Solving two-phase freezing Stefan problems: Stability and monotonicity

[EN] The two-phase Stefan problems with phase formation and depletion are special cases ofmoving boundary problemswith interest in science and industry. In this work, we study a solidification problem, introducing a front-fixing transformation. The resulting non-linear partial differential system involves singularities, both at the beginning of the freezing process and when the depletion is complete, that are treated with special attention in the numerical modelling. The problem is decomposed in three stages, in which implicit and explicit finite difference schemes are used. Numerical analysis reveals qualitative properties of the numerical solution spatial monotonicity of both solid and liquid temperatures and the evolution of the solidification front. Numerical experiments illustrate the behaviour of the temperatures profiles with time, as well as the dynamics of the solidification front.Ministerio de Ciencia, Innovacion y Universidades, Grant/Award Number: MTM2017-89664-P.Piqueras, MA.; Company Rossi, R.; Jódar Sánchez, LA. (2020). Solving two-phase freezing Stefan problems: Stability and monotonicity. Mathematical Methods in the Applied Sciences. 43(14):7948-7960. https://doi.org/10.1002/mma.5787S794879604314Schmidt, A. (1996). Computation of Three Dimensional Dendrites with Finite Elements. Journal of Computational Physics, 125(2), 293-312. doi:10.1006/jcph.1996.0095Singh, S., & Bhargava, R. (2014). Simulation of Phase Transition During Cryosurgical Treatment of a Tumor Tissue Loaded With Nanoparticles Using Meshfree Approach. Journal of Heat Transfer, 136(12). doi:10.1115/1.4028730Company, R., Egorova, V. N., & Jódar, L. (2014). Solving American Option Pricing Models by the Front Fixing Method: Numerical Analysis and Computing. Abstract and Applied Analysis, 2014, 1-9. doi:10.1155/2014/146745Griewank, P. J., & Notz, D. (2013). Insights into brine dynamics and sea ice desalination from a 1-D model study of gravity drainage. Journal of Geophysical Research: Oceans, 118(7), 3370-3386. doi:10.1002/jgrc.20247Javierre, E., Vuik, C., Vermolen, F. J., & van der Zwaag, S. (2006). A comparison of numerical models for one-dimensional Stefan problems. Journal of Computational and Applied Mathematics, 192(2), 445-459. doi:10.1016/j.cam.2005.04.062Briozzo, A. C., Natale, M. F., & Tarzia, D. A. (2007). Explicit solutions for a two-phase unidimensional Lamé–Clapeyron–Stefan problem with source terms in both phases. Journal of Mathematical Analysis and Applications, 329(1), 145-162. doi:10.1016/j.jmaa.2006.05.083Caldwell, J., & Chan, C.-C. (2000). Spherical solidification by the enthalpy method and the heat balance integral method. Applied Mathematical Modelling, 24(1), 45-53. doi:10.1016/s0307-904x(99)00031-1Chantasiriwan, S., Johansson, B. T., & Lesnic, D. (2009). The method of fundamental solutions for free surface Stefan problems. Engineering Analysis with Boundary Elements, 33(4), 529-538. doi:10.1016/j.enganabound.2008.08.010Hon, Y. C., & Li, M. (2008). A computational method for inverse free boundary determination problem. International Journal for Numerical Methods in Engineering, 73(9), 1291-1309. doi:10.1002/nme.2122RIZWAN-UDDIN. (1999). A Nodal Method for Phase Change Moving Boundary Problems. International Journal of Computational Fluid Dynamics, 11(3-4), 211-221. doi:10.1080/10618569908940875Caldwell, J., & Kwan, Y. Y. (2003). On the perturbation method for the Stefan problem with time-dependent boundary conditions. International Journal of Heat and Mass Transfer, 46(8), 1497-1501. doi:10.1016/s0017-9310(02)00415-5Stephan, K., & Holzknecht, B. (1976). Die asymptotischen lösungen für vorgänge des erstarrens. International Journal of Heat and Mass Transfer, 19(6), 597-602. doi:10.1016/0017-9310(76)90042-9Savović, S., & Caldwell, J. (2003). Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. International Journal of Heat and Mass Transfer, 46(15), 2911-2916. doi:10.1016/s0017-9310(03)00050-4Kutluay, S., Bahadir, A. R., & Özdeş, A. (1997). The numerical solution of one-phase classical Stefan problem. Journal of Computational and Applied Mathematics, 81(1), 135-144. doi:10.1016/s0377-0427(97)00034-4Asaithambi, N. S. (1997). A variable time step Galerkin method for a one-dimensional Stefan problem. Applied Mathematics and Computation, 81(2-3), 189-200. doi:10.1016/0096-3003(95)00329-0Landau, H. G. (1950). Heat conduction in a melting solid. Quarterly of Applied Mathematics, 8(1), 81-94. doi:10.1090/qam/33441Churchill, S. W., & Gupta, J. P. (1977). Approximations for conduction with freezing or melting. International Journal of Heat and Mass Transfer, 20(11), 1251-1253. doi:10.1016/0017-9310(77)90134-xKutluay, S., & Esen, A. (2004). An isotherm migration formulation for one-phase Stefan problem with a time dependent Neumann condition. Applied Mathematics and Computation, 150(1), 59-67. doi:10.1016/s0096-3003(03)00197-8Esen, A., & Kutluay, S. (2004). A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method. Applied Mathematics and Computation, 148(2), 321-329. doi:10.1016/s0096-3003(02)00846-9Mitchell, S. L., & Vynnycky, M. (2016). On the accurate numerical solution of a two-phase Stefan problem with phase formation and depletion. Journal of Computational and Applied Mathematics, 300, 259-274. doi:10.1016/j.cam.2015.12.021Meek, P. C., & Norbury, J. (1984). Nonlinear Moving Boundary Problems and a Keller Box Scheme. SIAM Journal on Numerical Analysis, 21(5), 883-893. doi:10.1137/0721057Tarzia, D. (2017). Relationship between Neumann solutions for two-phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions. Thermal Science, 21(1 Part A), 187-197. doi:10.2298/tsci140607003tPlemmons, R. J. (1977). M-matrix characterizations.I—nonsingular M-matrices. Linear Algebra and its Applications, 18(2), 175-188. doi:10.1016/0024-3795(77)90073-8Axelsson, O. (1994). Iterative Solution Methods. doi:10.1017/cbo978051162410

Use of Novel Drying Technologies to Improve the Retention of Infused Olive Leaf Polyphenols

The infusion of phenolic extracts in dried fruits constitutes an interesting means of improving their nutritional content. However, drying can affect the further process of impregnation. In this work, different drying treatments (air temperature and ultrasound application) were applied to apple samples and impregnated with olive leaf extract. The application of ultrasound during drying did not significantly (p<0.05) affect the infusion capacity of samples, but the ultrasonically assisted dried samples showed a greater antioxidant capacity than those conventionally dried. The highest content of oleuropein and verbascoside was found in samples dried at low temperature using ultrasound.The authors acknowledge the financial support of the Spanish Ministerio de Economia y Competitividad (MINECO) and FEDER, and the Generalitat Valenciana (from the projects DPI2012-37466-CO3-03, PROMETEO/2010/062, and the FPI fellowship granted to J.V. Santacatalina).Santacatalina Bonet, JV.; Ahmad-Qasem Mateo, MH.; Barrajón-Catalán, E.; Micol, V.; García Pérez, JV.; Cárcel Carrión, JA. (2015). Use of Novel Drying Technologies to Improve the Retention of Infused Olive Leaf Polyphenols. Drying Technology. 33(9):1051-1060. https://doi.org/10.1080/07373937.2014.982251S1051106033

A patient tumour-on-a-chip system for personalised investigation of radiotherapy based treatment regimens

Development of personalised cancer models to predict response to radiation would benefit patient care; particularly in malignancies where treatment resistance is prevalent. Herein, a robust, easy to use, tumour-on-a-chip platform which maintains precision cut head and neck cancer for the purpose of ex vivo irradiation is described. The device utilises sintered discs to separate the biopsy and medium, mimicking in vivo microvascular flow and diffusion, maintaining tissue viability for 68 h. Integrity of tissues is demonstrated by the low levels of lactate dehydrogenase release and retained histology, accompanied by assessment of cell viability by trypan blue exclusion and flow cytometry; fluid dynamic modelling validates culture conditions. An irradiation jig is described for reproducible delivery of clinically-relevant doses (5 × 2 Gy) to newly-presenting primary tumours (n = 12); the addition of concurrent cisplatin is also investigated (n = 8) with response analysed by immunohistochemistry. Fractionated irradiation reduced proliferation (BrdU, p = 0.0064), increased DNA damage (ƴH2AX, p = 0.0043) and caspase-dependent apoptosis (caspase-cleaved cytokeratin-18) compared to control; caspase-dependent apoptosis was further increased by concurrent cisplatin compared to control (p = 0.0063). This is a proof of principle study showing the response of cancer tissue to irradiation ex vivo in a bespoke system. The novel platform described has the potential to personalise treatment for patients in a cost-effective manner with applicability to any solid tumour

Macroscopic models for superconductivity

This paper reviews the derivation of some macroscopic models for superconductivity and also some of the mathematical challenges posed by these models. The paper begins by exploring certain analogies between phase changes in superconductors and those in solidification and melting. However, it is soon found that there are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models

Modeling Ultrasonically Assisted Convective Drying of Eggplan

[EN] Modeling constitutes a fundamental tool with which to analyze the influence of ultrasound on mass transfer phenomena during drying. In this work, the study of the effect of power ultrasound application on the drying kinetics of eggplant was addressed by using different models based on theoretical (diffusion) or empirical approaches. Drying kinetics of eggplant cylinders (height 20mm and diameter 24 mm) were carried at 40°C and 1 m/s applying different ultrasonic powers: 0, 6, 12, 19, 25, 31, and 37 kW/m 3. The experiments were carried out at least three times at each different ultrasonic power. Shrinkage and sorption isotherms were also addressed in order to attain an optimal description of eggplant drying. Applying ultrasound sped up the drying kinetics. The ultrasonic power was identified as having a significant (p<0.05) influence on both the effective moisture diffusivity and the mass transfer coefficient, which was well explained by linear relationships. The most complex model, which considered both external resistance and shrinkage to be significant phenomena, provided the best agreement with experimental data, giving percentages of explained variance of over 99.9% and mean relative errors of under 1.2% in every case. According to these results, ultrasound technology could have the potential to improve the convective drying of eggplant at an industrial scale. © 2011 Taylor & Francis Group, LLC.García Pérez, JV.; Ozuna López, C.; Ortuño Cases, C.; Carcel Carrión, JA.; Mulet Pons, A. (2011). Modeling Ultrasonically Assisted Convective Drying of Eggplan. Drying Technology. 29(13):1499-1509. doi:10.1080/07373937.2011.576321S149915092913Mujumdar, A. S. (2006). An overview of innovation in industrial drying: current status and R&D needs. Transport in Porous Media, 66(1-2), 3-18. doi:10.1007/s11242-006-9018-yChou, S. K., & Chua, K. J. (2001). New hybrid drying technologies for heat sensitive foodstuffs. Trends in Food Science & Technology, 12(10), 359-369. doi:10.1016/s0924-2244(01)00102-9Lewicki, P. P. (2006). Design of hot air drying for better foods. Trends in Food Science & Technology, 17(4), 153-163. doi:10.1016/j.tifs.2005.10.012Santos, P. H. S., & Silva, M. A. (2009). Kinetics ofL-Ascorbic Acid Degradation in Pineapple Drying under Ethanolic Atmosphere. Drying Technology, 27(9), 947-954. doi:10.1080/07373930902901950Suvarnakuta, P., Devahastin, S., & Mujumdar, A. S. (2005). Drying Kinetics and β-Carotene Degradation in Carrot Undergoing Different Drying Processes. Journal of Food Science, 70(8), s520-s526. doi:10.1111/j.1365-2621.2005.tb11528.xMayor, L., & Sereno, A. M. (2004). Modelling shrinkage during convective drying of food materials: a review. Journal of Food Engineering, 61(3), 373-386. doi:10.1016/s0260-8774(03)00144-4Gallego-Juarez, J. A. (2010). High-power ultrasonic processing: Recent developments and prospective advances. Physics Procedia, 3(1), 35-47. doi:10.1016/j.phpro.2010.01.006De la Fuente-Blanco, S., Riera-Franco de Sarabia, E., Acosta-Aparicio, V. M., Blanco-Blanco, A., & Gallego-Juárez, J. A. (2006). Food drying process by power ultrasound. Ultrasonics, 44, e523-e527. doi:10.1016/j.ultras.2006.05.181García-Pérez, J. V., Cárcel, J. A., Riera, E., & Mulet, A. (2009). Influence of the Applied Acoustic Energy on the Drying of Carrots and Lemon Peel. Drying Technology, 27(2), 281-287. doi:10.1080/07373930802606428García-Pérez, J. V., Cárcel, J. A., Clemente, G., & Mulet, A. (2008). Water sorption isotherms for lemon peel at different temperatures and isosteric heats. LWT - Food Science and Technology, 41(1), 18-25. doi:10.1016/j.lwt.2007.02.010Mulet, A. (1994). Drying modelling and water diffusivity in carrots and potatoes. Journal of Food Engineering, 22(1-4), 329-348. doi:10.1016/0260-8774(94)90038-8Cunha, L. M., Oliveira, F. A. R., & Oliveira, J. C. (1998). Optimal experimental design for estimating the kinetic parameters of processes described by the Weibull probability distribution function. Journal of Food Engineering, 37(2), 175-191. doi:10.1016/s0260-8774(98)00085-5Azzouz, S., Guizani, A., Jomaa, W., & Belghith, A. (2002). Moisture diffusivity and drying kinetic equation of convective drying of grapes. Journal of Food Engineering, 55(4), 323-330. doi:10.1016/s0260-8774(02)00109-7Simal, S., Femenia, A., Garau, M. C., & Rosselló, C. (2005). Use of exponential, Page’s and diffusional models to simulate the drying kinetics of kiwi fruit. Journal of Food Engineering, 66(3), 323-328. doi:10.1016/j.jfoodeng.2004.03.025Maroulis, Z. B., Saravacos, G. D., Panagiotou, N. M., & Krokida, M. K. (2001). MOISTURE DIFFUSIVITY DATA COMPILATION FOR FOODSTUFFS: EFFECT OF MATERIAL MOISTURE CONTENT AND TEMPERATURE. International Journal of Food Properties, 4(2), 225-237. doi:10.1081/jfp-100105189Simal, S., Femenia, A., Garcia-Pascual, P., & Rosselló, C. (2003). Simulation of the drying curves of a meat-based product: effect of the external resistance to mass transfer. Journal of Food Engineering, 58(2), 193-199. doi:10.1016/s0260-8774(02)00369-2Queiroz, M. R., & Nebra, S. A. (2001). Theoretical and experimental analysis of the drying kinetics of bananas. Journal of Food Engineering, 47(2), 127-132. doi:10.1016/s0260-8774(00)00108-4Hassini, L., Azzouz, S., Peczalski, R., & Belghith, A. (2007). Estimation of potato moisture diffusivity from convective drying kinetics with correction for shrinkage. Journal of Food Engineering, 79(1), 47-56. doi:10.1016/j.jfoodeng.2006.01.025Hernández, J. A., Pavón, G., & Garcı́a, M. A. (2000). Analytical solution of mass transfer equation considering shrinkage for modeling food-drying kinetics. Journal of Food Engineering, 45(1), 1-10. doi:10.1016/s0260-8774(00)00033-9Souma, S., Tagawa, A., & Iimoto, M. (2004). Structural Properties for Fruits and Vegetables during Drying. NIPPON SHOKUHIN KAGAKU KOGAKU KAISHI, 51(11), 577-584. doi:10.3136/nskkk.51.577García-Pérez, J. V., Cárcel, J. A., de la Fuente-Blanco, S., & Riera-Franco de Sarabia, E. (2006). Ultrasonic drying of foodstuff in a fluidized bed: Parametric study. Ultrasonics, 44, e539-e543. doi:10.1016/j.ultras.2006.06.059Cárcel, J. A., García-Pérez, J. V., Riera, E., & Mulet, A. (2007). Influence of High-Intensity Ultrasound on Drying Kinetics of Persimmon. Drying Technology, 25(1), 185-193. doi:10.1080/07373930601161070Blasco, M., García-Pérez, J. V., Bon, J., Carreres, J. E., & Mulet, A. (2006). Effect of Blanching and Air Flow Rate on Turmeric Drying. Food Science and Technology International, 12(4), 315-323. doi:10.1177/1082013206067352Garau, M. C., Simal, S., Femenia, A., & Rosselló, C. (2006). Drying of orange skin: drying kinetics modelling and functional properties. Journal of Food Engineering, 75(2), 288-295. doi:10.1016/j.jfoodeng.2005.04.017Wu, L., Orikasa, T., Ogawa, Y., & Tagawa, A. (2007). Vacuum drying characteristics of eggplants. Journal of Food Engineering, 83(3), 422-429. doi:10.1016/j.jfoodeng.2007.03.030Chaves , M. ; Sgroppo , S.C. ; Avanza , J.R. Cinéticas de secado de berenjena (Solanum melongenaL.). Comunicaciones Científicas y Tecnológicas (Universidad Nacional del Nordeste Corrientes Argentina),2003,Resumen E-060 .Akpinar, E. K., & Bicer, Y. (2005). Modelling of the drying of eggplants in thin-layers. International Journal of Food Science and Technology, 40(3), 273-281. doi:10.1111/j.1365-2621.2004.00886.xDe Lima, A. (2002). Simultaneous moisture transport and shrinkage during drying of solids with ellipsoidal configuration. Chemical Engineering Journal, 86(1-2), 85-93. doi:10.1016/s1385-8947(01)00276-5RAHMAN, N., & KUMAR, S. (2007). INFLUENCE OF SAMPLE SIZE AND SHAPE ON TRANSPORT PARAMETERS DURING DRYING OF SHRINKING BODIES. Journal of Food Process Engineering, 30(2), 186-203. doi:10.1111/j.1745-4530.2007.00104.

Velocity profile of granular flows inside silos and hoppers

We measure the flow of granular materials inside a quasi-two dimensional silo as it drains and compare the data with some existing models. The particles inside the silo are imaged and tracked with unprecedented resolution in both space and time to obtain their velocity and diffusion properties. The data obtained by varying the orifice width and the hopper angle allows us to thoroughly test models of gravity driven flows inside these geometries. All of our measured velocity profiles are smooth and free of the shock-like discontinuities ("rupture zones") predicted by critical state soil mechanics. On the other hand, we find that the simple Kinematic Model accurately captures the mean velocity profile near the orifice, although it fails to describe the rapid transition to plug flow far away from the orifice. The measured diffusion length $b$, the only free parameter in the model, is not constant as usually assumed, but increases with both the height above the orifice and the angle of the hopper. We discuss improvements to the model to account for the differences. From our data, we also directly measure the diffusion of the particles and find it to be significantly less than predicted by the Void Model, which provides the classical microscopic derivation of the Kinematic Model in terms of diffusing voids in the packing. However, the experimental data is consistent with the recently proposed Spot Model, based on a simple mechanism for cooperative diffusion. Finally, we discuss the flow rate as a function of the orifice width and hopper angles. We find that the flow rate scales with the orifice size to the power of 1.5, consistent with dimensional analysis. Interestingly, the flow rate increases when the funnel angle is increased.Comment: 17 pages, 8 figure

A mathematical model of melt lake development on an ice shelf

The accumulation of surface meltwater on ice shelves can lead to the formation of melt lakes. Melt lakes have been implicated in ice shelf collapse; Antarctica's Larsen B Ice Shelf was observed to have a large amount of surface melt lakes present preceding its collapse in 2002. Such collapse can affect ocean circulation and temperature, cause habitat loss and contribute to sea level rise through the acceleration of tributary glaciers. We present a mathematical model of a surface melt lake on an idealised ice shelf. The model incorporates a calculation of the ice shelf surface energy balance, heat transfer through the firn, the production and percolation of meltwater into the firn, the formation of ice lenses and the development and refreezing of surface melt lakes. The model is applied to the Larsen C Ice Shelf, where melt lakes have been observed. This region has warmed several times the global average over the last century and the Larsen C firn layer could become saturated with meltwater by the end of the century. When forced with weather station data, our model produces surface melting, meltwater accumulation, and melt lake development consistent with observations. We examine the sensitivity of lake formation to uncertain parameters, and provide evidence of the importance of processes such as lateral meltwater transport. We conclude that melt lakes impact surface melt and firn density and warrant inclusion in dynamic-thermodynamic models of ice shelf evolution within climate models, of which our model could form the basis for the thermodynamic component

Symmetries and modelling functions for diffusion processes

A constructive approach to theory of diffusion processes is proposed, which is based on application of both the symmetry analysis and method of modelling functions. An algorithm for construction of the modelling functions is suggested. This algorithm is based on the error functions expansion (ERFEX) of experimental concentration profiles. The high-accuracy analytical description of the profiles provided by ERFEX approximation allows a convenient extraction of the concentration dependence of diffusivity from experimental data and prediction of the diffusion process. Our analysis is exemplified by its employment to experimental results obtained for surface diffusion of lithium on the molybdenum (112) surface pre-covered with dysprosium. The ERFEX approximation can be directly extended to many other diffusion systems.Comment: 19 pages, 8 figure

Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme

We discuss the numerical solution of nonlinear parabolic partial differential equations, exhibiting finite speed of propagation, via a strongly implicit finite-difference scheme with formal truncation error $\mathcal{O}\left[(\Delta x)^2 + (\Delta t)^2 \right]$. Our application of interest is the spreading of viscous gravity currents in the study of which these type of differential equations arise. Viscous gravity currents are low Reynolds number (viscous forces dominate inertial forces) flow phenomena in which a dense, viscous fluid displaces a lighter (usually immiscible) fluid. The fluids may be confined by the sidewalls of a channel or propagate in an unconfined two-dimensional (or axisymmetric three-dimensional) geometry. Under the lubrication approximation, the mathematical description of the spreading of these fluids reduces to solving the so-called thin-film equation for the current's shape $h(x,t)$. To solve such nonlinear parabolic equations we propose a finite-difference scheme based on the Crank--Nicolson idea. We implement the scheme for problems involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or spherically-symmetric three-dimensional currents) on an equispaced but staggered grid. We benchmark the scheme against analytical solutions and highlight its strong numerical stability by specifically considering the spreading of non-Newtonian power-law fluids in a variable-width confined channel-like geometry (a "Hele-Shaw cell") subject to a given mass conservation/balance constraint. We show that this constraint can be implemented by re-expressing it as nonlinear flux boundary conditions on the domain's endpoints. Then, we show numerically that the scheme achieves its full second-order accuracy in space and time. We also highlight through numerical simulations how the proposed scheme accurately respects the mass conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements and corrections; to appear as a contribution in "Applied Wave Mathematics II