On the uniqueness for weak solutions of steady double-phase fluids

We consider a double-phase non-Newtonian fluid, described by a stress tensor which is the sum of a p-Stokes and a q-Stokes stress tensor, with 1 < p<2 < q<∞. For a wide range of parameters (p, q), we prove the uniqueness of small solutions. We use the p < 2 features to obtain quadratic-type estimates for the stress-tensor, while we use the improved regularity coming from the term with q > 2 to justify calculations for weak solutions. Results are obtained through a careful use of the symmetries of the convective term and are also valid for rather general (even anisotropic) stress-tensors.

Kink and anti-kink wave solutions for the generalized KdV equation with Fisher-type nonlinearity

This paper proposes a new dispersion-convection-reaction model, which is called the gKdV-Fisher equation, to obtain the travelling wave solutions by using the Riccati equation method. The proposed equation is a third-order dispersive partial differential equation combining the purely nonlinear convective term with the purely nonlinear reactive term. The obtained global and blow-up solutions, which might be used in the further numerical and analytical analyses of such models, are illustrated with suitable parameters.

Saturated Fronts in Crowds Dynamics

We consider a degenerate scalar parabolic equation, in one spatial dimension, of flux-saturated type. The equation also contains a convective term. We study the existence and regularity of traveling-wave solutions; in particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, and entropic solutions turn out to be the vanishing-diffusion limits of traveling-wave solutions to the equation with an additional non-degenerate diffusion. Applications to crowds dynamics, which motivated the present research, are also provided.

On connection between the order of a stationary one-dimensional dispersive equation and the growth of its convective term

A boundary value problem for a stationary nonlinear dispersive equation of 2l+1 order with a convective term in the form u^ku_x, k\in N was considered on an interval (0,L). The existence, uniqueness and continuous dependence of a regular solution as well as a relation between the order l and critical values of k of the equation have been established.

Thermomass Theory in the Framework of GENERIC

Thermomass theory was developed to deal with the non-Fourier heat conduction phenomena involving the influence of heat inertia. However, its structure, derived from an analogy to fluid mechanics, requires further mathematical verification. In this paper, General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework, which is a geometrical and mathematical structure in nonequilibrium thermodynamics, was employed to verify the thermomass theory. At first, the thermomass theory was introduced briefly; then, the GENERIC framework was applied in the thermomass gas system with state variables, thermomass gas density ρh and thermomass momentum mh, and the time evolution equations obtained from GENERIC framework were compared with those in thermomass theory. It was demonstrated that the equations generated by GENERIC theory were the same as the continuity and momentum equations in thermomass theory with proper potentials and eta-function. Thermomass theory gives a physical interpretation to the GENERIC theory in non-Fourier heat conduction phenomena. By combining these two theories, it was found that the Hamiltonian energy in reversible process and the dissipation potential in irreversible process could be unified into one formulation, i.e., the thermomass energy. Furthermore, via the framework of GENERIC, thermomass theory could be extended to involve more state variables, such as internal source term and distortion matrix term. Numerical simulations investigated the influences of the convective term and distortion matrix term in the equations. It was found that the convective term changed the shape of thermal energy distribution and enhanced the spreading behaviors of thermal energy. The distortion matrix implies the elasticity and viscosity of the thermomass gas.

On the regularity of solution to the time-dependent p-Stokes system

In this paper we consider the time evolutionary \(p\)-Stokes problem in a smooth and bounded domain. This system models the unsteady motion or certain non-Newtonian incompressible fluids in the regime of slow motions, when the convective term is negligible. We prove results of space/time regularity, showing that first-order time-derivatives and second-order space-derivatives of the velocity and first-order space-derivatives of the pressure belong to rather natural Lebesgue spaces.

ASPECT RATIO EFFECTS ON BOTTOM HEATED 2D CAVITY USING ENERGY STREAMLINES AND FIELD SYNERGY PRINCIPLE

In the present work free convective air flow in the two-dimensional cavity with three different aspect ratios (AR) are investigated using direct numerical simulation. The bottom wall is assumed to be kept at a uniform higher temperature than that of the top wall and the other two vertical walls are assumed to be thermally insulated. The computations are conducted for Rayleigh number (Ra) values from 103 to 106. Convective schemes are compared and Self Filtered Central Differencing Scheme is used to discretize convective term. Parallel computing MPI code is adapted to run the simulations. An attempt has been made to gather the visualization techniques such as streamlines, isotherms, energy streamlines and field synergy principle to analyse the flow behaviour inside the cavity. When Ra is small, the vertical energy streamlines are observed in the cavity. As Ra further increased, the free energy streamlines observed at the boundary and the trapped energy streamlines at the centre in the horizontal direction. For a fixed Ra, and increasing AR, the average synergy angle increases. This indicates synergy or the coordination between velocity magnitude and temperature field gets decreased and leads to the growth of heat transfer rate. The field synergy principle implies by enhancing the synergy between the velocity vector and temperature gradient.