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
The effects of axisymmetric flow of a Powell-Eyring fluid over an impermeable radially stretching surface are presented. Characteristics of the heat transfer process are analyzed with a more realistic condition named ...The effects of axisymmetric flow of a Powell-Eyring fluid over an impermeable radially stretching surface are presented. Characteristics of the heat transfer process are analyzed with a more realistic condition named the convective boundary condition. Governing equations for the flow problem are derived by the boundary layer approximations. The modeled highly coupled partial differential system is converted into a system of ordinary differential equations with acceptable similarity transformations. The convergent series solutions for the resulting system are constructed and analyzed. Optimal values are obtained and presented in a numerical form using an optimal homotopy analysis method （OHAM）. The rheological characteristics of different parameters of the velocity and temperature profiles are presented graphically. Tabular variations of the skin friction coefficient and the Nusselt number are also calculated. It is observed that the temperature distribution shows opposite behavior for Prandtl and Biot numbers. Furthermore, the rate of heating/cooling is higher for both the Prandtl and Biot numbers.展开更多
The aim of this research is the improvement towards the consumption of energy in the field of engineering and industry. The efforts have been paid to the enhancement of heat transmission and cooling process through a ...The aim of this research is the improvement towards the consumption of energy in the field of engineering and industry. The efforts have been paid to the enhancement of heat transmission and cooling process through a nanofluid coating of a nonlinear stretching disc. The combination of Water(H2 O) and multiple walled carbon nanotubes(MWCNT)/single walled carbon nanotubes(SWCNT) have been used as a nanofluid. The spreading of a thin nano-layer with variable thickness over a nonlinear and radially stretching surface has been considered. The estimated results of the problem have been accomplished using the Optimal Homotopy Analysis Method(OHAM). The residual errors of the OHAM method have been shown physically and numerically. The important physical parameters of skin friction and Nusselt number have been calculated and discussed. The other embedding parameters like generalized magnetic parameter, Prantl number, nanofluid volume fraction and Eckert number have been intended and discussed. The obtained results have been compared with the Numerical(ND-Solve) method for both sorts of CNTs. The closed agreement of both methods has been achieved.展开更多
The impact of the Marangoni convection over the thin film flow on an expanding cylinder has been examined in this study. The diverse effect of the embedded constraints has been detected during the liquid film flow. It...The impact of the Marangoni convection over the thin film flow on an expanding cylinder has been examined in this study. The diverse effect of the embedded constraints has been detected during the liquid film flow. It has been examined that the behavior of the physical parameters altered after the small intervals and diverse from the traditional approach. The similarity variables have been utilized to alter the basic flow equations into the nonlinear ordinary differential equations. The result of the transformed equations is computed by BVPh 2.0 package. The performance of different constraints, for flow motion and temperature distributions are plotted and conferred. It has been observed that under the Marangoni convection the impact of the physical parameters varies after the point of inflection and the diverse impact of the embedding constraints provide space for the variation of the point of inflection for the desired spray analysis.展开更多
In this contemporary study,theoritical investigation of nanofluidic model is thought-out.Two-dimensional nanomaterials based mixed flow is considered here.Convective solar radiative heat transport properties have been...In this contemporary study,theoritical investigation of nanofluidic model is thought-out.Two-dimensional nanomaterials based mixed flow is considered here.Convective solar radiative heat transport properties have been investigated over a nonlinearly stretched wall in the presence of magneto-hydrodynamic(MHD),by innovative application of semi analytical“optimal homotopy asymptotic method(OHAM)”.OHAM does not require any discretization,linearization and small parameter assumption.OHAM describes extremely precise 1^(st)/2^(nd) order solutions without the need of computing further higher order terms,therefore,fast convergence is observed.Nanofluidic governing model is transformed into system of ordinary differential equations(ODEs)by exploitation of similarity transformation.To study the significance of radiation parameter alongwith thermophoresis parameter,a semi analytical solver is applied to the transformed system.In this work,Brownianmotion,influence ofmagnetic field,Lewis number,Prandtl number,Eckert number and Biot number have investigated on velocity,temperature and nanoparticle concentration profiles.The study provides sufficient number of graphical representations to demonstrate the inspiration of mentioned parameters.展开更多
The steady, laminar, incompressible and two dimensional micropolar flow between two porous disks was investigated using optimal homotopy asymptotic method(OHAM) and fourth order Runge–Kutta numerical method. Comparis...The steady, laminar, incompressible and two dimensional micropolar flow between two porous disks was investigated using optimal homotopy asymptotic method(OHAM) and fourth order Runge–Kutta numerical method. Comparison between OHAM and numerical method shows that OHAM is an exact and high efficient method for solving these kinds of problems. The results are presented to study the velocity and rotation profiles for different physical parameters such as Reynolds number, vortex viscosity parameter, spin gradient viscosity and microinertia density parameter. As an important outcome, the magnitude of the microrotation increases with an increase in the values of injection velocity while it decreases by increasing the values of suction velocity.展开更多
In this paper,the magnetohydrodynamic 3 D flow of Prandtl nanoliquid subject to convectively heated extendable surface has been discussed.A linear stretching surface makes the flow.Thermophoretic and Brownian motion i...In this paper,the magnetohydrodynamic 3 D flow of Prandtl nanoliquid subject to convectively heated extendable surface has been discussed.A linear stretching surface makes the flow.Thermophoretic and Brownian motion impacts are explored.Heat transfer for convective procedure is considered.Prandtl liquid is taken electrically conducted through applied magnetic field.Suitable non-dimensional variables lead to strong nonlinear ordinary differential system.The obtained nonlinear differential systems are solved through optimal homotopic technique.Physical quantities like skin friction coefficients and Nusselt number are explored via plots.It is observed that effects of Hartman parameter and Biot number on temperature and concentration are quite similar.Both temperature and concentration are enhanced for larger values of Hartman parameter and Biot number.展开更多
In the present article Optimal Homotopy Asymptotic Method(OHAM)is used to obtain the solutions of momentum and heat transfer equations of non-Newtonian fluid flow in an axisymmetric channel with porous wall for turbin...In the present article Optimal Homotopy Asymptotic Method(OHAM)is used to obtain the solutions of momentum and heat transfer equations of non-Newtonian fluid flow in an axisymmetric channel with porous wall for turbine cooling applications.Numerical method is used for validity of this analytical method and excellent agreement is observed between the solutions obtained from OHAM and numerical results.Trusting to this validity,effects of some other parameters are discussed.The results show that Nusselt number increases with increase of Reynolds number,Prandtl number and power law index.展开更多
The viscous dissipation and heat transfer in the Darcy-Forchheimer flow by a rotating disk are examined. The partial slip conditions are invoked. The optimal series solutions are computed via the optimal homotopic ana...The viscous dissipation and heat transfer in the Darcy-Forchheimer flow by a rotating disk are examined. The partial slip conditions are invoked. The optimal series solutions are computed via the optimal homotopic analysis method(OHAM). The thermophoresis and Brownian motions are studied. The Darcy-Forchheimer relation characterizes the porous space. The roles of influential variables on the physical quantities are graphically examined. A reduction in the local Nusselt number is observed through thermophoresis and thermal slip parameters. The local Sherwood number depicts an increasing trend for the higher Brownian motion and concentration slip parameters.展开更多
In recent years,the emergence of nanotechnology experienced incredible development in the field of medical sciences.During the past decade,investigating the characteristics of nanoparticles during fluid flow has been ...In recent years,the emergence of nanotechnology experienced incredible development in the field of medical sciences.During the past decade,investigating the characteristics of nanoparticles during fluid flow has been one of the intriguing issues.Nanoparticle distribution and uniformity have emerged as substantial criteria in both medical and engineering applications.Adverse effects of chemotherapy on healthy tissues are known to be a significant concern during cancer therapy.A novel treatment method of magnetic drug targeting(MDT)has emerged as a promising topical cancer treatment along with some attractive advantages of improving efficacy,fewer side effects,and reduce drug dose.During magnetic drug targeting,the appropriate movement of nanoparticles(magnetic)as carriers is essential for the therapeutic process in the blood clot removal,infection treatment,and tumor cell treatment.In this study,we have numerically investigated the behavior of an unsteady blood flowinfused with magnetic nanoparticles during MDT under the influence of a uniform external magnetic field in a microtube.An optimal homotopy asymptotic method(OHAM)is employed to compute the governing equation for unsteady electromagnetohydrodynamics flow.The influence of Hartmann number(Ha),particle mass parameter(G),particle concentration parameter(R),and electro-osmotic parameter(k)is investigated on the velocity of magnetic nanoparticles and blood flow.Results obtained show that the electro-osmotic parameter,along with Hartmann’s number,dramatically affects the velocity of magnetic nanoparticles,blood flow velocity,and flow rate.Moreover,results also reveal that at a higher Hartman number,homogeneity in nanoparticles distribution improved considerably.The particle concentration andmass parameters effectively influence the capturing effect on nanoparticles in the blood flow using a micro-tube for magnetic drug targeting.Lastly,investigation also indicates that the OHAM analysis is efficient and quick to handle the system of nonlinear equations.展开更多
The melting phenomenon in two-dimensional(2 D)flow of fourth-grade material over a stretching surface is explored.The flow is created via a stretching surface.A Darcy-Forchheimer(D-F)porous medium is considered in the...The melting phenomenon in two-dimensional(2 D)flow of fourth-grade material over a stretching surface is explored.The flow is created via a stretching surface.A Darcy-Forchheimer(D-F)porous medium is considered in the flow field.The heat transport is examined with the existence of the Cattaneo-Christov(C-C)heat flux.The fourth-grade material is electrically conducting subject to an applied magnetic field.The governing partial differential equations(PDEs)are reduced into ordinary differential equations(ODEs)by appropriate transformations.The solutions are constructed analytically through the optimal homotopy analysis method(OHAM).The fluid velocity,temperature,and skin friction are examined under the effects of various involved parameters.The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter,porosity parameter,and Forchheimer number.The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number,magnetic parameter,and thermal relaxation parameter.Furthermore,the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters,thermal relaxation parameter,and Forchheimer number.展开更多
An analysis of the mixed convective flow of viscous fluids induced by a nonlinear inclined stretching surface is addressed.Heat and mass transfer phenomena are analyzed with additional effects of heat generation/absor...An analysis of the mixed convective flow of viscous fluids induced by a nonlinear inclined stretching surface is addressed.Heat and mass transfer phenomena are analyzed with additional effects of heat generation/absorption and activation energy,respectively.The nonlinear Darcy-Forchheimer relation is deliberated.The dimensionless problem is obtained through appropriate transformations.Convergent series solutions are obtained by utilizing an optimal homotopic analysis method(OHAM).Graphs depicting the consequence of influential variables on physical quantities are presented.Enhancement in the velocity is observed through the local mixed convection parameter while an opposite trend of the concentration field is noted for the chemical reaction rate parameter.展开更多
This article studies the unsteady thin film flow of a fourth grade fluid over a moving and oscillating vertical belt.The problem is modeled in terms of non-nonlinear partial differential equations with some physical c...This article studies the unsteady thin film flow of a fourth grade fluid over a moving and oscillating vertical belt.The problem is modeled in terms of non-nonlinear partial differential equations with some physical conditions.Both problems of lift and drainage are studied.Two different techniques namely the adomian decomposition method(ADM)and the optimal homotopy asymptotic method(OHAM)are used for finding the analytical solutions.These solutions are compared and found in excellent agreement.For the physical analysis of the problem,graphical results are provided and discussed for various embedded flow parameters.展开更多
Laminar,isothermal,incompressible and viscous flow in a rectangular domain bounded by two moving porous walls,w hich enable the fuid to enter or exit during successive expansions or contractions is investigated analyt...Laminar,isothermal,incompressible and viscous flow in a rectangular domain bounded by two moving porous walls,w hich enable the fuid to enter or exit during successive expansions or contractions is investigated analytically using optimal homotopy asymptotic method(OHAM).OHAM is a powerful method for solving nonlinear problems without depending to the small parameter.The concept of this method is briefly introduced,and it's application for this problem is studied.Then,the results are compared with numerical results and the validity of these methods is shown.After this verification,we analyze the effects of some physical applicable parameters to show the efficiency of OHAM for this type of problems.Graphical results are presented to investigate the influence of the non-dimensional wall dilation rate(a)and pemeation Reynolds number(Re)on the velocity,normal pressure distribution and wall shear stress.The present problem for slowly expanding or contracting walls with weak permeability is a simple model for the transport of biological fuids through contracting or expanding vessels.展开更多
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The effects of axisymmetric flow of a Powell-Eyring fluid over an impermeable radially stretching surface are presented. Characteristics of the heat transfer process are analyzed with a more realistic condition named the convective boundary condition. Governing equations for the flow problem are derived by the boundary layer approximations. The modeled highly coupled partial differential system is converted into a system of ordinary differential equations with acceptable similarity transformations. The convergent series solutions for the resulting system are constructed and analyzed. Optimal values are obtained and presented in a numerical form using an optimal homotopy analysis method （OHAM）. The rheological characteristics of different parameters of the velocity and temperature profiles are presented graphically. Tabular variations of the skin friction coefficient and the Nusselt number are also calculated. It is observed that the temperature distribution shows opposite behavior for Prandtl and Biot numbers. Furthermore, the rate of heating/cooling is higher for both the Prandtl and Biot numbers.
文摘The aim of this research is the improvement towards the consumption of energy in the field of engineering and industry. The efforts have been paid to the enhancement of heat transmission and cooling process through a nanofluid coating of a nonlinear stretching disc. The combination of Water(H2 O) and multiple walled carbon nanotubes(MWCNT)/single walled carbon nanotubes(SWCNT) have been used as a nanofluid. The spreading of a thin nano-layer with variable thickness over a nonlinear and radially stretching surface has been considered. The estimated results of the problem have been accomplished using the Optimal Homotopy Analysis Method(OHAM). The residual errors of the OHAM method have been shown physically and numerically. The important physical parameters of skin friction and Nusselt number have been calculated and discussed. The other embedding parameters like generalized magnetic parameter, Prantl number, nanofluid volume fraction and Eckert number have been intended and discussed. The obtained results have been compared with the Numerical(ND-Solve) method for both sorts of CNTs. The closed agreement of both methods has been achieved.
文摘The impact of the Marangoni convection over the thin film flow on an expanding cylinder has been examined in this study. The diverse effect of the embedded constraints has been detected during the liquid film flow. It has been examined that the behavior of the physical parameters altered after the small intervals and diverse from the traditional approach. The similarity variables have been utilized to alter the basic flow equations into the nonlinear ordinary differential equations. The result of the transformed equations is computed by BVPh 2.0 package. The performance of different constraints, for flow motion and temperature distributions are plotted and conferred. It has been observed that under the Marangoni convection the impact of the physical parameters varies after the point of inflection and the diverse impact of the embedding constraints provide space for the variation of the point of inflection for the desired spray analysis.
文摘In this contemporary study,theoritical investigation of nanofluidic model is thought-out.Two-dimensional nanomaterials based mixed flow is considered here.Convective solar radiative heat transport properties have been investigated over a nonlinearly stretched wall in the presence of magneto-hydrodynamic(MHD),by innovative application of semi analytical“optimal homotopy asymptotic method(OHAM)”.OHAM does not require any discretization,linearization and small parameter assumption.OHAM describes extremely precise 1^(st)/2^(nd) order solutions without the need of computing further higher order terms,therefore,fast convergence is observed.Nanofluidic governing model is transformed into system of ordinary differential equations(ODEs)by exploitation of similarity transformation.To study the significance of radiation parameter alongwith thermophoresis parameter,a semi analytical solver is applied to the transformed system.In this work,Brownianmotion,influence ofmagnetic field,Lewis number,Prandtl number,Eckert number and Biot number have investigated on velocity,temperature and nanoparticle concentration profiles.The study provides sufficient number of graphical representations to demonstrate the inspiration of mentioned parameters.
文摘The steady, laminar, incompressible and two dimensional micropolar flow between two porous disks was investigated using optimal homotopy asymptotic method(OHAM) and fourth order Runge–Kutta numerical method. Comparison between OHAM and numerical method shows that OHAM is an exact and high efficient method for solving these kinds of problems. The results are presented to study the velocity and rotation profiles for different physical parameters such as Reynolds number, vortex viscosity parameter, spin gradient viscosity and microinertia density parameter. As an important outcome, the magnitude of the microrotation increases with an increase in the values of injection velocity while it decreases by increasing the values of suction velocity.
基金the Deanship of Scientific Research at King Khalid University for funding this work through Research Groups Program under grant number (R.G.P2./19/40)
文摘In this paper,the magnetohydrodynamic 3 D flow of Prandtl nanoliquid subject to convectively heated extendable surface has been discussed.A linear stretching surface makes the flow.Thermophoretic and Brownian motion impacts are explored.Heat transfer for convective procedure is considered.Prandtl liquid is taken electrically conducted through applied magnetic field.Suitable non-dimensional variables lead to strong nonlinear ordinary differential system.The obtained nonlinear differential systems are solved through optimal homotopic technique.Physical quantities like skin friction coefficients and Nusselt number are explored via plots.It is observed that effects of Hartman parameter and Biot number on temperature and concentration are quite similar.Both temperature and concentration are enhanced for larger values of Hartman parameter and Biot number.
文摘In the present article Optimal Homotopy Asymptotic Method(OHAM)is used to obtain the solutions of momentum and heat transfer equations of non-Newtonian fluid flow in an axisymmetric channel with porous wall for turbine cooling applications.Numerical method is used for validity of this analytical method and excellent agreement is observed between the solutions obtained from OHAM and numerical results.Trusting to this validity,effects of some other parameters are discussed.The results show that Nusselt number increases with increase of Reynolds number,Prandtl number and power law index.
文摘The viscous dissipation and heat transfer in the Darcy-Forchheimer flow by a rotating disk are examined. The partial slip conditions are invoked. The optimal series solutions are computed via the optimal homotopic analysis method(OHAM). The thermophoresis and Brownian motions are studied. The Darcy-Forchheimer relation characterizes the porous space. The roles of influential variables on the physical quantities are graphically examined. A reduction in the local Nusselt number is observed through thermophoresis and thermal slip parameters. The local Sherwood number depicts an increasing trend for the higher Brownian motion and concentration slip parameters.
基金the research grant of Jeju National University in 2020,the Basic Science Research Program through the National Research Foundation of Korea(NRF)grant funded by the Korea Government(Ministry of Science and ICT)(NRF-2018R1A4A1025998)Higher Education Commission of Pakistan(Project No.210-3800/NRPU/R&D/HEC/1530).
文摘In recent years,the emergence of nanotechnology experienced incredible development in the field of medical sciences.During the past decade,investigating the characteristics of nanoparticles during fluid flow has been one of the intriguing issues.Nanoparticle distribution and uniformity have emerged as substantial criteria in both medical and engineering applications.Adverse effects of chemotherapy on healthy tissues are known to be a significant concern during cancer therapy.A novel treatment method of magnetic drug targeting(MDT)has emerged as a promising topical cancer treatment along with some attractive advantages of improving efficacy,fewer side effects,and reduce drug dose.During magnetic drug targeting,the appropriate movement of nanoparticles(magnetic)as carriers is essential for the therapeutic process in the blood clot removal,infection treatment,and tumor cell treatment.In this study,we have numerically investigated the behavior of an unsteady blood flowinfused with magnetic nanoparticles during MDT under the influence of a uniform external magnetic field in a microtube.An optimal homotopy asymptotic method(OHAM)is employed to compute the governing equation for unsteady electromagnetohydrodynamics flow.The influence of Hartmann number(Ha),particle mass parameter(G),particle concentration parameter(R),and electro-osmotic parameter(k)is investigated on the velocity of magnetic nanoparticles and blood flow.Results obtained show that the electro-osmotic parameter,along with Hartmann’s number,dramatically affects the velocity of magnetic nanoparticles,blood flow velocity,and flow rate.Moreover,results also reveal that at a higher Hartman number,homogeneity in nanoparticles distribution improved considerably.The particle concentration andmass parameters effectively influence the capturing effect on nanoparticles in the blood flow using a micro-tube for magnetic drug targeting.Lastly,investigation also indicates that the OHAM analysis is efficient and quick to handle the system of nonlinear equations.
文摘The melting phenomenon in two-dimensional(2 D)flow of fourth-grade material over a stretching surface is explored.The flow is created via a stretching surface.A Darcy-Forchheimer(D-F)porous medium is considered in the flow field.The heat transport is examined with the existence of the Cattaneo-Christov(C-C)heat flux.The fourth-grade material is electrically conducting subject to an applied magnetic field.The governing partial differential equations(PDEs)are reduced into ordinary differential equations(ODEs)by appropriate transformations.The solutions are constructed analytically through the optimal homotopy analysis method(OHAM).The fluid velocity,temperature,and skin friction are examined under the effects of various involved parameters.The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter,porosity parameter,and Forchheimer number.The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number,magnetic parameter,and thermal relaxation parameter.Furthermore,the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters,thermal relaxation parameter,and Forchheimer number.
文摘An analysis of the mixed convective flow of viscous fluids induced by a nonlinear inclined stretching surface is addressed.Heat and mass transfer phenomena are analyzed with additional effects of heat generation/absorption and activation energy,respectively.The nonlinear Darcy-Forchheimer relation is deliberated.The dimensionless problem is obtained through appropriate transformations.Convergent series solutions are obtained by utilizing an optimal homotopic analysis method(OHAM).Graphs depicting the consequence of influential variables on physical quantities are presented.Enhancement in the velocity is observed through the local mixed convection parameter while an opposite trend of the concentration field is noted for the chemical reaction rate parameter.
文摘This article studies the unsteady thin film flow of a fourth grade fluid over a moving and oscillating vertical belt.The problem is modeled in terms of non-nonlinear partial differential equations with some physical conditions.Both problems of lift and drainage are studied.Two different techniques namely the adomian decomposition method(ADM)and the optimal homotopy asymptotic method(OHAM)are used for finding the analytical solutions.These solutions are compared and found in excellent agreement.For the physical analysis of the problem,graphical results are provided and discussed for various embedded flow parameters.
文摘Laminar,isothermal,incompressible and viscous flow in a rectangular domain bounded by two moving porous walls,w hich enable the fuid to enter or exit during successive expansions or contractions is investigated analytically using optimal homotopy asymptotic method(OHAM).OHAM is a powerful method for solving nonlinear problems without depending to the small parameter.The concept of this method is briefly introduced,and it's application for this problem is studied.Then,the results are compared with numerical results and the validity of these methods is shown.After this verification,we analyze the effects of some physical applicable parameters to show the efficiency of OHAM for this type of problems.Graphical results are presented to investigate the influence of the non-dimensional wall dilation rate(a)and pemeation Reynolds number(Re)on the velocity,normal pressure distribution and wall shear stress.The present problem for slowly expanding or contracting walls with weak permeability is a simple model for the transport of biological fuids through contracting or expanding vessels.
