Time-fractional quenching problem: Blow-up of DtαuD_{t}^{\alpha}u at the quenching point

In this paper, we establish the occurrence of blow-up in the time-fractional term at the quenching point. By demonstrating that the quenching points are contained within a compact subset of the designated spatial interval, we employ this finding to prove the blow-up of the Caputo fractional time-derivative at the quenching point.Comment: Some corrections have been made in Section 4. The notation mistake in previous versions is not helpful or instructive to readers. Thank you for your understandin

Current status of turbulent dynamo theory: From large-scale to small-scale dynamos

Several recent advances in turbulent dynamo theory are reviewed. High resolution simulations of small-scale and large-scale dynamo action in periodic domains are compared with each other and contrasted with similar results at low magnetic Prandtl numbers. It is argued that all the different cases show similarities at intermediate length scales. On the other hand, in the presence of helicity of the turbulence, power develops on large scales, which is not present in non-helical small-scale turbulent dynamos. At small length scales, differences occur in connection with the dissipation cutoff scales associated with the respective value of the magnetic Prandtl number. These differences are found to be independent of whether or not there is large-scale dynamo action. However, large-scale dynamos in homogeneous systems are shown to suffer from resistive slow-down even at intermediate length scales. The results from simulations are connected to mean field theory and its applications. Recent work on helicity fluxes to alleviate large-scale dynamo quenching, shear dynamos, nonlocal effects and magnetic structures from strong density stratification are highlighted. Several insights which arise from analytic considerations of small-scale dynamos are discussed.Comment: 36 pages, 11 figures, Spa. Sci. Rev., submitted to the special issue "Magnetism in the Universe" (ed. A. Balogh

Front Propagation in Random Media

This PhD thesis deals with the problem of the propagation of fronts under random circumstances. A statistical model to represent the motion of fronts when are evolving in a media characterized by microscopical randomness is discussed and expanded, in order to cope with three distinct applications: wild-land fire simulation, turbulent premixed combustion, biofilm modeling. In the studied formalism, the position of the average front is computed by making use of a sharp-front evolution method, such as the level set method. The microscopical spread of particles which takes place around the average front is given by the probability density function linked to the underlying diffusive process, that is supposedly known in advance. The adopted statistical front propagation framework allowed a deeper understanding of any studied field of application. The application of this model introduced eventually parameters whose impact on the physical observables of the front spread have been studied with Uncertainty Quantification and Sensitivity Analysis tools. In particular, metamodels for the front propagation system have been constructed in a non intrusive way, by making use of generalized Polynomial Chaos expansions and Gaussian Processes.The Thesis received funding from Basque Government through the BERC 2014-2017 program. It was also funded by the Spanish Ministry of Economy and Competitiveness MINECO via the BCAM Severo Ochoa SEV-2013-0323 accreditation. The PhD is fundend by La Caixa Foundation through the PhD grant “La Caixa 2014”. Funding from “Programma Operativo Nazionale Ricerca e Innovazione” (PONRI 2014-2020) , “Innotavive PhDs with Industrial Characterization” is kindly acknowledged for a research visit at the department of Mathematics and Applications “Renato Caccioppoli” of University “Federico II” of Naples

Front propagation in random media.

244 p.This PhD thesis deals with the problem of the propagation of fronts under random circumstances. Astatistical model to represent the motion of fronts when are evolving in a media characterized bymicroscopical randomness is discussed and expanded, in order to cope with three distinctapplications: wild-land fire simulation, turbulent premixed combustion, biofilm modeling. In thestudied formalism, the position of the average front is computed by making use of a sharp-frontevolution method, such as the level set method. The microscopical spread of particles which takesplace around the average front is given by the probability density function linked to the underlyingdiffusive process, that is supposedly known in advance. The adopted statistical front propagationframework allowed a deeper understanding of any studied field of application. The application ofthis model introduced eventually parameters whose impact on the physical observables of the frontspread have been studied with Uncertainty Quantification and Sensitivity Analysis tools. Inparticular, metamodels for the front propagation system have been constructed in a non intrusiveway, by making use of generalized Polynomial Chaos expansions and Gaussian Processes.bcam:basque center for applied mathematic

The emergence of biofilms:Computational and experimental studies

The response of biofilms to any external stimuli is a cumulative response aggregated from individual bacteria residing within the biofilm. The organizational complexity of biofilm can be studied effectively by understanding bacterial interactions at cell level. The overall aim of the thesis is to explore the complex evolutionary behaviour of bacterial biofilms. This thesis is divided into three major studies based on the type of perturbation analysed in the study. The first study analyses the physics behind the development of mushroom-shaped structures from the influence of nutrient cues in biofilms. Glazier-Graner-Hogeweg model is used to simulate the cell characteristics. From the study, it is observed that chemotaxis of bacterial cells towards nutrient source is one of the major precursors for formation of mushroom-shaped structures. The objective of the second study is to analyse the impact of environmental conditions on the inter-biofilm quorum sensing (QS) signalling. Using a hybrid convection-diffusion-reaction model, the simulations predict the diffusivity of QS molecules, the spatiotemporal variations of QS signal concentrations and the competition outcome between QS and quorum quenching mutant bacterial communities. The mechanical effects associated with the fluid-biofilm interaction is addressed in the third study. A novel fluid-structure interaction model based on fluid dynamics and structural energy minimization is developed in the study. Model simulations are used to analyse the detachment and surface effects of the fluid stresses on the biofilm. In addition to the mechanistic models described, a separate study is carried out to estimate the computational efficiency of the biofilm simulation models

Mean flow and spiral defect chaos in Rayleigh-Benard convection

We describe a numerical procedure to construct a modified velocity field that does not have any mean flow. Using this procedure, we present two results. Firstly, we show that, in the absence of mean flow, spiral defect chaos collapses to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wavenumbers that approach those uniquely selected by focus-type singularities, which, in the absence of mean flow, lie at the zig-zag instability boundary. The quenched patterns also have larger correlation lengths and are comprised of rolls with less curvature. Secondly, we describe how mean flow can contribute to the commonly observed phenomenon of rolls terminating perpendicularly into lateral walls. We show that, in the absence of mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with Rayleigh number.Comment: 14 pages, 19 figure