Modeling of stratified two-phase flows with non-uniform evaporation based on the exact solution of convection equations

Within the framework of the Oberbeck–Boussinesq approximation, the exact solution of equations of thermoconcentration convection is studied, which has the group origin. The issue of applicability of the exact solution for describing steady-state convective flows of a liquid and a co-current gas flux under the conditions of inhomogeneous evaporation of the diffusive type in a flat horizontal channel is discussed. Algorithms for obtaining analytical representations of the required functions for various types of the boundary conditions for the temperature function on the outer channel wall are proposed. The influence of the external thermal load and boundary thermal conditions on the structure of the velocity and temperature fields, evaporation mass flow rate, and vapor content in the gas layer is investigated at an example of the HFE-7100–nitrogen system. The solution correctly predicts hydrodynamical, temperature, and concentration parameters of convective regimes appearing in the two-phase system. The main characteristics are compared with the characteristics of the system in the case of uniform evaporation with a constant intensity.     

On the stabilizing effect of convection in three‐dimensional incompressible flows

We investigate the stabilizing effect of convection in three‐dimensional incompressible Euler and Navier‐Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new three‐dimensional model that is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full three‐dimensional Euler or Navier‐Stokes equations except for the convection term, which is neglected in our model. If we added the convection term back to our model, we would recover the full Navier‐Stokes equations. We will present numerical evidence that seems to support that the three‐dimensional model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new three‐dimensional model and how the convection term in the full Euler and Navier‐Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time. © 2008 Wiley Periodicals, Inc.     

A regularity condition of strong solutions to the two‐dimensional equations of compressible nematic liquid crystal flows

This paper is concerned with the short time strong solutions for Cauchy problem to a simplified Ericksen–Leslie system of compressible nematic liquid crystals in two dimensions with vacuum as far field density. We establish a blow‐up criterion for possible breakdown of such solutions at a finite time, which is analogous to the well‐known Serrin's blow‐up criterion for the incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Navier–Stokes equations with free convection in three‐dimensional unbounded triangular channels

The quaternionic calculus is a powerful tool for treating the Navier–Stokes equations very elegantly and in a compact form, through the evaluation of two types of integral operators: the Teodorescu operator and the quaternionic Bergman projector. While the integral kernel of the Teodorescu transform is universal for all domains, the kernel function of the Bergman projector, called the Bergman kernel, depends on the geometry of the domain. In this paper, we use special variants of quaternionic‐holomorphic multiperiodic functions in order to obtain explicit formulas for unbounded three‐dimensional parallel plate channels, rectangular block domains and regular triangular channels. Copyright © 2007 John Wiley & Sons, Ltd.     

On the blow-up of solutions of equations of hydrodynamic type under special boundary conditions

We use the nonlinear capacity method to prove the blow-up of solutions of initial-boundary value problems of hydrodynamic type in bounded domains. We present sufficient boundary conditions ensuring the blow-up of the solution of an equation that is globally solvable under the classical boundary conditions. We estimate the blow-up time of solutions under given initial conditions. Note that it is the first result concerning blow-up for one of the problems considered.     

Exponential decay of the power spectrum and finite dimensionality for solutions of the three dimensional primitive equations

In this article we estimate the number of modes, volumes and nodes, sufficient to describe well the solution of the three dimensional primitive equations; the physical meaning of these estimates is also discussed. We also study the exponential decay of the spatial power spectrum for the three dimensional primitive equations.     

Compactification of supports of energy weak solutions to quasilinear parabolic equations of nonstationary filtration type with strong convection

The paper deals with localization properties of solutions to the Cauchy problem with the initial data u₀(x) ∈ L₂(ℝ_n) for a wide class of equations in the divergence form. This class contains, e.g., the following equation:

Asymptotic behavior of solutions of equations of the Emden-Fowler type at infinity

We obtain an asymptotic representation of solutions of equations of the Emden-Fowler type with “supercritical” exponent and prove the existence of solutions with a given asymptotics. The methods used include the construction of supersolutions for deriving a priori estimates and the use of Kondrat’ev’s results for weighted spaces. The existence of solutions is proved by the Leray-Schauder method.     

A note on the regularity of solutions of infinite dimensional Riccati equations

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smooths solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.     

On the temporal decay of solutions to the two‐dimensional nematic liquid crystal flows

We consider the temporal decay estimates for weak solutions to the two‐dimensional nematic liquid crystal flows, and we show that the energy norm of a global weak solution has non‐uniform decay under suitable conditions on the initial data. We also show the exact rate of the decay (uniform decay) of the energy norm of the global weak solution.     

On the regularity of the axially symmetric solutions to the three‐dimensional MHD equations

We present the improved three‐dimensional axially symmetric incompressible magnetohydrodynamics (MHD) equations with nonzero swirl. We consider three kinds of smooth axially symmetric particular solutions to the MHD equations: (1) u^θ=0,B^r=B^z=0, (2) B^r=B^z=0, and (3) B^θ=0. In particular, we derive new regularity criteria for these three kinds of the three‐dimensional axially symmetric smooth solutions to the MHD equations. Our results also reveal some interesting dynamic behavior of the interaction by the angular vorticity field ω^θ and the angular current density field j^θ. Copyright © 2016 John Wiley & Sons, Ltd.     

Long‐time existence of solutions to the Navier–Stokes equations with inflow–outflow and heat convection

Long time existence of regular solutions to the Navier–Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in cylindrical pipe with inflow and outflow is shown. We assume the slip boundary conditions for velocity and the Neumann conditions for temperature. First, an appropriate estimate is shown, and next the existence of solutions is proved by the Leray–Schauder fixed point theorem. The estimate is obtained for a long time, which is possible because L₂ norms of derivatives in the direction along the cylinder of the initial velocity, initial temperature and the external force are sufficiently small. Copyright © 2012 John Wiley & Sons, Ltd.     

A note on the periodic solutions of a Mathieu–Duffing type equations

By means of a new change of variable we prove the existence of a positive 2π‐periodic solution for the Mathieu–Duffing type equations having its nonlinearity a super‐linear growth. As result we can guarantee the existence of 2π‐periodic solutions even assuming that the parameter of the associated Mathieu equation is in the contentious zone of resonance.     

An improved blow‐up criterion for smooth solutions of the two‐dimensional MHD equations

Our main object is to establish a regularity criterion with p≥q > 1 for the incompressible magnetohydrodynamics equations with zero magnetic diffusivity. Copyright © 2016 John Wiley & Sons, Ltd.     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.     

Global existence of classical solutions to two‐dimensional Navier–Stokes equations with Cauchy data containing vacuum

In this paper, the Cauchy problem to the two‐dimensional isentropic compressible Navier–Stokes equations with smooth initial data containing vacuum is investigated. If the initial data are of small energy but possibly large oscillations, we obtain the global well‐posedness of classical solutions in the case of initially nonvacuum far fields. In particular, the smallness of the energy only depends on the norm of the initial velocity, where β can be arbitrary close to 0. In the case of compactly supported initial density, a blow‐up example is given. Copyright © 2013 John Wiley & Sons, Ltd.     

Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity

In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L² norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.