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
The basic physics of unsteady Hele-Shaw flow at high Reynolds numbers is mainly studied by an experimental measurement. In order to confirm the Darcy′s law in Hele-Shaw cell, since there is an analogy between flow in...The basic physics of unsteady Hele-Shaw flow at high Reynolds numbers is mainly studied by an experimental measurement. In order to confirm the Darcy′s law in Hele-Shaw cell, since there is an analogy between flow in cells and that in porous media, progressive water waves are utilized to build an unsteady flow in a Hele-Shaw cell, and which complex wave number is measured by a wave height gauge. Meanwhile, theoretical analyses are used to compare with experimental data. Result shows Darcy′s Law is not exactly correct for unsteady Hele-Shaw flows, and it is expected to conduct a modified Darcy′s Law.展开更多
Viscous fingering in a modified Hele-Shaw cell is numerically investigated. The cell allows periodic variation of depth in the lateral direction. The wavenumber n of the depth perturbation has great influence on finge...Viscous fingering in a modified Hele-Shaw cell is numerically investigated. The cell allows periodic variation of depth in the lateral direction. The wavenumber n of the depth perturbation has great influence on fingering patterns. For n = 1, the fingering pattern due to the interface instability remains the same as that in the conventional Hele- Shaw cell, while the depth variation causes the steady finger to be a little narrower. For n = 2, four different fingering patterns are captured, similar to the available experimental observations in a modified Hele-Shaw cell containing a centered step-like occlusion. It is found that new fingering patterns appear as n further increases, among which, two patterns with spatial oscillation along both edges of the finger are particularly interesting. One is a symmetric oscillatory finger for n = 3, and the other is an asymmetric one for n = 4. The influence of capillary number on fingering patterns is studied for n = 3 and 4. We find that spatial oscillation of the finger nearly ceases at moderate capillary numbers and occurs again as the capillary number increases further. Meanwhile, the wide finger shifts to the narrow one. It is accompanied by a sudden decrease in the finger width which otherwise decreases continuously as the capillary number increases. The wavenumber and the amplitude of depth perturbation have little effect on the finger width.展开更多
In this paper, we study the dissolution problems occurring in laterally large 3D systems with very small dimensions along the third coordinate, such as fractures or Hele-Shaw cells. On the basis of the scale separatio...In this paper, we study the dissolution problems occurring in laterally large 3D systems with very small dimensions along the third coordinate, such as fractures or Hele-Shaw cells. On the basis of the scale separation assumption, we apply upscaling to the 3D pore-scale model using the volume averaging method to develop 2D averaged equations. The influence of the choice of momentum equations on the accuracy of the 2D Hele-Shaw model is discussed, and we show that the results obtained using Darcy-Brinkman equations are better than those obtained using Darcy’s law, because of the consideration of the viscous boundary layer. The validity and accuracy of the resulting 2D model are assessed based on comparisons with full 3D solutions for problems corresponding to the existence of geometrical 3D features to which a simple averaging procedure along a line(i.e., the height of the gap) perpendicular to the 2D plane cannot be applied, such as the dissolution of pillars. The results show that when Péclet and Reynolds numbers exceed certain limits, 3D effects must be considered. Moreover, natural convection effects are important when the Rayleigh number is large.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
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
With the utilization of underground space,backward erosion piping(BEP)has been observed in many underground structures(e.g.,shield tunnels)founded on sandy aquifers.However,due to invisibility,the geometry of the erod...With the utilization of underground space,backward erosion piping(BEP)has been observed in many underground structures(e.g.,shield tunnels)founded on sandy aquifers.However,due to invisibility,the geometry of the eroded pipe and its spatial evolution with time during the piping process was still not clear.In this study,we developed a Hele-Shaw cell to visualize the dynamic progression of BEP.With imaging process technology,we obtained a typical process of BEP(the erosion process can be divided into a piping progression phase and a piping stabilization phase),quantitatively characterized the formation of erosion pipes,and compared the patterns of erosion(e.g.,the erosion area A and the maximum erosion radius R)that spontaneously develop under different fluxes of water.The most interesting finding is that the sand grains in a thicker Hele-Shaw model are easier to dislodge,which is possibly due to the granular system in a thicker model having more degrees of freedom,reducing the stability of the sand grains.展开更多
The effect of container geometry on the Faraday waves in Hele-Shaw cells has been investigated. The wave heights increase with the width of the cell and a linear function is selected to express the relation between th...The effect of container geometry on the Faraday waves in Hele-Shaw cells has been investigated. The wave heights increase with the width of the cell and a linear function is selected to express the relation between these data and parameters. The wave lengths also increase with the width and are in good agreement with the dispersion relation. In order to reveal the real nature behind these phenomena, we have developed a gap-averaged model to numerically solve this issue and give an analysis of the result to show how these Faraday waves are formed in a Hele-Shaw cell.展开更多
The Advection-Diffusion Reaction (ADR) equation appears in many problems in nature. This constitutes a general model that is useful in various scenarios, from porous media to atmospheric processes. Particularly, it is...The Advection-Diffusion Reaction (ADR) equation appears in many problems in nature. This constitutes a general model that is useful in various scenarios, from porous media to atmospheric processes. Particularly, it is used at the interface between two fluids where different types of instabilities due to surface mobility may appear. Together with the ADR equation, the Darcy-Brinkman model describes the phenomena known as fingering that appear in different contexts. The study of this type of system gains in complexity when the number of chemical species dissolved in both fluids increases. With more solutes, the increasing complexity of this phenomenon generally requires much computational power. To face the need for more computational resources, we build a solver tool based on an Alternating Direction Implicit (ADI) scheme that can be run in Central Processing Unit (CPU) and Graphic Processing Unit (GPU) architectures on any notebook. The implementation is done using the MATLAB platform to compare both versions. It is shown that using the GPU version strongly saves both resources and calculation times.展开更多
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).
文摘The basic physics of unsteady Hele-Shaw flow at high Reynolds numbers is mainly studied by an experimental measurement. In order to confirm the Darcy′s law in Hele-Shaw cell, since there is an analogy between flow in cells and that in porous media, progressive water waves are utilized to build an unsteady flow in a Hele-Shaw cell, and which complex wave number is measured by a wave height gauge. Meanwhile, theoretical analyses are used to compare with experimental data. Result shows Darcy′s Law is not exactly correct for unsteady Hele-Shaw flows, and it is expected to conduct a modified Darcy′s Law.
基金North Dakota industrial commission oil and gas research program(G-041-081)UND vice president for research&economic development postdoctoral funding program
基金Project supported by the National Natural Science Foundation of China(No.11232011)the 111 Project of China(No.B07033)
文摘Viscous fingering in a modified Hele-Shaw cell is numerically investigated. The cell allows periodic variation of depth in the lateral direction. The wavenumber n of the depth perturbation has great influence on fingering patterns. For n = 1, the fingering pattern due to the interface instability remains the same as that in the conventional Hele- Shaw cell, while the depth variation causes the steady finger to be a little narrower. For n = 2, four different fingering patterns are captured, similar to the available experimental observations in a modified Hele-Shaw cell containing a centered step-like occlusion. It is found that new fingering patterns appear as n further increases, among which, two patterns with spatial oscillation along both edges of the finger are particularly interesting. One is a symmetric oscillatory finger for n = 3, and the other is an asymmetric one for n = 4. The influence of capillary number on fingering patterns is studied for n = 3 and 4. We find that spatial oscillation of the finger nearly ceases at moderate capillary numbers and occurs again as the capillary number increases further. Meanwhile, the wide finger shifts to the narrow one. It is accompanied by a sudden decrease in the finger width which otherwise decreases continuously as the capillary number increases. The wavenumber and the amplitude of depth perturbation have little effect on the finger width.
基金support from the National Natural Science Foundation of China (Grant No. 12102371)Natural Science Foundation of Sichuan Province,China (Grant No. 2022NSFSC1932)。
文摘In this paper, we study the dissolution problems occurring in laterally large 3D systems with very small dimensions along the third coordinate, such as fractures or Hele-Shaw cells. On the basis of the scale separation assumption, we apply upscaling to the 3D pore-scale model using the volume averaging method to develop 2D averaged equations. The influence of the choice of momentum equations on the accuracy of the 2D Hele-Shaw model is discussed, and we show that the results obtained using Darcy-Brinkman equations are better than those obtained using Darcy’s law, because of the consideration of the viscous boundary layer. The validity and accuracy of the resulting 2D model are assessed based on comparisons with full 3D solutions for problems corresponding to the existence of geometrical 3D features to which a simple averaging procedure along a line(i.e., the height of the gap) perpendicular to the 2D plane cannot be applied, such as the dissolution of pillars. The results show that when Péclet and Reynolds numbers exceed certain limits, 3D effects must be considered. Moreover, natural convection effects are important when the Rayleigh number is large.
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 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the National Engineering Laboratory for Digital Construction and Evaluation Technology of Urban Rail Transit(No.2021GY01)the National Natural Science Foundation of China(No.41630641)。
文摘With the utilization of underground space,backward erosion piping(BEP)has been observed in many underground structures(e.g.,shield tunnels)founded on sandy aquifers.However,due to invisibility,the geometry of the eroded pipe and its spatial evolution with time during the piping process was still not clear.In this study,we developed a Hele-Shaw cell to visualize the dynamic progression of BEP.With imaging process technology,we obtained a typical process of BEP(the erosion process can be divided into a piping progression phase and a piping stabilization phase),quantitatively characterized the formation of erosion pipes,and compared the patterns of erosion(e.g.,the erosion area A and the maximum erosion radius R)that spontaneously develop under different fluxes of water.The most interesting finding is that the sand grains in a thicker Hele-Shaw model are easier to dislodge,which is possibly due to the granular system in a thicker model having more degrees of freedom,reducing the stability of the sand grains.
基金supported by the National Natural Science Foundation of China(Grant No.11702099)the China Postdoctoral Science Foundation(Grant No.2017M612670)
文摘The effect of container geometry on the Faraday waves in Hele-Shaw cells has been investigated. The wave heights increase with the width of the cell and a linear function is selected to express the relation between these data and parameters. The wave lengths also increase with the width and are in good agreement with the dispersion relation. In order to reveal the real nature behind these phenomena, we have developed a gap-averaged model to numerically solve this issue and give an analysis of the result to show how these Faraday waves are formed in a Hele-Shaw cell.
文摘The Advection-Diffusion Reaction (ADR) equation appears in many problems in nature. This constitutes a general model that is useful in various scenarios, from porous media to atmospheric processes. Particularly, it is used at the interface between two fluids where different types of instabilities due to surface mobility may appear. Together with the ADR equation, the Darcy-Brinkman model describes the phenomena known as fingering that appear in different contexts. The study of this type of system gains in complexity when the number of chemical species dissolved in both fluids increases. With more solutes, the increasing complexity of this phenomenon generally requires much computational power. To face the need for more computational resources, we build a solver tool based on an Alternating Direction Implicit (ADI) scheme that can be run in Central Processing Unit (CPU) and Graphic Processing Unit (GPU) architectures on any notebook. The implementation is done using the MATLAB platform to compare both versions. It is shown that using the GPU version strongly saves both resources and calculation times.
