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We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the capillary surface. Using a single graph representation, we interpret this geometric motion as a gradient flow on a Hilbert manifold. We propose unconditionally stable first/second order numerical schemes to simulate this geometric motion of the droplet, which is described using motion by mean curvature coupled with moving contact lines. The schemes are based on (i) explicit moving boundaries, which decouple the dynamic updates of the contact lines and the capillary surface, (ii) a semi-Lagrangian method on moving grids and (iii) a predictor-corrector method with a nonlinear elliptic solver upto second order accuracy. For the case of quasi-static dynamics with continuous spatial variable in the numerical schemes, we prove the stability and convergence of the first/second order numerical schemes. To demonstrate the accuracy and long-time validation of the proposed schemes, several challenging computational examples - including breathing droplets, droplets on inhomogeneous rough surfaces and quasi-static Kelvin pendant droplets - are constructed and compared with exact solutions to quasi-static dynamics obtained by desingularized differential-algebraic system of equations (DAEs)., 8 figures, 48 pages

This work presents a novel strategy for the construction of a capillary column with high resolving performance and inertness for gas chromatographic (GC) analysis by combining the proposed surface modification with a selective stationary phase. Through the modified biphasic sol–gel reaction inside a capillary GC column, porous modification was made on the capillary surface (PM capillary) prior to coating with a stationary phase. Herein, we designed and synthesized a new triptycene-based stationary phase (TPPG) for the study. Using the proposed strategy, the PM-TPPG column was fabricated with a column efficiency of 4950 plates per m determined by n-dodecane at 120 °C and then investigated for its separation performance, inertness, repeatability, loadability and thermal stability. As demonstrated, the PM-TPPG column showed distinct advantages in these aspects over two reference columns (TPPG column and commercial PEG column). Particularly, the PM-TPPG column well resolved all the components in the Grob mixture and exhibited distinctly improved peak shapes for carboxylic acids and heterocyclic bases. Moreover, it exhibited excellent repeatability and reproducibility and higher column loadability than the two reference columns. Its applications to purity monitoring of chemical products demonstrate its good potential for practical GC analysis.

The paper considers three-dimensional (3D) traveling surface waves on water of finite depth under the forces of gravity and surface tension using the exact governing equations, also called Euler eq...

Шляхом математичного моделювання квазістатики капілярних поверхонь типу лежача крапля отримана методологічна база дослідження поверхнево-активних речовин за динамікою поверхневого натягу методом пульсуючого меніска. Сутність методу полягає в тому, що, шляхом прямої і зворотної подачі газу у меніск, реалізується процес пульсації меніска в околі максимального тиску в ньому, значення якого однозначно визначає поверхневий натяг на кожному циклі. Ефект полягає в тому, що меніск, при кожному циклі вимірювання, не захлопується після моменту досягнення максимального тиску у ньому, а переходить у новий стійкий стан рівноваги, тобто процес адсорбції поверхнево-активних речовин не припиняється. Завдяки цьому запропонований метод відповідає точності вимірювання поверхневого натягу за методом максимального тиску у бульбашці і оперативності методу лежачої краплі. Можливість реалізації запропонованого методу теоретично обґрунтовано шляхом розробки математичної моделі процесу пульсації меніска в околі максимального тиску у ньому в системі замкнутій відносно кількості газової фази. На основі теоретичних досліджень результатів моделювання запропоновано структурну схему і розроблено алгоритм процесувизначення динамічних характеристик поверхневого натягу., By mathematical modeling of quasi-statics of capillary surfaces of the lying drop type, the methodological basis for the study of surfactants on the dynamics of surface tension by the method of pulsating meniscus was obtained. The essence of the method is that, by direct and reverse supply of gas to the meniscus, the process of pulsation of the meniscus in the vicinity of the maximum pressure in it, the value of which uniquely determines the surface tension on each cycle. The effect is that the meniscus, with each measurement cycle, does not close after reaching the maximum pressure in it, but goes into a new steady state of equilibrium, ie the process of adsorption of surfactants does not stop. Due to this, the proposed method corresponds to the accuracy of measuring the surface tension by the method of maximum pressure in the bubble and the efficiency of the lying drop method. The possibility of implementing the proposed method is theoretically justified by developing a mathematical model of the process of pulsation of the meniscus in the vicinity of the maximum pressure in the system closed relative to the amount of gas phase. Based on theoretical studies of the simulation results, a block diagram is proposed and an algorithm for the process of determining the dynamic characteristics of surface tension is developed.

Nanoparticles are widely used in capillary electrophoresis as stationary phases adsorbed on the internal capillary walls for the separation and concentration of analytes in capillary electrochromatography. The fastest and simplest approach for the formation of coatings is a physical adsorption of nanoparticles. Nevertheless, the formed coatings frequently possess low stability. The layer-by-layer approach for the formation of stabile and dense coating of internal capillary walls based on negatively charged nanosized cation-exchanger was proposed. The method included the sequential alteration of appositively charged layers of nanosized ion-exchangers. The proposed “anion-exchanger – cation-exchanger” bilayer coating possesses high stability in wide pH range (2-10) and provides up to 120 analyses without the need of re-coating. The coating was applied for the separation and on-line concentration of catecholamines and amino acids in capillary electrochromatography mode. High efficiencies were achieved (N = 450-720 th. t.p./m and N = 400-520 th. t.p./m for cathecholamies and amino acids, respectively), while the analysis time was significantly decreased. It was established, that high concentration of negatively charged functional groups on the capillary surface led to the increase of stacking efficiency factors due to the interactions between analytes and functional groups of the modifier on the capillary walls. It contributed to the 2-4 times reduced detection limits (LODs) of analytes compared to the mono-layer coatings (LODs of catecholamines = 3-4 ng/mL, LODs of amino acids = 40-100 ng/mL).Keywords: nano-sized anion-exchanger, nano-sized cation-exchanger, bilayer coatings, capillary electrochromatography, biogenic amines, amino acids, on-line concentration DOI: http://dx.doi.org/10.15826/analitika.2021.25.3.004 Makeeva D.V., Polikarpova D.A., Kartsova L.A. Saint-Petersburg State University, Institute of Chemistry, Наночастицы активно используются в капиллярном электрофорезе в качестве стационарных фаз, адсорбированных на поверхности кварцевого капилляра, в целях улучшения электрофоретического разделения аналитов в режиме капиллярной электрохроматографии. Формирование покрытий за счет физической адсорбции полимерных наночастиц – наиболее быстрый и простой в исполнении метод. Однако полученные покрытия зачастую характеризуются низкой стабильностью. В работе предложен подход к формированию плотного и стабильного покрытия стенок кварцевого капилляра с внешним отрицательно-заряженным слоем нанокатионита, заключающийся в последовательном нанесении за счет физической адсорбции противоположно-заряженных слоев нанормазмерных ионообменников. Сформировано двухслойное покрытие «наноанионит-нанокатионит», отличающееся высокой стабильностью в диапазоне рН от 2 до 10, и позволяющее проводить до 120 циклов анализа без обновления покрытия. Полученные снимки внутренней поверхности капилляров на сканирующем электронном микроскопе позволили провести сравнение плотности и равномерности одно- и двухслойных покрытий на основе наноионитов. Аналитические возможности двухслойного покрытия продемонстрированы при анализе смесей биогенных аминов и аминокислот в режиме капиллярной электрохроматографии. Достигнуты высокие значения эффективности (N = 450-720 тыс т.т./м и N = 400-520 тыс т.т./м для катехоламинов и аминокислот, соответственно). Высокая концентрация отрицательно заряженных ионогенных групп на поверхности капилляра способствует увеличению степеней концентрирования, достигаемых в результате электростэкинга, поскольку к общему механизму концентрирования добавляется взаимодействие аналита с функциональными центрами поверхности капилляра. Это позволило снизить пределы обнаружения аналитов в 2-4 раза по сравнению с однослойным НИК-покрытием (3-4 нг/мл в случае катехоламинов и 40-100 нг/мл для аминокислот).Ключевые слова: наноанионит, нанокатионит, двухслойные покрытия, капиллярная электрохроматография, биогенные амины, аминокислоты, концентрированиеDOI: http://dx.doi.org/10.15826/analitika.2021.25.3.004

The novel drug delivery system refers to the formulations and technologies for transporting a pharmaceutical compound in the body as it is needed to safely achieve its desired therapeutic effects. In this study, the onset vibrational amplitude of capillary surface waves for ultrasonic atomization spray is explained based on Faraday instability. Using ultrasonic frequency, the vibrational amplitude approached a critical point, and the liquid surface broke up into tiny drops. The micro-droplets were are steadily and continuously formed after the liquid feeding rate was optimized. The simulation study reported a minimum vibrational amplitude or onset value of 0.38 μm at 500 kHz frequency. The required minimum energy to atomize the drops was simulated by COMSOL Multiphysics simulation software. The simulation result agreed well with the numerical results of a subharmonic vibrational model that ocurred at 250 kHz frequency on the liquid surface. This newly designed single frequency ultrasonic atomizer showed its true physical characteristic of resonance on the fluid surface plane. Hence, this research will contribute to the future development of a single-frequency ultrasonic nebulizer and mechatronics for the generation of uniform atomized droplets.

Streaming potential is created when an electrolyte solution is forced to flow pass a charged surface. For an uncoated fused silica capillary, the streaming potential is measured between the inlet and outlet vials while applying a pressure across the capillary. The changes in streaming potential can be used to characterize the properties of the capillary inner surface. In this work, HCl, NaCl, and NaOH solutions ranging from 0.4 to 6 mM were used as the background electrolyte (BGE) at temperatures of 15 to 35 °C for the mesurements. The streaming potential decreases with the increase in BGE concentration, and the trend is amplified at higher temperatures. When buffer solutions in the pH range of 1.5 to 12.7 were used as the BGE, streaming potential was shown to be sensitive to changes in pH but reaches a maximum at around 9.5. At pH

The impact of temperature inhomogeneity, surface charge and surface mass densities on the stability analysis of charged surface waves at the interface between dense, incompressible, radiative, self-gravitating magnetized electron–ion plasma and vacuum is investigated. For such an incompressible plasma system, the temperature inhomogeneity is governed by an energy balance equation. Adopting the one-fluid magnetohydrodynamic (MHD) approximation, a general dispersion relation is obtained for capillary surface waves, which takes into account gravitational, radiative and magnetic field effects. The dispersion relation is analysed to obtain the conditions under which the plasma–vacuum interface may become unstable. In the absence of electromagnetic (EM) pressure, astrophysical objects undergo gravitational collapse through Jeans surface oscillations in contrast to the usual central contraction of massive objects due to enhanced gravity. EM radiation does not affect the dispersion relation much, but actually tends to stabilize the Jeans surface instability. In certain particular cases, pure gravitational radiation may propagate on the plasma vacuum interface. The growth rate of radiative dissipative instability is obtained in terms of the wavevector. Our theoretical model of the Jeans surface instability is applicable in astrophysical environments and also in laboratory plasmas.

A capillary surface bound by a solid rectangular channel exhibits dynamic wetting effects characterized by a constitutive law relating the dynamic contact-angle to the contact-line speed through the contact-line mobility $$\Lambda $$ parameter. Limiting cases correspond to the free ( $$\Lambda =0$$ ) and pinned ( $$\Lambda =\infty $$ ) contact-line. Viscous potential flow is used to derive the governing integrodifferential equation from a boundary integral approach. The spectrum is determined from a boundary value problem where the eigenvalue parameter appears in the boundary condition. Here we introduce a new frequency scan approach to compute the spectrum, whereby we scan the complex frequency plane and compute the system response from which we identify the complex resonant frequency. Damping effects due to viscosity and Davis dissipation from finite $$\Lambda $$ do not attenuate signal response, but rather shift the response poles into the complex plane. Our new approach is verified against an analytical solution in the appropriate limit. We identify the critical mobility that maximizes Davis dissipation and the critical Ohnesorge number (viscosity) where the transition from underdamped to overdamped oscillations occurs, as it depends upon the static contact-angle $$\alpha $$ . Our approach is applied to a rectangular channel, but is suitable for a myriad of geometric supports.

In this study, the onset amplitude of the initial capillary surface wave for ultrasonic atomization of fluids has been implemented. The design and characterization of 485 kHz microfabricated silicon-based ultrasonic nozzles are presented for the concept of economic energy development. Each nozzle is composed of a silicon resonator and a piezoelectric drive section consisting of three Fourier horns. The required minimum energy to atomize liquid droplets is verified by COMSOL Multiphysics simulation software to clarify experimental data. The simulation study reports a minimum vibrational amplitude (onset) of 0.365 μm at the device bottom under the designated frequency of 485 kHz. The experimental study agrees well with the suggested frequency and the amplitude concerning the corresponding surface vibrational velocity in simulation. While operating, the deionized water was initially atomized into microdroplets at the given electrode voltage of 5.96 V. Microdroplets are steadily and continuously formed after the liquid feeding rate is optimized. This newly designed ultrasonic atomizer facilitates the development of capillary surface wave resonance at a designated frequency. A required vibrational amplitude and finite electric driving voltage promote not only the modern development in the green energy industry, but also the exploration of noninvasive, microencapsulated drug delivery and local spray needs.

A direct numerical simulation of the process of binary alloy separation in a thin nonuniformly heated circular capillary is carried out. The key point of this process is the assumption of the existence of a thin gas layer between the melt and the boundary of the reservoir due to the non-wetting conditions. This effect was described using the equations of interphase hydrodynamics, which made it possible to construct a phenomenological model of the processes at the melt-solid interface for mixtures of liquid metals. The problem was solved by the finite difference method in combination with an explicit scheme on the PGNIU-Kepler supercomputer at the Research and Education Center “Parallel and Distributed Computing” of the Perm State National Research University. The velocity and temperature fields, as well as the concentration of melt components in the bulk and on the surface, were determined through numerical simulation. The dynamics of the separation process is described. It is shown that the longitudinal temperature gradient and the non-wetting conditions on the lateral surface create a downward motion of the melt along the boundary, which, together with the adsorption and desorption effects, leads to the formation of volumetric nonuniformity of the concentration of the mixture components along the capillary. The experimentally observed time dependence of the difference in the volume concentrations of the components at the ends of the capillary is reproduced. The distribution of volume concentrations of the components and the surface phase along the capillary is analyzed at various values of the adsorption and desorption coefficients and the Marangoni number. The dependence of the concentration difference for the melt components at the ends of the capillary as function of its length is studied. It is found that the separation effect intensifies with an increase in the capillary length. A qualitative and quantitative comparison of most of the characteristics demonstrates a good correlation of the calculation results with the data previously obtained for the plane problem and the available experimental data. A numerical experiment shows that the motion in the surface layer is rather intense. The heavy component is transported by the convective flow to the lower part of the capillary so that its concentration increases by almost an order of magnitude on 1/8 of the capillary surface.

Flow behavior is of major importance in the extrusion processing of rubber compounds. It is evaluated by means of a series of tests on a high-pressure capillary viscometer (HCV). Adhesion between the polymer melt and the capillary wall is assumed in all current calculation models, although such adhesion does not always pertain to the case of rubber compounds. To date, no uniform model discussed in the literature on the topic extensively describes the wall slippage behavior of rubber compounds. The phenomenon of wall slippage is analyzed by determining the power-law parameters n (flow exponent) and K (consistency factor) from the flow curve in the subcritical flow range. This makes it possible to explicitly calculate first the slip velocity and then the slippage ratio relative to the total volume flow as a function of the given shear rate and temperature. The work is based on the testing of EPDM raw polymers of different molecular weights in the HCV. In addition, EPDM compounds containing either a carbon black or a softener were analyzed with regard to their flow behavior. The rheological analysis was carried out on three variously coated flow channels. It was observed that with attainment of a critical wall shear stress, the wall slippage effect becomes more pronounced; thus, occurrences of flow anomalies such as slip-stick or shark-skin significantly influence processing and flow behavior. Wall slippage effects are noticeable, however, even before the critical wall shear stress is attained.

Electroosmotic flow (EOF) plays a pivotal role in optimization of capillary electrophoresis (CE) separations of (bio)molecules and (bio)particles. EOF velocity is directly related to analysis time, peak resolution and separation efficiency. Here, we report a concept of charged polymer coatings of the inner fused silica capillary wall, which allows anodic EOF with mobility ranging from 0 to ∼(30–40) × 10−9 m2V−1s−1. The capillary wall is modified by covalently bound cationic copolymer poly(acrylamide-co-(3-acrylamidopropyl)trimethylammonium chloride) (PAMAPTAC) containing variable ratio of the charged monomer in the 0–60 mol. % interval. The EOF mobility showed minor variability with composition of background electrolyte (BGE) and pH in the 2–10 interval. The coatings were evaluated by CE-UV and nanospray CE-MS in the counter-EOF arrangement for a series of basic drug molecules in acetic acid based acidic BGE. Tunable EOF velocity was demonstrated as a useful tool for optimization of peak resolution, separation efficiency and migration time of analytes. Electrostatic repulsion of positively charged capillary surface was shown as beneficial for suppression of analyte adsorption, notably for hydrophobic cationic analytes.

The transitory change of fluid and solute transport parameters occurring during the initial phase of a peritoneal dialysis dwell is a well-documented phenomenon; however, its physiological interpretation is rather hypothetical and has been disputed. Two different explanations were proposed: (1) the prevailing view—supported by several experimental and clinical studies—is that a vasodilatory effect of dialysis fluid affects the capillary surface area available for dialysis, and (2) a recently presented alternative explanation is that the molecular radius of glucose increases due to the high glucose concentration in fresh dialysis fluid and that this change affects peritoneal transport parameters. The experimental bases for both phenomena are discussed as well as the problem of the accuracy necessary for a satisfactory description of clinical data when the three-pore model of peritoneal transport is applied. We show that the correction for the change of transport parameters with dwell time provides a better fit with clinical data when applying the three-pore model. Our conclusion is in favor of the traditional interpretation namely that the transitory change of transport parameters with dwell time during peritoneal dialysis is primarily due to the vasodilatory effect of dialysis fluids.

The focusing property of an ellipsoidal monocapillary has been characterized using the ptychography method with a 405 nm laser beam. The recovered wavefront gives a 12.5×10.4μm2 focus. The reconstructed phase profile of the focused beam can be used to estimate the height error of the capillary surface. The obtained height error shows a Gaussian distribution with a standard deviation of 1.3 μm. This approach can be used as a quantitative tool for evaluating the inner functional surfaces of reflective optics, complementary to conventional metrology methods.

Modifications to the flow profile used in open tube capillary liquid chromatography (OT-CLC) include using slip-flow walls and using electroosmosis as a fluid pump as practiced in electrochromatography. These modifications are implemented experimentally by changing the capillary surface and solvent conditions which results in the change of boundary conditions at the capillary wall. In this paper we employ a theory-based study and compare the zone broadening of simple solutes using parabolic flow from a liquid pump, slip-flow from a highly hydrophobic inner surface with water eluent, and electroosmosis for the conditions of pure water and dilute salt utilizing 2 µm inner diameter OT capillaries. In general, the two types of flow other than parabolic exhibit thin zones in the early part of the chromatogram, consistent with previous studies of slip-flow and electroosmotic flow used in electrochromatography. Electrochromatography is shown to yield higher efficiency and less zone broadening than parabolic and slip-flow conditions used in this study. Nonetheless, it is found that the zone standard deviations are shown to be similar for these flow profiles as is the number of plates for these different flow profiles under the conditions utilized here. It is revealed that these modifications do not warrant the effort to maintain the special solvent conditions when compared to gradient elution OT-CLC, which gives a nearly constant peak width throughout the chromatogram, is easiest to implement, and is the method of choice for complex analysis.

Pattern-forming instability in various fields is an interesting research topic because of its complex physical nature and numerous applications. In this paper, we experimentally study capillary surface waves and patterns formed on a liquid film, cast on a plane substrate without physical walls, but pinned to the substrate edges, and subjected to multi-axis horizontal (lateral) oscillations (55–333 Hz). The effect of single-axis ultrasonic horizontal vibrations (20–170 kHz) was also investigated. We show that using substrates with different geometrical shapes and various travelling paths created by multi-axis vibrations with a phase angle difference between the axes produce a plethora of standing and travelling wave patterns on the liquid film surface. We report perfect standing square and spiral-like patterns for low-frequency multi-axis horizontal vibrations, which are commonly observed for vertical vibrations, while the mechanisms of momentum transfer to the liquid film from the vibrating substrate are different in vertical and horizontal vibrations. Other patterns forming on the liquid film surface in our experiments include lines/stripes, circles, swirls, pentagons, triangles, etc. It is also reported that low-frequency excitations create harmonic travelling waves and standing patterns, while the frequency of response waves generated by the application of ultrasonic horizontal vibrations is several orders of magnitude less than the excitation frequency. No subharmonic cross-waves are observed in this study, which strengthens the idea that plane substrates (without walls) are a good approximation for the theoretical case of a horizontally vibrated liquid film with infinite lateral length.

The guiding of highly charged ions through a single nanocapillary is simulated in comparison with previous experiments performed with highly insulating polyethylene terephthalate (PET). The simulations are carried out using 3-keV Ne7+ ions injected into capillaries with diameters ranging from 100 nm to 400 nm. In the calculations, non-linear effects are applied to model the charge transport along the capillary surface and into the bulk depleting the deposited charges from the capillary walls. In addition to the surface carrier mobility, the non-linear effects are also implemented into the bulk conductivity. A method is presented to determine the parameters of the surface charge transport and the bulk conductivity by reproducing the oscillatory structure of the mean emission angle. A common set of charge depletion rates are determined with relatively high accuracy providing confidence in the present theoretical analysis. Significant differences in the oscillatory structures, experimentally observed, are explained by the calculations. Experimental and theoretical results of the guiding power for capillaries of different diameters are compared. Finally, dynamic non-linear effects on the surface and bulk relaxation rates are determined from the simulations.

Background: Waniewski postulated a transient increase in peritoneal capillary surface area to fit their model predictions to experimental data of Heimburger measured in renal failure (RF) patients undergoing peritoneal dialysis (PD) but with only a 3.86% glucose dialysis fluid. The present aim is to propose a new mathematical model of the patient PD procedure that could closely fit the complete Heimburger measurement set without this postulate. Methods: The three-pore model of Rippe was used to describe transient changes in peritoneal volume and solute concentrations during a PD dwell. The predialysis, RF patient, plasma solute concentrations were assumed to remain constant during the dwell. The model was validated using the 3.86% glucose Heimburger measurements. Permeability surface area product parameters were chosen to match only the end-dwell peritoneal fluid glucose concentration and the end-dwell amounts of urea, creatinine, and Na+ removed from this simulated patient group. Then, this model was used to predict additional measurements by Heimburger on two other patient groups dialyzed with glucose concentrations of 2.27% and 1.36%, respectively. Parameters were unchanged when simulating these other patient groups. Results: To match the shape of the transient changes in drained volume and dialysis fluid glucose concentration for the 3.86% glucose group, it was necessary for only one parameter, the effective radius of glucose, to vary linearly in proportion to the dialysis fluid glucose concentration. This description was unchanged in the other two groups. Conclusion: Postulated transient increases in peritoneal capillary surface area were unnecessary to predict the entire Heimburger measurements.

The paper concerns the controllability and stabilization of surface water waves in a two-dimensional rectangular basin under the forces of gravity and surface tension. The surface waves are generated by a wave-maker placed at the left side-boundary and it is physical relevant to see whether the surface waves are controllable or can be stabilized using appropriate motion of the wave-maker. Due to the surface tension, an edge condition must be imposed at the contact point between the free surface and a solid boundary. Two types of wave-makers are considered: "flexible" or "rigid". It is shown that the surface waves are approximately controllable, but not exactly controllable, for both "flexible" and "rigid" wave-makers. In addition, under a static feedback to control a "rigid" wave-maker, the strong stability of feedback control system is obtained. Correction: The page numbers on each page of the PDF file have been corrected. We apologize for any inconvenience this may cause.

Numerous experimental studies have documented that injecting low-salinity water into an oil reservoir can increase the amount of oil recovered. However, owing to the complexity of the chemical interactions involved in this process, there has been much debate over the dominant mechanism causing this effect. In order to further understand one proposed mechanism, multicomponent ionic exchange, we study the motion of an oil slug through a clay pore throat filled with saline water. The pore throat is modelled as a capillary tube connecting two bulk regions of water. We assume that the surfaces of the oil and the capillary are negatively charged and that, due to repulsion between these surfaces, the oil slug is separated from the capillary surface by a thin film of water. Ion interactions at the oil–water and clay–water interfaces are modelled using the law of mass action. By using lubrication theory to describe the thin-film flow in the water layer separating the oil from the clay surface, and the macroscopic flow through the capillary, we derive expressions for the thickness of the wetting film, and the velocity of the oil slug, given a pressure difference across the ends of the capillary. Numerical results show that the thickness of the water layer and the velocity of the oil slug increase as the salinity of the water is reduced, suggesting that this mechanism contributes to the low-salinity effect. An analytical solution is presented in the limit in which the applied pressure is small.

Direct numerical simulation of the process of binary alloy separation in a thin non-uniformly heated circular capillary is carried out. The key point of the process under consideration is the assumption of the existence of a thin gas layer between the melt and the capillary wall due to non-wetting condition. Equations of interfacial hydrodynamics are used to describe the effect of separation, which allows us to develop a phenomenological model of the processes at the melt - solid boundary interface for mixtures of liquid metals. The finite difference method in combination with the explicit scheme is used for solving the problem. The calculations have been carried out on the supercomputer “PSU-Kepler” of the Scientific and Educational Center “Parallel and Distributed Calculations” in Perm State National Research University. Velocity and temperature fields, as well as the concentration of melt components in the volume and on the surface, are determined through numerical simulations. The dynamics of the separation process is described in detail. It has been found that the longitudinal temperature gradient and the non-wetting condition on the lateral capillary side generate a downward movement along the external boundary, which, together with adsorption and desorption effects, lead to the formation of volume concentration non-uniformity along the capillary. The temporal evolution of the volume concentrations difference along the capillary observed in experiments is reproduced numerically. The distributions of volume concentrations and the surface phase are studied as a function of both Marangoni number and adsorption-desorption coefficients. Concentration differences for alloy components are studied in relation to the capillary length. It has been shown that an increase in the capillary length leads to the strengthening of the separation effect. The qualitative and quantitative comparison of most characteristics demonstrates that the calculated data are in good agreement with the numerical results of the plane problem considered earlier and with the available experimental data. Numerical experiments show that the movement in the surface layer is quite intensive. The heavy component is transported by the convection flow towards the lower part of the capillary so that its concentration increases by the order of magnitude on the 1/8 part of the capillary surface.

Defined mathematically as critical points of surface area subject to a volume constraint, constant mean curvatures (CMC) surfaces are idealizations of interfaces occurring between two immiscible fluids. Their behavior elucidates phenomena seen in many microscale systems of applied science and engineering; however, explicitly computing the shapes of CMC surfaces is often impossible, especially when the boundary of the interface is fixed and parameters, such as the volume enclosed by the surface, vary. In this work, we propose a novel method for computing discrete versions of CMC surfaces based on solving a quasilinear, elliptic partial differential equation that is derived from writing the unknown surface as a normal graph over another known CMC surface. The partial differential equation is then solved using an arc-length continuation algorithm, and the resulting algorithm produces a continuous family of CMC surfaces for varying volume whose physical stability is known. In addition to providing details of the algorithm, various test examples are presented to highlight the efficacy, accuracy and robustness of the proposed approach., Comment: 19 pages, 3 figures

The principle existence theorem (i.e. Theorem 1) of "Existence and Behavior of the Radial Limits of a Bounded Capillary Surface at a Corner" (Pacific J. Math. Vol. 176, No. 1 (1996), 165-194) is extended to the case of a contact angle $\gamma$ which is not bounded away from $0$ and $\pi$ (and depends on position in a bounded domain $\Omega\in {\bf R}^{2}$ with a convex corner at ${\cal O}=(0,0)$). The lower bound on the size of "side fans" (i.e. Theorem 2 in the above paper) is extended to case of such contact angles for convex and nonconvex corners., Comment: 20 pages, 3 figures, removed Example 3, added more details in the proofs, slight modifications, submitted to the Pacific Journal of Mathematics

A functionally graded half-plane with shear modulus varying exponentially along the direction normal to the surface and surface effects accounted through Gurtin–Murdoch model, indented by long rigid smooth cylindrical indenter is solved to understand the effect of material inhomogeneity and surface effects on indentation response. The Green’s function relating surface load to surface displacement under plane strain condition is obtained semi-analytically through the combination of Airy stress function approach and Fourier transforms and utilized to solve the contact problem. The solution is used to study the effect of inhomogeneity through grading parameter and surface effects through residual surface tension based intrinsic length scale on the contact pressure, contact size and in-plane normal stress on the surface responsible for cracks during indentation.