Journal articles: 'Capillarity equation'

1

Figliuzzi, B., and C. R. Buie. "Rise in optimized capillary channels." Journal of Fluid Mechanics 731 (August 14, 2013): 142–61. http://dx.doi.org/10.1017/jfm.2013.373.

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AbstractMany technological applications rely on the phenomenon of wicking flow induced by capillarity. However, despite a continuing interest in the subject, the influence of the capillary geometry on the wicking dynamics remains underexploited. In numerous applications, the ability to promote wicking in a capillary is a key issue. In this article, a model describing the capillary rise of a liquid in a capillary of varying circular cross-section is presented. The wicking dynamics is described by an ordinary differential equation with a term dependent upon the shape of the capillary channel. Using optimal control theory, we were able to design optimized capillaries which promote faster wicking than uniform cylinders. Numerical simulations show that the height of the rising liquid was up to 50 % greater with the optimized shapes than with a uniform cylinder of optimal radius. Experiments on specially designed capillaries with silicone oil show a good agreement with the theory. The methods presented can be useful in the design and optimization of systems employing capillary-driven transport including micro-heat pipes or oil extracting devices.

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Bhatnagar, Rajat, and Robert Finn. "On the Capillarity Equation in Two Dimensions." Journal of Mathematical Fluid Mechanics 18, no. 4 (May 4, 2016): 731–38. http://dx.doi.org/10.1007/s00021-016-0257-6.

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Liang, Fei-Tsen. "Global boundedness, interior gradient estimates, and boundary regularity for the mean curvature equation with boundary conditions." International Journal of Mathematics and Mathematical Sciences 2004, no. 18 (2004): 913–48. http://dx.doi.org/10.1155/s0161171204307039.

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We obtain global estimates for the modulus, interior gradient estimates, and boundary Hölder continuity estimates for solutionsuto the capillarity problem and to the Dirichlet problem for the mean curvature equation merely in terms of the mean curvature, together with the boundary contact angle in the capillarity problem and the boundary values in the Dirichlet problem.

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Tritscher, Peter. "An integrable fourth-order nonlinear evolution equation applied to surface redistribution due to capillarity." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 38, no. 4 (April 1997): 518–41. http://dx.doi.org/10.1017/s0334270000000849.

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AbstractMembers of an hierarchy of integrable nonlinear evolution equations, related to the well-known linearizable diffusion equation which has the diffusivity form as the reciprocal of the square of the concentration, are adapted to derive a new integrable nonlinear equation which models the surface evolution of an arbitrarily-oriented theoretical anisotropic material by the concomitant action of evaporation-condensation and surface diffusion. The constitutive relations are explicitly formulated and these show that the theoretical anisotropic material behaves like a liquid crystal. The integrable nonlinear equation may be used to advantage as test cases for numerical schemes. Its form has many attributes of the nonlinear governing equation for an isotropic material. Closed-form solutions are constructed for the evolution of a ramped surface by concomitant evaporation-condensation and surface diffusion.

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Roubíček, Tomáš. "Cahn-Hilliard equation with capillarity in actual deforming configurations." Discrete & Continuous Dynamical Systems - S 14, no. 1 (2021): 41–55. http://dx.doi.org/10.3934/dcdss.2020303.

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Rodr guez-Valverde, M. A., M. A. Cabrerizo-V lchez, and R. Hidalgo- lvarez. "The Young Laplace equation links capillarity with geometrical optics." European Journal of Physics 24, no. 2 (February 10, 2003): 159–68. http://dx.doi.org/10.1088/0143-0807/24/2/356.

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Lu, Ning. "Generalized Soil Water Retention Equation for Adsorption and Capillarity." Journal of Geotechnical and Geoenvironmental Engineering 142, no. 10 (October 2016): 04016051. http://dx.doi.org/10.1061/(asce)gt.1943-5606.0001524.

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Ning, Tao, Meng Xi, Bingtao Hu, Le Wang, Chuanqing Huang, and Junwei Su. "Effect of Viscosity Action and Capillarity on Pore-Scale Oil–Water Flowing Behaviors in a Low-Permeability Sandstone Waterflood." Energies 14, no. 24 (December 7, 2021): 8200. http://dx.doi.org/10.3390/en14248200.

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Water flooding technology is an important measure to enhance oil recovery in oilfields. Understanding the pore-scale flow mechanism in the water flooding process is of great significance for the optimization of water flooding development schemes. Viscous action and capillarity are crucial factors in the determination of the oil recovery rate of water flooding. In this paper, a direct numerical simulation (DNS) method based on a Navier–Stokes equation and a volume of fluid (VOF) method is employed to investigate the dynamic behavior of the oil–water flow in the pore structure of a low-permeability sandstone reservoir in depth, and the influencing mechanism of viscous action and capillarity on the oil–water flow is explored. The results show that the inhomogeneity variation of viscous action resulted from the viscosity difference of oil and water, and the complex pore-scale oil–water two-phase flow dynamic behaviors exhibited by capillarity play a decisive role in determining the spatial sweep region and the final oil recovery rate. The larger the viscosity ratio is, the stronger the dynamic inhomogeneity will be as the displacement process proceeds, and the greater the difference in distribution of the volumetric flow rate in different channels, which will lead to the formation of a growing viscous fingering phenomenon, thus lowering the oil recovery rate. Under the same viscosity ratio, the absolute viscosity of the oil and water will also have an essential impact on the oil recovery rate by adjusting the relative importance between viscous action and capillarity. Capillarity is the direct cause of the rapid change of the flow velocity, the flow path diversion, and the formation of residual oil in the pore space. Furthermore, influenced by the wettability of the channel and the pore structure’s characteristics, the pore-scale behaviors of capillary force—including the capillary barrier induced by the abrupt change of pore channel positions, the inhibiting effect of capillary imbibition on the flow of parallel channels, and the blockage effect induced by the newly formed oil–water interface—play a vital role in determining the pore-scale oil–water flow dynamics, and influence the final oil recovery rate of the water flooding.

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Hua, Wei, Wei Wang, Weidong Zhou, Ruige Wu, and Zhenfeng Wang. "Experiment—Simulation Comparison in Liquid Filling Process Driven by Capillarity." Micromachines 13, no. 7 (July 12, 2022): 1098. http://dx.doi.org/10.3390/mi13071098.

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This paper studies modifications made to the Bosanquet equation in order to fit the experimental observations of the liquid filling process in circular tubes that occurs by capillary force. It is reported that there is a significant difference between experimental observations and the results predicted by the Bosanquet equation; hence, it is reasonable to investigate these differences intensively. Here, we modified the Bosanquet equation such that it could consider more factors that contribute to the filling process. First, we introduced the air flowing out of the tube as the liquid inflow. Next, we considered the increase in hydraulic resistance due to the surface roughness of the inner tube. Finally, we further considered the advancing contact angle, which varies during the filling process. When these three factors were included, the modified Bosanquet equation was well correlated with the experimental results, and the R square—which indicates the fitting quality between the simulation and the experiment—significantly increased to above 0.99.

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10

Li, Shi-Ming, and Danesh K. Tafti. "A Mean-Field Pressure Formulation for Liquid-Vapor Flows." Journal of Fluids Engineering 129, no. 7 (December 28, 2006): 894–901. http://dx.doi.org/10.1115/1.2742730.

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A nonlocal pressure equation is derived from mean-field free energy theory for calculating liquid-vapor systems. The proposed equation is validated analytically by showing that it reduces to van der Waals’ square-gradient approximation under the assumption of slow density variations. The proposed nonlocal pressure is implemented in the mean-field free energy lattice Boltzmann method (LBM). The LBM is applied to simulate equilibrium liquid-vapor interface properties and interface dynamics of capillary waves and oscillating droplets in vapor. Computed results are validated with Maxwell constructions of liquid-vapor coexistence densities, theoretical relationship of variation of surface tension with temperature, theoretical planar interface density profiles, Laplace’s law of capillarity, dispersion relationship between frequency and wave number of capillary waves, and the relationship between radius and the oscillating frequency of droplets in vapor. It is shown that the nonlocal pressure formulation gives excellent agreement with theory.

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Caro, A., D. Schwen, J. Hetherly, and E. Martinez. "The capillarity equation at the nanoscale: Gas bubbles in metals." Acta Materialia 89 (May 2015): 14–21. http://dx.doi.org/10.1016/j.actamat.2015.01.048.

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12

HASPOT, BORIS. "CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY." Mathematical Models and Methods in Applied Sciences 20, no. 07 (July 2010): 1049–87. http://dx.doi.org/10.1142/s0218202510004532.

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In this paper, we consider the compressible Navier–Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier–Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

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Archer, N. A. L., M. Bonell, A. M. MacDonald, and N. Coles. "A constant head well permeameter formula comparison: its significance in the estimation of field-saturated hydraulic conductivity in heterogeneous shallow soils." Hydrology Research 45, no. 6 (February 18, 2014): 788–805. http://dx.doi.org/10.2166/nh.2014.159.

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We evaluate the application and investigate various formulae (and the associated parameter sensitivities) using the constant head well permeameter method to estimate field-saturated hydraulic conductivity (Kfs) in a previously glaciated temperate landscape in the Scottish Borders where shallow soils constrain the depth of augering. In finer-textured soils, the Glover equation provided Kfs estimates nearly twice those of the Richards equation. For this environment, we preferred the Glover equation with a correction factor for the effect of gravity, which does not include soil capillarity effects because: (1) the low depth to diameter ratio of the auger holes (AH) required in the shallow stratified soils of temperate glaciated environment needs a correction for gravity; (2) the persistently moist environment and the use of long pre-wetting times before measurements seem to reduce the effect of soil capillarity; (3) the Richards equation is dependent on accurate α* values, but the measured AH intersected soil horizon boundaries that had different soil structure and texture, causing difficulty in selecting the most appropriate α* value; (4) when comparing the different solutions to estimate Kfs using the constant-head well permeameter method against the AH method and ponded permeameter measurements, the Glover solution with a correction for gravity gave the best comparable result in fine-textured soil.

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14

Hoffmann, Karl-Heinz, and Piotr Rybka. "On convergence of solutions to the equation of viscoelasticity with capillarity." Communications in Partial Differential Equations 25, no. 9-10 (January 2000): 1845–90. http://dx.doi.org/10.1080/03605300008821570.

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15

Antanovskii, L. K., C. Rogers, and W. K. Schief. "A note on a capillarity model and the nonlinear Schrödinger equation." Journal of Physics A: Mathematical and General 30, no. 16 (August 21, 1997): L555—L557. http://dx.doi.org/10.1088/0305-4470/30/16/001.

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16

Zhang, Chao, Zhen Liu, and Peng Deng. "Using molecular dynamics to unravel phase composition behavior of nano-size pores in frozen soils: Does Young–Laplace equation apply in low temperature range?" Canadian Geotechnical Journal 55, no. 8 (August 2018): 1144–53. http://dx.doi.org/10.1139/cgj-2016-0150.

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The phase composition curve of frozen soils is a fundamental relationship in understanding permafrost and seasonally frozen soils. However, due to the complex interplay between adsorption and capillarity, a clear physically based understanding of the phase composition curve in the low temperature range, i.e., <265 K, is still absent. Especially, it is unclear whether the Young–Laplace equation corresponding to capillarity still holds in nano-size pores where adsorption could dominate. In this paper, a framework based on molecular dynamics was developed to investigate the phase transition behavior of water confined in nano-size pores. A series of simulations was conducted to unravel the effects of the pore size and wettability on the freezing and melting of pore water. This is the first time that the phase composition behavior of frozen soils is analyzed using molecular dynamics. It is found that the Young–Laplace equation may not apply in the low temperature range.

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Khodabocus, M. I., M. Sellier, and V. Nock. "Slug Self-Propulsion in a Capillary Tube Mathematical Modeling and Numerical Simulation." Advances in Mathematical Physics 2016 (2016): 1–16. http://dx.doi.org/10.1155/2016/1234642.

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A composite droplet made of two miscible fluids in a narrow tube generally moves under the action of capillarity until complete mixture is attained. This physical situation is analysed here on a combined theoretical and numerical analysis. The mathematical framework consists of the two-phase flow phase-field equation set, an advection-diffusion chemical concentration equation, and closure relationships relating the surface tensions to the chemical concentration. The numerical framework is composed of the COMSOL Laminar two-phase flow phase-field method coupled with an advection-diffusion chemical concentration equation. Through transient studies, we show that the penetrating length of the bidroplet system into the capillary tube is linear at early-time regime and exponential at late-time regime. Through parametric studies, we show that the rate of penetration of the bidroplet system into the capillary tube is proportional to a time-dependent exponential function. We also show that this speed obeys the Poiseuille law at the early-time regime. A series of position, speed-versus-property graphs are included to support the analysis. Finally, the overall results are contrasted with available experimental data, grouped together to settle a general mathematical description of the phenomenon, and explained and concluded on this basis.

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De La Fuente, Maria, Jean Vaunat, and Héctor Marín-Moreno. "Modelling Methane Hydrate Saturation in Pores: Capillary Inhibition Effects." Energies 14, no. 18 (September 7, 2021): 5627. http://dx.doi.org/10.3390/en14185627.

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Experimental and field observations evidence the effects of capillarity in narrow pores on inhibiting the thermodynamic stability of gas hydrates and controlling their saturation. Thus, precise estimates of the gas hydrate global inventory require models that accurately describe gas hydrate stability in sediments. Here, an equilibrium model for hydrate formation in sediments that accounts for capillary inhibition effects is developed and validated against experimental data. Analogous to water freezing in pores, the model assumes that hydrate formation is controlled by the sediment pore size distribution and the balance of capillary forces at the hydrate–liquid interface. To build the formulation, we first derive the Clausius–Clapeyron equation for the thermodynamic equilibrium of methane and water chemical potentials. Then, this equation is combined with the van Genuchten’s capillary pressure to relate the thermodynamic properties of the system to the sediment pore size distribution and hydrate saturation. The model examines the influence of the sediment pore size distribution on hydrate saturation through the simulation of hydrate formation in sand, silt, and clays, under equilibrium conditions and without mass transfer limitations. The results show that at pressure–temperature conditions typically found in the seabed, capillary effects in very fine-grained clays can limit the maximum hydrate saturation below 20% of the host sediment porosity.

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Timsina, Ramesh Chandra, Harihar Khanal, Andrei Ludu, and Kedar Nath Uprety. "A Mathematical Model for Transport and Growth of Microbes in Unsaturated Porous Soil." Mathematical Problems in Engineering 2021 (December 27, 2021): 1–13. http://dx.doi.org/10.1155/2021/6278126.

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In this work, we develop a mathematical model for transport and growth of microbes by natural (rain) water infiltration and flow through unsaturated porous soil along the vertical direction under gravity and capillarity by coupling a system of advection diffusion equations (for concentration of microbes and their growth-limiting substrate) with the Richards equation. The model takes into consideration several major physical, chemical, and biological mechanisms. The resulting coupled system of PDEs together with their boundary conditions is highly nonlinear and complicated to solve analytically. We present both a partial analytic approach towards solving the nonlinear system and finding the main type of dynamics of microbes, and a full-scale numerical simulation. Following the auxiliary equation method for nonlinear reaction-diffusion equations, we obtain a closed form traveling wave solution for the Richards equation. Using the propagating front solution for the pressure head, we reduce the transport equation to an ODE along the moving frame and obtain an analytic solution for the history of bacteria concentration for a specific test case. To solve the system numerically, we employ upwind finite volume method for the transport equations and stabilized explicit Runge–Kutta–Legendre super-time-stepping scheme for the Richards equation. Finally, some numerical simulation results of an infiltration experiment are presented, providing a validation and backup to the analytic partial solutions for the transport and growth of bacteria in the soil, stressing the occurrence of front moving solitons in the nonlinear dynamics.

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Bresch, Didier, Mathieu Colin, Khawla Msheik, Pascal Noble, and Xi Song. "Lubrication and shallow-water systems Bernis-Friedman and BD entropies." ESAIM: Proceedings and Surveys 69 (2020): 1–23. http://dx.doi.org/10.1051/proc/202069001.

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This paper concerns the results recently announced by the authors, in C.R. Acad. Sciences Maths volume 357, Issue 1, 1-6 (2019), which make the link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF in our paper) dissipative entropy introduced to study the lubrication equations. More precisely different dissipative BF entropies are obtained from the BD entropies playing with drag terms and capillarity formula for viscous shallow water type equations. This is the main idea in the paper which makes the link between two communities. The limit processes employ the standard compactness arguments taking care of the control in the drag terms. It allows in one dimension for instance to prove global existence of nonnegative weak solutions for lubrication equations starting from the global existence of nonnegative weak solutions for appropriate viscous shallow-water equations (for which we refer to appropriate references). It also allows to prove global existence of nonnegative weak solutions for fourth-order equation including the Derrida-Lebowitz-Speer-Spohn equation starting from compressible Navier-Stokes type equations.

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Wei, Li, Liling Duan, and Haiyun Zhou. "Study on the Existence and Uniqueness of Solution of Generalized Capillarity Problem." Abstract and Applied Analysis 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/154307.

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By using the perturbation theories on sums of ranges of nonlinear accretive mappings of Calvert and Gupta (1978), the abstract result on the existence and uniqueness of the solution inLp(Ω)of the generalized Capillarity equation with nonlinear Neumann boundary value conditions, where2N/(N+1)<p<+∞andN≥1denotes the dimension ofRN, is studied. The equation discussed in this paper and the methods here are a continuation of and a complement to the previous corresponding results. To obtain the results, some new techniques are used in this paper.

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Pasandideh-Fard, M., P. Chen, J. Mostaghimi, and A. W. Neumann. "The generalized Laplace equation of capillarity I. Thermodynamic and hydrostatic considerations of the fundamental equation for interfaces." Advances in Colloid and Interface Science 63 (January 1996): 151–77. http://dx.doi.org/10.1016/0001-8686(95)00282-0.

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Bidaut-Véron, Marie-Françoise. "Global existence and uniqueness results for singular solutions of the capillarity equation." Pacific Journal of Mathematics 125, no. 2 (December 1, 1986): 317–33. http://dx.doi.org/10.2140/pjm.1986.125.317.

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Slemrod, Marshall. "Hilbert’s sixth problem and the failure of the Boltzmann to Euler limit." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2118 (March 19, 2018): 20170222. http://dx.doi.org/10.1098/rsta.2017.0222.

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This paper addresses the main issue of Hilbert’s sixth problem, namely the rigorous passage of solutions to the mesoscopic Boltzmann equation to macroscopic solutions of the Euler equations of compressible gas dynamics. The results of the paper are that (i) in general Hilbert’s program will fail because of the appearance of van der Waals–Korteweg capillarity terms in a macroscopic description of motion of a gas, and (ii) the van der Waals–Korteweg theory itself might satisfy Hilbert’s quest for a map from the ‘atomistic view’ to the laws of motion of continua. This article is part of the theme issue ‘Hilbert’s sixth problem’.

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H. M. Zahran, Emad, and Maha S. M. Shehat. "The new solitary solutions of the foam drainage & (2+1) dimensional breaking soliton equations." International Journal of Basic and Applied Sciences 7, no. 3 (June 27, 2018): 39. http://dx.doi.org/10.14419/ijbas.v7i3.8792.

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In this study, the modified extended tanh-function method is handling to obtain many new solitary wave solutions of two important models in nonlinear physics. The first one is the foam drainage equation which is a simple model for describing the flow of liquid through channels and nodes between the bubbles, driven by gravity and capillarity. The second is (2+1)-dimensional breaking soliton equation which describe the interaction of a Riemann wave propagating along the y-axis with along the x-axis. The obtained results are compared with that obtained in previous work.

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Kralchevsky, P. A., V. N. Paunov, and Kuniaki Nagayama. "Lateral capillary interaction between particles protruding from a spherical liquid layer." Journal of Fluid Mechanics 299 (September 25, 1995): 105–32. http://dx.doi.org/10.1017/s0022112095003442.

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The lateral capillary interaction between two particles immersed in a spherical thin liquid film is investigated. The interfacial shape, the lateral capillary force and the interparticle energy are calculated by using the numerical solution of the linearized Laplace equation of capillarity. Orthogonal bipolar coordinates on a sphere (inducing biconical coordinates in space) are introduced as a helpful instrument for solving this problem and other problems of similar geometry. We consider two types of boundary conditions at the particle surfaces: fixed contact angle and fixed contact line. We established that for particles of fixed contact angle the capillary interaction energy depends monotonically on the interparticle distance whereas for particles of fixed contact line the interaction energy exhibits a maximum. The numerical results show that in both cases the capillary interaction is much larger than the thermal energy kT and can induce aggregation and ordering of submicrometre particles. These theoretical findings can be important for understanding the properties of Pickering emulsions (stabilized by particles) and liposomes or biomembranes containing incorporated membrane proteins.

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Chen, P., S. S. Susnar, M. Pasandideh-Fard, J. Mostaghimi, and A. W. Neumann. "The generalized Laplace equation of capillarity II. Hydrostatic and thermodynamic derivations of the Laplace equation for high curvatures." Advances in Colloid and Interface Science 63 (January 1996): 179–93. http://dx.doi.org/10.1016/0001-8686(95)00283-9.

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Zloshchastiev, Konstantin G. "Temperature-driven dynamics of quantum liquids: Logarithmic nonlinearity, phase structure and rising force." International Journal of Modern Physics B 33, no. 17 (July 10, 2019): 1950184. http://dx.doi.org/10.1142/s0217979219501844.

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We study a large class of strongly interacting condensate-like materials, which can be characterized by a normalizable complex-valued function. A quantum wave equation with logarithmic nonlinearity is known to describe such systems, at least in a leading-order approximation, wherein the nonlinear coupling is related to temperature. This equation can be mapped onto the flow equations of an inviscid barotropic fluid with intrinsic surface tension and capillarity; the fluid is shown to have a nontrivial phase structure controlled by its temperature. It is demonstrated that in the case of a varying nonlinear coupling an additional force occurs, which is parallel to a gradient of the coupling. The model predicts that the temperature difference creates a direction in space in which quantum liquids can flow, even against the force of gravity. We also present arguments explaining why superfluids, be it superfluid components of liquified cold gases or Cooper pairs inside superconductors, can affect closely positioned acceleration-measuring devices.

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Andreianov, Boris, and Clément Cancès. "Vanishing capillarity solutions of Buckley–Leverett equation with gravity in two-rocks’ medium." Computational Geosciences 17, no. 3 (January 8, 2013): 551–72. http://dx.doi.org/10.1007/s10596-012-9329-8.

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França, A. C., L. R. Carrocci, and A. F. Siqueira. "VELOCITY PROFILE VISUALIZATION OF WATER NATURAL PERCOLATION IN A POROUS MEDIUM." Revista de Engenharia Térmica 8, no. 1 (June 30, 2009): 31. http://dx.doi.org/10.5380/reterm.v8i1.61878.

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This paper aims to show the profile and the behavior of the velocity of the water flow through a porous medium composed of clay and sand aggregated by burning in an oil furnace. The work models mathematics based on the Navier-Stokes differential equation, which represents the behavior of the water velocity flow in porous medium taking into account parameters of a low velocity laminar flow, increased load loss value and Number of Reynolds > 1. Physical phenomena such as porosity, permeability, particles arrangement, radius and wet perimeter are considered in the equation. The study shows the three-dimensional profile of the water percolation velocity which, originated from the capillary phenomenon, causes a sum of the tensions of increased values able to produce cracks in the medium structure. And, differently from filtration phenomenon, which overcomes the capillarity of the medium by the gravitational force or by efforts applied aiming to increase the flow velocity, the natural percolation opposes to the gravity and to the surrounding pressure moving slowly, reaching the flow at 30 and 40 centimeters depending on the permeability of the porous medium.

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BEDJAOUI, N., C. CHALONS, F. COQUEL, and P. G. LeFLOCH. "NON-MONOTONIC TRAVELING WAVES IN VAN DER WAALS FLUIDS." Analysis and Applications 03, no. 04 (October 2005): 419–46. http://dx.doi.org/10.1142/s0219530505000649.

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We investigate the existence and properties of traveling wave solutions for the hyperbolic-elliptic system of conservation laws describing the dynamics of van der Waals fluids. The model is based on a constitutive equation of state containing two inflection points and incorporates nonlinear viscosity and capillarity terms. A global description of the traveling wave solutions is provided. We distinguish between classical and non-classical trajectories and, for the latter, the existence and properties of kinetic functions is investigated. An earlier work in this direction (cf. [4]) was restricted to dealing with one inflection point only. Specifically, given any left-hand state and any shock speed (within some admissible range), we prove the existence of a non-classical traveling wave for a sequence of parameter values representing the ratio of viscosity and capillarity. Our analysis exhibits a surprising lack of monotonicity of traveling waves. The behavior of these non-classical trajectories is also investigated numerically.

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Rybka, Piotr, and Karl-Heinz Hoffmann. "Convergence of Solutions to the Equation of Quasi-Static Approximation of Viscoelasticity with Capillarity." Journal of Mathematical Analysis and Applications 226, no. 1 (October 1998): 61–81. http://dx.doi.org/10.1006/jmaa.1998.6066.

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33

Miles, John W. "The evolution of a weakly nonlinear, weakly damped, capillary-gravity wave packet." Journal of Fluid Mechanics 187 (February 1988): 141–54. http://dx.doi.org/10.1017/s0022112088000370.

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Longuet-Higgins's (1976) analysis of energy transfer within a narrow spectrum of gravity waves with approximately uncorrelated phases is generalized to accommodate capillarity and weak damping. The analysis is based on the corresponding generalization of Zakharov's (1968) evolution equation for weakly nonlinear, deep-water gravity-wave packets. The results for a symmetric normal spectrum are expressed in terms of elliptic integrals and depend, after appropriate scaling, on a single similarity parameter and on the sign of the curvature of the linear dispersion relation. Energy transfer is away from the peak of that spectrum if kl* < 0.393, where k is the wavenumber and l* is the capillary length (2.8 mm for water), but may be towards the peak if 0.343 < kl* < 0.707 (4.5 cm > 2π/k > 2.5 cm for water). The formulation is based on energy exchange through resonant quartets and is not valid in the neighbourhood of kl* = 0.707; at which the second harmonic of a capillary-gravity wave resonates with its fundamental (Wilton's ripples). The modulational instability of a weakly damped capillary-gravity wave is examined in an Appendix.

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WEI, HSIEN-HUNG, and DAVID S. RUMSCHITZKI. "The linear stability of a core–annular flow in an asymptotically corrugated tube." Journal of Fluid Mechanics 466 (September 10, 2002): 113–47. http://dx.doi.org/10.1017/s0022112002001210.

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This paper examines the core–annular flow of two immiscible fluids in a straight circular tube with a small corrugation, in the limit where the ratio ε of the mean undisturbed annulus thickness to the mean core radius and the corrugation (characterized by the parameter σ) are both asymptotically small and where the surface tension is small. It is motivated by the problems of liquid–liquid displacement in irregular rock pores such as occur in secondary oil recovery and in the evolution of the liquid film lining the bronchii in the lungs whose diameters vary over different generations of branching. We investigate the asymptotic base flow in this limit and consider the linear stability of its leading order (in the corrugation parameter) solution. For the chosen scalings of the non-dimensional parameters the core's base flow slaves that of the annulus. The equation governing the leading-order interfacial position for a given wall corrugation function shows a competition between shear and capillarity. The former tends to align the interface shape with that of the wall and the latter tends to introduce a phase shift, which can be of either sign depending on whether the circumferential or the longitudinal component of capillarity dominates. The asymptotic linear stability of this leading-order base flow reduces to a single partial differential equation with non-constant coefficients deriving from the non-uniform base flow for the time evolution of an interfacial disturbance. Examination of a single mode k wall function allows the use of Floquet theory to analyse this equation. Direct numerical solutions of the above partial differential equation agree with the predictions of the Floquet analysis. The resulting spectrum is periodic in α- space, α being the disturbance wavenumber space. The presence of a small corrugation not only modifies (at order σ2) the primary eigenvalue of the system. In addition, short-wave order-one disturbances that would be stabilized flowing to capillarity in the absence of corrugation can, in the presence of corrugation and over time scales of order ln(1/σ), excite higher wall harmonics (α±nk) leading to the growth of unstable long waves. Similar results obtain for more complicated wall shape functions. The main result is that a small corrugation makes a core–annular flow unstable to far more disturbances than would destabilize the same uncorrugated flow system. A companion paper examines that competition between this added destabilization due to pore corrugation with the wave steepening and stabilization in the weakly nonlinear regime.

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Dan, Han-Cheng, Pei Xin, Ling Li, Liang Li, and David Lockington. "Capillary effect on flow in the drainage layer of highway pavement." Canadian Journal of Civil Engineering 39, no. 6 (June 2012): 654–66. http://dx.doi.org/10.1139/l2012-050.

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This paper aims to examine capillarity effect on flows in the drainage layer of highway pavement. A two-dimensional (2-D) model based on the Richards equation was used to simulate saturated and unsaturated flows in the drainage layer. For comparison, flows were also simulated using a 1-D Boussinesq equation based model and a 2-D model based on the Laplace equation, both assuming saturated flow only. The drainage layer was modeled with sand and gravel, which possess similar hydraulic properties to those of commonly used filling materials in practice. The results showed that the two saturated flow models agreed well with each other, indicating the dominance of horizontal flow in the drainage layer. However, their predictions differed significantly from those of the variably saturated flow models. The latter model predicted significant flow activities in a relatively large unsaturated zone, especially for a sandy drainage layer. Such unsaturated flow contributes to and enhances the capacity of the drainage layer. With the unsaturated flow neglected, the saturated flow models over-predicted the extent of the saturated zone and hence the groundwater table elevation. As the current engineering design of the drainage layer is typically based on the groundwater table elevation predicted by the saturated flow models, the finding of this study suggests that the design criterion is likely to lead to over-design of the drainage system. Further work is also required to prove the practical significance of the capillary effect and account for other factors.

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WEI, HSIEN-HUNG, and DAVID S. RUMSCHITZKI. "The weakly nonlinear interfacial stability of a core–annular flow in a corrugated tube." Journal of Fluid Mechanics 466 (September 10, 2002): 149–77. http://dx.doi.org/10.1017/s0022112002001222.

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A core–annular flow, the concurrent axial flow of two immiscible fluids in a circular tube or pore with one fluid in the core and the other in the wetting annular region, is frequently used to model technologically important flows, e.g. in liquid–liquid displacements in secondary oil recovery. Most of the existing literature assumes that the pores in which such flows occur are uniform circular cylinders, and examine the interfacial stability of such systems as a function of fluid and interfacial properties. Since real rock pores possess a more complex geometry, the companion paper examined the linear stability of core–annular flows in axisymmetric, corrugated pores in the limit of asymptotically weak corrugation. It found that short-wave disturbances that were stable in straight tubes could couple to the wall's periodicity to excite unstable long waves. In this paper, we follow the evolution of the axisymmetric, linearly unstable waves for fluids of equal densities in a corrugated tube into the weakly nonlinear regime. Here, we ask whether this continual generation of new disturbances by the coupling to the wall's periodicity can overcome the nonlinear saturation mechanism that relies on the nonlinear (kinematic-condition-derived) wave steepening of the Kuramoto–Sivashinsky (KS) equation. If it cannot, and the unstable waves still saturate, then do these additional excited waves make the KS solutions more likely to be chaotic, or does the dispersion introduced into the growth rate correction by capillarity serve to regularize otherwise chaotic motions?We find that in the usual strong surface tension limit, the saturation mechanism of the KS mechanism remains able to saturate all disturbances. Moreover, an additional capillary-derived nonlinear term seems to favour regular travelling waves over chaos, and corrugation adds a temporal periodicity to the waves associated with their periodical traversing of the wall's crests and troughs. For even larger surface tensions, capillarity dominates over convection and a weakly nonlinear version of Hammond's no-flow equation results; this equation, with or without corrugation, suggests further growth. Finally, for a weaker surface tension, the leading-order base flow interface follows the wall's shape. The corrugation-derived excited waves appear able to push an otherwise regular travelling wave solution to KS to become chaotic, whereas its dispersive properties in this limit seem insufficiently strong to regularize chaotic motions.

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Obersnel, Franco, and Pierpaolo Omari. "Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation." Discrete & Continuous Dynamical Systems - A 33, no. 1 (2013): 305–20. http://dx.doi.org/10.3934/dcds.2013.33.305.

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Dan, Han-Cheng, Pei Xin, Ling Li, Liang Li, and David Lockington. "Boussinesq Equation-Based Model for Flow in the Drainage Layer of Highway with Capillarity Correction." Journal of Irrigation and Drainage Engineering 138, no. 4 (April 2012): 336–48. http://dx.doi.org/10.1061/(asce)ir.1943-4774.0000404.

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Li, L., D. A. Barry, F. Stagnitti, and J. Y. Parlange. "Groundwater waves in a coastal aquifer: A new governing equation including vertical effects and capillarity." Water Resources Research 36, no. 2 (February 2000): 411–20. http://dx.doi.org/10.1029/1999wr900307.

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van Duijn, C. J., K. Mitra, and I. S. Pop. "Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure." Nonlinear Analysis: Real World Applications 41 (June 2018): 232–68. http://dx.doi.org/10.1016/j.nonrwa.2017.10.015.

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Duruk, Selin, Edouard Boujo, and Mathieu Sellier. "Thin Liquid Film Dynamics on a Spinning Spheroid." Fluids 6, no. 9 (September 6, 2021): 318. http://dx.doi.org/10.3390/fluids6090318.

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The present work explores the impact of rotation on the dynamics of a thin liquid layer deposited on a spheroid (bi-axial ellipsoid) rotating around its vertical axis. An evolution equation based on the lubrication approximation was derived, which takes into account the combined effects of the non-uniform curvature, capillarity, gravity, and rotation. This approximate model was solved numerically, and the results were compared favorably with solutions of the full Navier–Stokes equations. A key advantage of the lubrication approximation is the solution time, which was shown to be at least one order of magnitude shorter than for the full Navier–Stokes equations, revealing the prospect of controlling film dynamics for coating applications. The thin film dynamics were investigated for a wide range of geometric, kinematic, and material parameters. The model showed that, in contrast to the purely gravity-driven case, in which the fluid drains downwards and accumulates at the south pole, rotation leads to a migration of the maximum film thickness towards the equator, where the centrifugal force is the strongest.

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Weissmüller, Jörg. "(Invited) Mechano-Chemical Coupling at Interfaces in Novel Hybrid Materials." ECS Meeting Abstracts MA2018-01, no. 32 (April 13, 2018): 1983. http://dx.doi.org/10.1149/ma2018-01/32/1983.

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The strong coupling between chemistry or electrochemistry and mechanics at interfaces may be exploited for designing materials with unexpected functional behavior. One example is provided by nanoporous-metal based hybrid materials that behave similar to piezoelectric ceramics: Bodies of nanoporous gold impregnated with electrolyte emit exceptionally robust electric signals when subjected to external load. The metal-based material may thus be considered as piezoelectric, in a literal interpretation of the term. Another example is a class of novel phenomena in which the mechanical response to load – which may be elastic or plastic – can be tuned by an electric potential. The relevant capillary parameters can be either, surface stress or surface tension. In each case, a coupling between the capillarity and electricity is crucial, and the coupling strengths can be quite different. While the energy-charge coupling at a surface is described by the Lippmann equation, a predictive understanding of the interfacial stress-charge coupling, which governs the apparent piezoelectricity, remains elusive. The talk will discuss relevant issues from the perspectives of experiment, phenomenological thermodynamics, and atomistics.

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Chapuis, Robert P. "Numerical modeling of rising-head permeability tests in monitoring wells after lowering the water level down to the screen." Canadian Geotechnical Journal 42, no. 3 (June 1, 2005): 705–15. http://dx.doi.org/10.1139/t05-003.

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To begin a rising-head permeability test in a monitoring well (MW), the water level is lowered in the pipe. If it is lowered down to the screen, the recovery graph may differ from the theoretical straight line, making it difficult to assess the mean field hydraulic conductivity. A numerical analysis (finite element method) of this type of test, considering the complete equations for saturated and unsaturated flow, is presented. The numerically obtained graphs are similar to those of real field tests. In an aquitard, when the filter pack is made of fine sand, which retains water by capillarity, the screen dewatering influence is hardly visible in the velocity graph and is undetected in the usual semilog graph. In an aquitard, when the filter pack cannot retain water by capillarity during dewatering, the semilog graph presents two straight-line portions. The velocity graph, a representation of the conservation equation, helps to distinguish the early time interval, when the groundwater fills the screen and the filter pack, and the later interval, when it fills only the pipe. The later portion of the graph must be used to calculate the hydraulic conductivity. In an unconfined aquifer, when there is no filter pack, dewatering down to the screen by pumping significantly lowers the water table around the MW. The usual semilog graph appears as a set of two straight lines. The velocity graph indicates that all calculations must consider a piezometric level that is lower than that measured before dewatering. In all cases, the velocity graph shows clearly what happened during the numerically simulated tests. The more complex case of an MW installed with a filter pack in an unconfined aquifer and tested using a mechanical slug was not numerically examined in this paper.Key words: hydraulic conductivity, rising head, monitoring well, numerical analysis.

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Hopper, Robert W. "Plane Stokes flow driven by capillarity on a free surface. Part 2. Further developments." Journal of Fluid Mechanics 230 (September 1991): 355–64. http://dx.doi.org/10.1017/s0022112091000824.

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For the free creeping viscous incompressible plane flow of a finite region, bounded by a simple smooth closed curve and driven solely by surface tension, analyzed previously, the shape evolution was described in terms of a time-dependent mapping function z = Ω(ζ,t) of the unit circle, conformal on |ζ| [les ] 1. An equation giving the time evolution of the map, typically in parametric form, was derived. In this article, the flow of the infinite region exterior to a hypotrochoid is given. This includes the elliptic hole, which shrinks at a constant rate with a constant aspect ratio. The theory is extended to a class of semi-infinite regions, mapped from Im ζ [les ] 0, and used to solve the flow in a half-space bounded by a certain groove. The depth of the groove ultimately decays inversely with time.

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45

Snyder, G. K. "Estimating diffusion distances in muscle." Journal of Applied Physiology 63, no. 5 (November 1, 1987): 2154–58. http://dx.doi.org/10.1152/jappl.1987.63.5.2154.

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Models of fibers and capillaries in cross sections of muscle were used to quantify the relationships between diffusion distances and tissue capillarity. The fibers were constructed as square and hexagonal arrays, and the placement of capillaries around the perimeters of the fibers ordered them in similar arrays. Diffusion distances were measured as the percent cumulative frequency of fiber area within a given distance of a capillary when capillary-to-fiber ratio was increased from 0.5 to 4.0. Equations fitted to the data make it possible to estimate diffusion distances in muscle and to correlate changes in diffusion distances with fiber growth, capillary growth, and the geometrical arrangement of capillaries in the muscle bed.

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Kačur, Jozef, and Patrik Mihala. "Heat and Mass Transport in Unsaturated-Saturated Porous Media Including Water Suspension and Inner Heat Exchange." Defect and Diffusion Forum 407 (March 2021): 138–46. http://dx.doi.org/10.4028/www.scientific.net/ddf.407.138.

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Water suspension with heat transport into unsaturated-saturated porous media is analyzed. The numerical modeling includes the infiltration of silt. Moreover, the heat energy of suspension is exchanged with the heat energy of the matrix. The deposited silt influence the porosity and hydraulic permeability. The flow model is based on Richard's type equation and empirical van Genuchten - Mualem model describing capillarity driving force and saturation-pressure relation governing the flow in unsaturated part of porous media. The developed numerical method is usable for solving inverse problems determining some model parameters. The numerical method is based on flexible time discretization and a finite volume method in space variables. The nonlinearity in the flow part of the model is solved by iterative linearization based on the idea in Celia et all. The correctness of the numerical approximation is justified also by a different numerical approximation based on space discretization leading to the reduction of the whole system to the solution of ordinary differential equations. But this method requires significantly more computation time. This is not suitable for solving inverse problems. The used method is justified in numerical experiments solving the direct problem.

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Kou, Jisheng, Shuyu Sun, and Bo Yu. "Multiscale Time-Splitting Strategy for Multiscale Multiphysics Processes of Two-Phase Flow in Fractured Media." Journal of Applied Mathematics 2011 (2011): 1–24. http://dx.doi.org/10.1155/2011/861905.

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The temporal discretization scheme is one important ingredient of efficient simulator for two-phase flow in the fractured porous media. The application of single-scale temporal scheme is restricted by the rapid changes of the pressure and saturation in the fractured system with capillarity. In this paper, we propose a multi-scale time splitting strategy to simulate multi-scale multi-physics processes of two-phase flow in fractured porous media. We use the multi-scale time schemes for both the pressure and saturation equations; that is, a large time-step size is employed for the matrix domain, along with a small time-step size being applied in the fractures. The total time interval is partitioned into four temporal levels: the first level is used for the pressure in the entire domain, the second level matching rapid changes of the pressure in the fractures, the third level treating the response gap between the pressure and the saturation, and the fourth level applied for the saturation in the fractures. This method can reduce the computational cost arisen from the implicit solution of the pressure equation. Numerical examples are provided to demonstrate the efficiency of the proposed method.

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Belay, Tsegay, Kim Chun IL, and Peter Schiavone. "Mechanics of a lipid bilayer subjected to thickness distension and membrane budding." Mathematics and Mechanics of Solids 23, no. 1 (September 13, 2016): 67–84. http://dx.doi.org/10.1177/1081286516666136.

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We study the distension-induced gradient capillarity in membrane bud formation. The budding process is assumed to be primarily driven by diffusion of transmembrane proteins and acting line tensions on the protein-concentrated interface. The proposed model, based on the Helfrich-type potential, is designed to accommodate inhomogeneous elastic responses of the membrane, non-uniform protein distributions over the membrane surface and, more importantly, the thickness distensions induced by bud formations in the membrane. The latter are employed via the augmented energy potential of bulk incompressibility in a weakened manner. By computing the variations of the proposed membrane energy potential, we obtained the corresponding equilibrium equation (membrane shape equation) describing the morphological transitions of the lipid membrane undergoing bud formation and the associated thickness distensions. The effects of lipid distension on the shape equation and the necessary adjustments to the accompanying boundary conditions are also derived in detail. The resulting shape equation is solved numerically for the parametric representation of the surface which has one-to-one-correspondence with the membrane surface under consideration. The proposed model successfully predicts the bud formation phenomenon on a flat lipid membrane and the associated thickness distensions of the membrane and demonstrates a smooth transition from one phase to the other (including necking domains). It is also found that the final deformed configuration is energetically favorable and therefore is stable. Finally, we show that the inhomogeneous thickness deformation on the membrane in response to transmembrane protein diffusion makes a significant contribution to the budding and necking processes of the membrane.

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Zhang, Tao, and Shuyu Sun. "Thermodynamics-Informed Neural Network (TINN) for Phase Equilibrium Calculations Considering Capillary Pressure." Energies 14, no. 22 (November 18, 2021): 7724. http://dx.doi.org/10.3390/en14227724.

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The thermodynamic properties of fluid mixtures play a crucial role in designing physically meaningful models and robust algorithms for simulating multi-component multi-phase flow in subsurface, which is needed for many subsurface applications. In this context, the equation-of-state-based flash calculation used to predict the equilibrium properties of each phase for a given fluid mixture going through phase splitting is a crucial component, and often a bottleneck, of multi-phase flow simulations. In this paper, a capillarity-wise Thermodynamics-Informed Neural Network is developed for the first time to propose a fast, accurate and robust approach calculating phase equilibrium properties for unconventional reservoirs. The trained model performs well in both phase stability tests and phase splitting calculations in a large range of reservoir conditions, which enables further multi-component multi-phase flow simulations with a strong thermodynamic basis.

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Obersnel, Franco, Pierpaolo Omari, and Sabrina Rivetti. "Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions." Nonlinear Analysis: Real World Applications 13, no. 6 (December 2012): 2830–52. http://dx.doi.org/10.1016/j.nonrwa.2012.04.012.

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Journal articles on the topic 'Capillarity equation'

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