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
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 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
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 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.
