This study investigates the flow and heat transfer of dusty Williamson (MHD) Nanofluid flow over a stretching permeable cylinder in a porous medium. Dusty Williamson Nanofluid was considered due to its thermal propert...This study investigates the flow and heat transfer of dusty Williamson (MHD) Nanofluid flow over a stretching permeable cylinder in a porous medium. Dusty Williamson Nanofluid was considered due to its thermal properties and potential benefits of increasing the heat transfer rate. Firstly, partial differential equations are transformed into coupled non-linear ordinary differential equations through a similarity variables transformation. The resulting set of dimensionless equations is solved analytically by using the Homogony Perturbation Method (HPM). The effects of the emerging parameters on the velocity and temperature profiles as well as skin-friction coefficient and Nusselt number are publicized through tables and graphs with appropriate discussions. The present result has been compared with published papers and found to be in agreement. To the best of author’s knowledge, there has been sparse research work in the literature that considers the effect of dust with Williamson Nanofluid and also solving the problem analytically. Therefore to the best of author’s knowledge, this is the first time analytical solution has been established for the problem. The results revealed that the fluid velocity of both the fluid and dust phases decreases as the Williamson parameter increases. Motivated by the above limitations and the gaps in past works, therefore, it is hoped that the present work will assist in providing accurate solutions to many practical problems in science, industry and engineering.展开更多
Amphiphilic graft copolymers having poly（phthalazinone ether sulfone ketone）（PPESK） backbones and poly（ethylene glycol） side chains were synthesized through a modified Williamson ether method.Initially,chloromet...Amphiphilic graft copolymers having poly（phthalazinone ether sulfone ketone）（PPESK） backbones and poly（ethylene glycol） side chains were synthesized through a modified Williamson ether method.Initially,chloromethylated PPESK（CMPPESK） was prepared using chloromethylether as chloromethylation agent.Then,PPESK-g-PEGs were synthesized via the coupling of CMPPESK with alkoxides formed from PEG mono methyl ether（PEG-OH） and sodium hydride.Two PEG-OHs with the average molecular weight of 350 g/mol and 750 g/mol,respectively were used as the grafting reactants.The grafting of PEG-OHs onto PPESK backbones was confirmed by FT-IR,1H-NMR and solid13C CP-MAS NMR.The grafting degrees of PEG-OH were calculated from1H-NMR analysis,and the obtained results were approximately consistent with those from TGA analysis.The resultant PPESK-g-PEG copolymers were hydrophilic,but water insoluble,rendering them potential candidates as surface modifying additive for PPESK porous membranes.展开更多
This study explores the effects of heat transfer on the Williamson fluid over a porous exponentially stretching surface. The boundary layer equations of the Williamson fluid model for two dimensional flow with heat tr...This study explores the effects of heat transfer on the Williamson fluid over a porous exponentially stretching surface. The boundary layer equations of the Williamson fluid model for two dimensional flow with heat transfer are presented. Two cases of heat transfer are considered, i.e., the prescribed exponential order surface temperature （PEST） case and the prescribed exponential order heat flux （PEHF） case. The highly nonlinear partial differential equations are simplified with suitable similar and non-similar variables, and finally are solved analytically with the help of the optimal homotopy analysis method （OHAM）. The optimal convergence control parameters are obtained, and the physical fea- tures of the flow parameters are analyzed through graphs and tables. The skin friction and wall temperature gradient are calculated.展开更多
This research elaborates magnetohydrodynamics (MHD) impact on non-Newtonian (Williamson) fluid flow by stretchable rotating disks.Both disks are rotating with different angular velocities and different stretching rate...This research elaborates magnetohydrodynamics (MHD) impact on non-Newtonian (Williamson) fluid flow by stretchable rotating disks.Both disks are rotating with different angular velocities and different stretching rates.Viscous dissipation aspect is considered for energy expression formulation.Entropy generation analysis is described via implementation of thermodynamic second law.Chemical processes (heterogeneous and homogeneous) subjected to entropy generation are introduced first time in literature.Boundary-layer approach is employed for modeling.Apposite variables are introduced for non-dimensionalization of governing systems.Homotopy procedure yields convergence of solutions subjected to computations of highly nonlinear expressions.The significant characteristics of sundry factors against thermal,velocity and solutal fields are described graphically.Besides,tabular results are addressed for engineering quantities (skin-friction coefficient,Nusselt number).The outcomes certify an increment in temperature distribution for Weissenberg (We) and Eckert (Ec) numbers.展开更多
The intention of this investigation is to study the effects of heat transfer and inclined magnetic field on the peristaltic flow of Williamson fluid in an asymmetric channel through porous medium. The governing two-di...The intention of this investigation is to study the effects of heat transfer and inclined magnetic field on the peristaltic flow of Williamson fluid in an asymmetric channel through porous medium. The governing two-dimensional equations are simplified under the assumption of long wavelength approximation. The simplified equations are solved for the stream function, temperature, and axial pressure gradient by using a regular perturbation method. The expression for pressure rise is computed numerically. The profiles of velocity, pressure gradient, temperature, heat transfer coefficient and stream function are sketched and interpreted for various embedded parameters and also the behavior of stream function for various wave forms is discussed through graphs. It is observed that the peristaltic velocity increases from porous medium to non-porous medium, the magnetic effects have increasing effect on the temperature, and the size of the trapped bolus decreases with the increasing of magnetic effects while the trend is reversed with the increasing of Darcy number. Moreover, limiting solutions of our problem are in close agreement with the corresponding results of the Newtonian fluid model.展开更多
文摘This study investigates the flow and heat transfer of dusty Williamson (MHD) Nanofluid flow over a stretching permeable cylinder in a porous medium. Dusty Williamson Nanofluid was considered due to its thermal properties and potential benefits of increasing the heat transfer rate. Firstly, partial differential equations are transformed into coupled non-linear ordinary differential equations through a similarity variables transformation. The resulting set of dimensionless equations is solved analytically by using the Homogony Perturbation Method (HPM). The effects of the emerging parameters on the velocity and temperature profiles as well as skin-friction coefficient and Nusselt number are publicized through tables and graphs with appropriate discussions. The present result has been compared with published papers and found to be in agreement. To the best of author’s knowledge, there has been sparse research work in the literature that considers the effect of dust with Williamson Nanofluid and also solving the problem analytically. Therefore to the best of author’s knowledge, this is the first time analytical solution has been established for the problem. The results revealed that the fluid velocity of both the fluid and dust phases decreases as the Williamson parameter increases. Motivated by the above limitations and the gaps in past works, therefore, it is hoped that the present work will assist in providing accurate solutions to many practical problems in science, industry and engineering.
文摘Amphiphilic graft copolymers having poly（phthalazinone ether sulfone ketone）（PPESK） backbones and poly（ethylene glycol） side chains were synthesized through a modified Williamson ether method.Initially,chloromethylated PPESK（CMPPESK） was prepared using chloromethylether as chloromethylation agent.Then,PPESK-g-PEGs were synthesized via the coupling of CMPPESK with alkoxides formed from PEG mono methyl ether（PEG-OH） and sodium hydride.Two PEG-OHs with the average molecular weight of 350 g/mol and 750 g/mol,respectively were used as the grafting reactants.The grafting of PEG-OHs onto PPESK backbones was confirmed by FT-IR,1H-NMR and solid13C CP-MAS NMR.The grafting degrees of PEG-OH were calculated from1H-NMR analysis,and the obtained results were approximately consistent with those from TGA analysis.The resultant PPESK-g-PEG copolymers were hydrophilic,but water insoluble,rendering them potential candidates as surface modifying additive for PPESK porous membranes.
基金supported by the Ph.D.Indigenous Scheme of the Higher Education Commission of Pakistan(No.112-21674-2PS1-576)
文摘This study explores the effects of heat transfer on the Williamson fluid over a porous exponentially stretching surface. The boundary layer equations of the Williamson fluid model for two dimensional flow with heat transfer are presented. Two cases of heat transfer are considered, i.e., the prescribed exponential order surface temperature （PEST） case and the prescribed exponential order heat flux （PEHF） case. The highly nonlinear partial differential equations are simplified with suitable similar and non-similar variables, and finally are solved analytically with the help of the optimal homotopy analysis method （OHAM）. The optimal convergence control parameters are obtained, and the physical fea- tures of the flow parameters are analyzed through graphs and tables. The skin friction and wall temperature gradient are calculated.
文摘This research elaborates magnetohydrodynamics (MHD) impact on non-Newtonian (Williamson) fluid flow by stretchable rotating disks.Both disks are rotating with different angular velocities and different stretching rates.Viscous dissipation aspect is considered for energy expression formulation.Entropy generation analysis is described via implementation of thermodynamic second law.Chemical processes (heterogeneous and homogeneous) subjected to entropy generation are introduced first time in literature.Boundary-layer approach is employed for modeling.Apposite variables are introduced for non-dimensionalization of governing systems.Homotopy procedure yields convergence of solutions subjected to computations of highly nonlinear expressions.The significant characteristics of sundry factors against thermal,velocity and solutal fields are described graphically.Besides,tabular results are addressed for engineering quantities (skin-friction coefficient,Nusselt number).The outcomes certify an increment in temperature distribution for Weissenberg (We) and Eckert (Ec) numbers.
文摘The intention of this investigation is to study the effects of heat transfer and inclined magnetic field on the peristaltic flow of Williamson fluid in an asymmetric channel through porous medium. The governing two-dimensional equations are simplified under the assumption of long wavelength approximation. The simplified equations are solved for the stream function, temperature, and axial pressure gradient by using a regular perturbation method. The expression for pressure rise is computed numerically. The profiles of velocity, pressure gradient, temperature, heat transfer coefficient and stream function are sketched and interpreted for various embedded parameters and also the behavior of stream function for various wave forms is discussed through graphs. It is observed that the peristaltic velocity increases from porous medium to non-porous medium, the magnetic effects have increasing effect on the temperature, and the size of the trapped bolus decreases with the increasing of magnetic effects while the trend is reversed with the increasing of Darcy number. Moreover, limiting solutions of our problem are in close agreement with the corresponding results of the Newtonian fluid model.
