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This article proposes a framework for estimating temperature fields in food-freezing applications that significantly reduces computational load while ensuring accurate temperature monitoring, representing a promising technological tool for optimizing and controlling food engineering processes. The strategy is based on (i) a mathematical model of a convection-dominated problem coupling thermal convection and turbulence and (ii) a least-squares approach for solving the inverse data assimilation problem, regularized by projecting the governing dynamics onto a reduced-order model (ROM). The unsteady freezing process considers an idealized salmon slice in a freezer cabinet, modeled with temperature-dependent thermophysical properties. The forward problem is approximated using a third-order WENO finite volume solver, including an optimized second-order backward scheme for time discretization. We employ our data assimilation framework to reconstruct the temperature field from a limited number of sensor data and to estimate temperature distributions within frozen food. Sensor placement is optimized using a new greedy algorithm, relying on maximizing the observability of the reduced-order dynamics for a fixed set of sensors. The proposed approach allows efficient extrapolation from external sensor measurements to the internal temperature of the food, which is crucial for maintaining food quality.

For the incompressible Navier--Stokes system with variable density and viscosity, we propose and analyse an IMEX framework treating the convective and diffusive terms semi-implicitly. This extends to variable density and second order in time some methods previously analysed for variable viscosity and constant density. We present three new schemes, both monolithic and fractional-step. All of them share the methodological novelty that the viscous term is treated in an implicit-explicit (IMEX) fashion, which allows decoupling the velocity components. Unconditional temporal stability is proved for all three variants. Furthermore, the system to solve at each time step is linear, thus avoiding the costly solution of nonlinear problems even if the viscosity follows a non-Newtonian rheological law. Our presentation is restricted to the semi-discrete case, only considering the time discretisation. In this way, the results herein can be applied to any spatial discretisation. We validate our theory through numerical experiments considering finite element methods in space. The tests range from simple manufactured solutions to complex two-phase viscoplastic flows., Comment: 23 pages

Hemodynamic parameters are often estimated assuming a constant Newtonian viscosity, even though blood exhibits shear-thinning behavior. This article investigates the influence of blood rheology and hematocrit (Hct) percentage on the estimation of Wall Shear Stress (WSS), rate of viscous Energy Loss ($\dot{E}_L$) at different points in the cardiac cycle, and the Oscillatory Shear Index (OSI). We focus on a hematocrit-dependent power-law non-Newtonian model, considering a wide range of Hct values at physiological temperature, with rheological parameters obtained from previously reported experimental data. In all cases, we systematically compared WSS, $\dot{E}_L$, and OSI using both Newtonian and power-law models, underscoring the crucial role of blood rheology in accurately assessing cardiovascular diseases. Our results show that, in in-silico experiments, differences in WSS and $\dot{E}_L$ across a wide range of Hct values can reach as high as 190\% and 113\% at systole, and as low as -72\% and -74\% at diastole, respectively. In in-vivo data, differences in WSS and $\dot{E}_L$ can reach up to -45\% and -60\% at systole, and range from -69\% to 73\% at diastole. This study enhances our understanding of the impact of blood rheology on hemodynamic parameter estimations using both in-silico and in-vivo aortic 4D Flow MRI data.

In this work we study different Implicit-Explicit (IMEX) schemes for incompressible flow problems with variable viscosity. Unlike most previous work on IMEX schemes, which focuses on the convective part, we here focus on treating parts of the diffusive term explicitly to reduce the coupling between the velocity components. We present different, both monolithic and fractional-step, IMEX alternatives for the variable-viscosity Navier--Stokes system, analysing their theoretical and algorithmic properties. Stability results are proven for all the methods presented, with all these results being unconditional, except for one of the discretisations using a fractional-step scheme, where a CFL condition (in terms of the problem data) is required for showing stability. Our analysis is supported by a series of numerical experiments.