Literature confirms the crucial influence on glacier and rock glacier flow of non-viscous deformations together with temperature impact. This observation suggests numerical glaciologists ought to reconsider the established mathematical modeling based on the representation of ice as a power-law viscous fluid and the Glen's law. Along this line, we propose the numerical solution of a two-dimensional rock-glacier flow model, based on a constitutive law of second grade of complexity two, as just published for a one-dimensional set-up by two of the authors. With the representation of the composition of the rocky ice as a mixture of ice and rock and sand grains, and the inclusion of the local impact of pressure and of thermal effects, this model has allowed the reproduction of borehole measurement data from alpine glacier internal sliding motion via a similarity solution of the flow governing equations. Here, the adopted numerical procedure uses a second order finite difference scheme and imposes the incompressibility constrain up to computer accuracy via the pressure method, that we have extended from Newtonian computational fluid dynamics. This method solves the governing equations for the flow in primitive variables with the advantage that no pre-/post-processing is required; in addition, it avoids splitted solution of the Poisson equation for pressure which might be source of undesired numerical mass unbalancing. The results of a numerical test on the Murtel-Corvatsch alpine glacier flow, reporting satisfactory matching with published on-field observations, are presented.

Recent literature confirms the crucial influence of non-viscous deformations together with temperature impact on glacier and rock glacier flow numerical simulation. Along this line, supported by the successful test on a one-dimensional set-up developed by two of the author, we propose the numerical solution of a two-dimensional rock-glacier flow model based on an ice constitutive law of second grade differential type . The procedure adopted uses a 2nd order finite difference scheme and imposes the incompressibility constrain up to computer precision via the pressure method, ex- tended from newtonian computational fluid dynamics. The governing equations are solved in primitive variables with the advantage to avoid pre-/post-processing; splitted solution of the derived Poisson equation for pressure, source of undesired numerical mass unbalancing, is avoided as well. Numerical results will be shown.The financial support of Piano Nazionale Ricerca Antartide (project PNRA16-0012) is acknowledged.

The present work is inspired by laboratory experiments, investigating the cross-talk between immune and cancer cells in a confined environment given by a microfluidic chip, the so called Organ-on-Chip (OOC). Based on a mathematical model in form of coupled reaction-diffusion-transport equations with chemotactic functions, our effort is devoted to the development of a parameter estimation methodology that is able to use real data obtained from the laboratory experiments to estimate the model parameters and infer the most plausible chemotactic function present in the experiment. In particular, we need to estimate the model parameters representing the convective and diffusive regimes included in the PDE model, in order to evaluate the diffusivity of the chemoattractant produced by tumor cells and its biasing effect on immune cells. The main issues faced in this work are the efficient calibration of the model against noisy synthetic data, available as macroscopic density of immune cells. A calibration algorithm is derived based on minimization methods which applies several techniques such as regularization terms and multigrids application to improve the results, which show the robustness and accuracy of the proposed algorithm.

The present work is inspired by the recent developments in laboratory experiments madeon chips, where the culturing of multiple cell species was possible. The model is based on coupledreaction-diffusion-transport equations with chemotaxis and takes into account the interactions amongcell populations and the possibility of drug administration for drug testing effects. Our effort isdevoted to the development of a simulation tool that is able to reproduce the chemotactic movementand the interactions between different cell species (immune and cancer cells) living in a microfluidicchip environment. The main issues faced in this work are the introduction of mass-preserving andpositivity-preserving conditions, involving the balancing of incoming and outgoing fluxes passingthrough interfaces between 2D and 1D domains of the chip and the development of mass-preservingand positivity preserving numerical conditions at the external boundaries and at the interfacesbetween 2D and 1D domains.

Up-to-date computational glaciology is very often basing its investigations about glacier flow on the intensive use of the large amount of data, gathered in (alpine or polar) on-field campaigns, and on the "brute force" adaptation of the Glen's law via phenomenological multi-parametrical functional factors and/or addenda. Although, reasonable to fully satisfactory numerical results have been being obtained with this approach adopted by the most popular open-source computational glaciology codes, a modelling effort is worth in order to include the normal stress gradient effects which are not covered by such a power law model and are indeed physically significant in the case of moraine ice and rock glaciers. In this trend Kannan, Mansutti and Rajagopal have proposed (2021) a mathematical numerical model which has been successfully challenged on the reproduction of borehole deformation measurements of the Murtel-Corvatsch rock glacier on the Grisons Alps, Switzerland. This case, and possibly other numerical results at the present time in progress, will be discussed.

t. The present work was inspired by the recent developments in laboratory experiments made on chip, where culturing of multiple cell species waspossible. The model is based on coupled reaction-diffusion-transport equationswith chemotaxis, and takes into account the interactions among cell populations and the possibility of drug administration for drug testing effects.Our effort was devoted to the development of a simulation tool that is able toreproduce the chemotactic movement and the interactions between differentcell species (immune and cancer cells) living in microfluidic chip environment.The main issues faced in this work are the introduction of mass-preservingand positivity-preserving conditions involving the balancing of incoming andoutgoing fluxes passing through interfaces between 2D and 1D domains of thechip and the development of mass-preserving and positivity preserving numerical conditions at the external boundaries and at the interfaces between 2Dand 1D domains