This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that ...This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.展开更多
The steady flow in a Hele-Shaw cell filled with fluids with a high viscosity contrast in the presence of fluid oscillations is experimentally studied.The control of oscillatory dynamics of multiphase systems with inte...The steady flow in a Hele-Shaw cell filled with fluids with a high viscosity contrast in the presence of fluid oscillations is experimentally studied.The control of oscillatory dynamics of multiphase systems with interfaces is a challenging technological problem.We consider miscible(water and glycerol)and immiscible(water and high-viscosity silicone oil PMS-1000)fluids under subsonic oscillations perpendicular to the interface.Observations show that the interface shape depends on the amplitude and frequency of oscillations.The interface is undisturbed only in the absence of oscillations.Under small amplitudes,the interface between water and glycerol widens due to mixing.When the critical amplitude is reached,the interface becomes unstable to the fingering instability:Aqueous fingers penetrate the high-viscosity glycerol and induce intensive mixing of miscible fluids and associated decay of the instability.After the disappearance of the fingers,the interface takes a U-shape in the central part of the cell.A similar effect is observed for immiscible fluids:The oscillating interface tends to bend to the side of a high-viscosity fluid.Again,when the critical amplitude is reached,the fingering instability arises at the convex interface.This paper focuses on the causes of bending of the initially undisturbed interface between miscible or immiscible fluids.For this purpose,we measure the steady flow velocity near the interface and in the bulk of a high-viscosity fluid using Particle Image Velocimetry(PIV).展开更多
We investigate the nonlinear dynamics of amoving interface in aHele-Shaw cell subject to an in-plane applied electric field.We develop a spectrally accurate numerical method for solving a coupled integral equation sys...We investigate the nonlinear dynamics of amoving interface in aHele-Shaw cell subject to an in-plane applied electric field.We develop a spectrally accurate numerical method for solving a coupled integral equation system.Although the stiffness due to the high order spatial derivatives can be removed using a small scale decomposition technique,the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands.We remove this physically imposed stiffness by employing a rescaling scheme,which accelerates the slow dynamics and reduces the computational cost.Our nonlinear results reveal that positive currents restrain finger ramification and promote the overall stabilization of patterns.On the other hand,negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region.When no fluid is injected,and a negative current is utilized,the interface tends to approach the origin and break up into several drops.We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law(t∗−t)b,where t∗is the estimated pinch-off time.展开更多
Electrowetting has been proposed as a technique for manipulating dropletssurrounded by air or oil. In this paper, we discuss the modeling and simulation of thedroplet fission process between two parallel plates inside...Electrowetting has been proposed as a technique for manipulating dropletssurrounded by air or oil. In this paper, we discuss the modeling and simulation of thedroplet fission process between two parallel plates inside an electrowetting on dielectric (EWOD) device. Since the gap between the plates is small, we use the two-phaseHele-Shaw flow as a model. While there are several high order methods around, suchas the immersed interface methods [1, 2], we decide to use two first-order methods forsimplicity. A ghost-fluid (GF) method is employed to solve the governing equationsand a local level set method is used to track the drop interface. For comparison purposes, the same set of two-phase Hele-Shaw equations are also solved directly usingthe immersed boundary (IB) method. Numerical results are consistent with experimental observations reported in the literature.展开更多
基金the NSF grants DMS-0410266 and DMS-0710831the China National Basic Research Program under the grant 2005CB321701+1 种基金the Program for the New Century Outstanding Talents in Universities of Chinathe Natural Science Foundation of Jiangsu Province under the grant BK2006511
文摘This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.
基金supported by the Ministry of Education of the Russian Federation(Project KPZU-2023-0002).
文摘The steady flow in a Hele-Shaw cell filled with fluids with a high viscosity contrast in the presence of fluid oscillations is experimentally studied.The control of oscillatory dynamics of multiphase systems with interfaces is a challenging technological problem.We consider miscible(water and glycerol)and immiscible(water and high-viscosity silicone oil PMS-1000)fluids under subsonic oscillations perpendicular to the interface.Observations show that the interface shape depends on the amplitude and frequency of oscillations.The interface is undisturbed only in the absence of oscillations.Under small amplitudes,the interface between water and glycerol widens due to mixing.When the critical amplitude is reached,the interface becomes unstable to the fingering instability:Aqueous fingers penetrate the high-viscosity glycerol and induce intensive mixing of miscible fluids and associated decay of the instability.After the disappearance of the fingers,the interface takes a U-shape in the central part of the cell.A similar effect is observed for immiscible fluids:The oscillating interface tends to bend to the side of a high-viscosity fluid.Again,when the critical amplitude is reached,the fingering instability arises at the convex interface.This paper focuses on the causes of bending of the initially undisturbed interface between miscible or immiscible fluids.For this purpose,we measure the steady flow velocity near the interface and in the bulk of a high-viscosity fluid using Particle Image Velocimetry(PIV).
基金the National Science Foundation,Division of Mathematical Sciences(NSF-DMS)grants DMS-1714973,1719960,1763272(J.L.)DMS-1720420(S.L.).J.L.thanks the support from the Simons Foundation(594598QN)for a NSF-Simons Center for Multiscale Cell Fate Research.J.L.also thanks the National Institutes of Health for partial support through grants 1U54CA217378-01A1 for a National Center in Cancer Systems Biology at UC Irvine and P30CA062203 for the Chao Family Comprehensive Cancer Center at UC Irvine.
文摘We investigate the nonlinear dynamics of amoving interface in aHele-Shaw cell subject to an in-plane applied electric field.We develop a spectrally accurate numerical method for solving a coupled integral equation system.Although the stiffness due to the high order spatial derivatives can be removed using a small scale decomposition technique,the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands.We remove this physically imposed stiffness by employing a rescaling scheme,which accelerates the slow dynamics and reduces the computational cost.Our nonlinear results reveal that positive currents restrain finger ramification and promote the overall stabilization of patterns.On the other hand,negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region.When no fluid is injected,and a negative current is utilized,the interface tends to approach the origin and break up into several drops.We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law(t∗−t)b,where t∗is the estimated pinch-off time.
基金The work was supported in part by Chinese NSF Project 10431030,NSERC and MITACS(Canada).
文摘Electrowetting has been proposed as a technique for manipulating dropletssurrounded by air or oil. In this paper, we discuss the modeling and simulation of thedroplet fission process between two parallel plates inside an electrowetting on dielectric (EWOD) device. Since the gap between the plates is small, we use the two-phaseHele-Shaw flow as a model. While there are several high order methods around, suchas the immersed interface methods [1, 2], we decide to use two first-order methods forsimplicity. A ghost-fluid (GF) method is employed to solve the governing equationsand a local level set method is used to track the drop interface. For comparison purposes, the same set of two-phase Hele-Shaw equations are also solved directly usingthe immersed boundary (IB) method. Numerical results are consistent with experimental observations reported in the literature.
