For stronger magnetic field,the similarity solution for the shrinking sheet case becomes unique.
文摘The present study considers the magnetohydrodynamic(MHD)stagnation point flow with chemical reaction effect over a permeable stretching/shrinking sheet.The partial differential equations are reduced to a set of ordinary differential equations using a similarity transformation.The transformed equations are then solved numerically by employing the bvp4c function available in the MATLAB software.The numerical results illustrate the effects of several parameters on the skin friction coefficient,local Nusselt number and the local Sherwood number.Dual solutions are obtained for a certain range of parameters.The temporal stability analysis is carried out to determine which one of these solutions is stable and thus physically reliable in a long run.
文摘We investigate the dual solutions for the MHD flow of micropolar fluid over a stretching/shrinking sheet with heat transfer. Suitable relations transform the partial differential equations into the ordinary differential equations.Closed forms solutions are also obtained in terms of confluent hypergeometric function. This is the first attempt to determine the exact solutions for the non-linear equations of MHD micropolar fluid model. It is demonstrated that the microrotation parameter helps in increasing Nusselt number and the dual solutions exist for all fluid flow parameters under consideration. The dual behavior of dimensionless velocity, temperature, microrotation, skin-friction coefficient,local Nusselt number is displayed on graphs and examined.
基金the Research Grant of University Kebangsaan Malaysia(No.GUP-2019-034)。
文摘The non-Newtonian fluid model reflects the behavior of the fluid flow in global manufacturing progress and increases product performance.Therefore,the present work strives to analyze the unsteady Maxwell hybrid nanofluid toward a stretching/shrinking surface with thermal radiation effect and heat transfer.The partial derivatives of the multivariable differential equations are transformed into ordinary differential equations in a specified form by applying appropriate transformations.The resulting mathematical model is clarified by utilizing the bvp4c technique.Different control parameters are investigated to see how they affect the outcomes.The results reveal that the skin friction coefficient increases by adding nanoparticles and suction parameters.The inclusion of the Maxwell parameter and thermal radiation effect both show a declining tendency in the local Nusselt number,and as a result,the thermal flow efficacy is reduced.The reduction of the unsteadiness characteristic,on the other hand,considerably promotes the improvement of heat transfer performance.The existence of more than one solution is proven,and this invariably leads to an analysis of solution stability,which validates the first solution viability.
