Buckley–Leverett Theory for a Forchheimer–Darcy Multiphase Flow Model with Phase Coupling

This paper is dedicated to the modeling, analysis, and numerical simulation of a two-phase non-Darcian flow through a porous medium with phase-coupling. Specifically, we introduce an extended Forchheimer–Darcy model where the interaction between phases is taken into consideration. From the modeling point of view, the extension consists of the addition to each phase equation of a term depending on the gradient of the pressure of the other phase, leading to a coupled system of differential equations. The obtained system is much more involved than the classical Darcy system since it involves the Forchheimer equation in addition to the Darcy one. This model is more appropriate when there is a substantial difference between the phases’ velocities, for instance in the case of gas/water phases, and applications in oil recovery using gas flooding. Based on the Buckley–Leverett theory, including capillary pressure, we derive an explicit expression of the phases’ velocities and fractional water flows in terms of the gradient of the capillary pressure, and the total constant velocity. Various scenarios are considered, and the respective numerical simulations are presented. In particular, comparisons with the classical models (without phase coupling) are provided in terms of breakthrough time among others. Eventually, we provide a post-processing method for the derivation of the solution of the new coupled system using the classical non-coupled system. This method is of interest for industry since it allows for including the phase coupling approach in existing numerical codes and software (designed for solving classical models) without major technical changes.

A New Ice Accretion Model for Aircraft Icing Based on Phase-Field Method

Aircraft icing presents a serious threat to the aerodynamic performance and safety of aircraft. The numerical simulation method for the accurate prediction of icing shape is an important method to evaluate icing hazards and develop aircraft icing protection systems. Referring to the phase-field method, a new ice accretion mathematical model is developed to predict the ice shape. The mass fraction of ice in the mixture is selected as the phase parameter, and the phase equation is established with a freezing coefficient. Meanwhile, the mixture thickness and temperature are determined by combining mass conservation and energy balance. Ice accretions are simulated under typical ice conditions, including rime ice, glaze ice and mixed ice, and the ice shape and its characteristics are analyzed and compared with those provided by experiments and LEWICE. The results show that the phase-field ice accretion model can predict the ice shape under different icing conditions, especially reflecting some main characteristics of glaze ice.

A double phase equation with convection

We consider a double phase problem with a gradient dependent reaction (convection). Using the theory of nonlinear operators of monotone type, we show the existence of a nontrivial, positive, bounded solution.

Analysis of Steady-State Oscillations in an Autonomous Oscillator under Multi-Frequency Excitation

The analysis of the behavior of an oscillator under multi-frequency excitation is considered in the paper. The investigation is based on the phase macromodel. The paper shows that three steady-state modes can exist in oscillator under multi-frequency excitation. The synchronized (locked) mode can be defined as the coincidence of the oscillator fundamentals with the excitation fundamentals in the region of sufficiently large excitation magnitude. The unsynchronized (unlocked) mode exists outside the synchronized region and its spectrum contains additional intrinsic fundamental besides the excitation ones. Singular points mode in some isolated points outside the synchronized region is characterized by the equality of the number of the oscillator fundamentals with the number of the excitation fundamentals. Performed numerical experiments confirmed the appearance of bifurcation points while transition of oscillator into the synchronization mode. The existence of singular points outside the synchronization region and their isolated character was also experimentally demonstrated. The problems of finding a steady-state solution of the phase equation of an excited oscillator by the Harmonic Balance (HB) method are considered. It is shown that main difficulties are connected with the presence of linear term in the steady-state solution. A transformation is proposed to provide the formation of HB equations for the phase micromodel in a standard form. Additional difficulties of HB simulations of synchronized oscillator phase equations are discussed.

Equivalence between distributional and viscosity solutions for the double-phase equation

We investigate the different notions of solutions to the double-phase equation - div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u + a ⁢ ( x ) ⁢ | D ⁢ u | q - 2 ⁢ D ⁢ u ) = 0 , -{\operatorname{div}(\lvert Du\rvert^{p-2}Du+a(x)\lvert Du\rvert^{q-2}Du)}=0, which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the A H ⁢ ( ⋅ ) \mathcal{A}_{H(\,{\cdot}\,)} -harmonic functions of nonlinear potential theory and then show that A H ⁢ ( ⋅ ) \mathcal{A}_{H(\,{\cdot}\,)} -harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.

On strategy for determining displacement from perturbed phase of self mixing interferometry when parameters are unknown

While self-mixing interferometry(SMI) has proven to be suitable for displacement measurement and other sensing applications, its characteristic self-mixing signal shape is strongly governed by the non-linear phase equation which forms relation between perturbed and unperturbed phase of self-mixing laser. Therefore, while it is desirable for robust estimation of displacement of moving target, the algorithms to achieve this must have an objective strategy that can be achieved by understanding the characteristic of extracting knowledge of the perturbed phase from the unperturbed phase. Therefore, it has been proved and shown that such a strategy must not involve sole methods where the perturbed phase is a continuous function of the unperturbed phase (e.g: Taylor series or fixed-point methods) or through successive displacements (e.g: variations of Gauss-Seidel method). The subset of this strategy is to perform spectral filtering of the perturbed phase followed by perturbative or homotopic deformation. A less computationally expensive approach of this strategy is adopted to achieve displacement with a mean error of 62.2nm covering all feedback regimes, when the coupling factor 'C' is unknown.

On strategy for determining displacement from perturbed phase of self mixing interferometry when parameters are unknown

While self-mixing interferometry(SMI) has proven to be suitable for displacement measurement and other sensing applications, its characteristic self-mixing signal shape is strongly governed by the non-linear phase equation which forms relation between perturbed and unperturbed phase of self-mixing laser. Therefore, while it is desirable for robust estimation of displacement of moving target, the algorithms to achieve this must have an objective strategy that can be achieved by understanding the characteristic of extracting knowledge of the perturbed phase from the unperturbed phase. Therefore, it has been proved and shown that such a strategy must not involve sole methods where the perturbed phase is a continuous function of the unperturbed phase (e.g: Taylor series or fixed-point methods) or through successive displacements (e.g: variations of Gauss-Seidel method). The subset of this strategy is to perform spectral filtering of the perturbed phase followed by perturbative or homotopic deformation. A less computationally expensive approach of this strategy is adopted to achieve displacement with a mean error of 62.2nm covering all feedback regimes, when the coupling factor 'C' is unknown.