Thermal convection in a Brinkman–Darcy–Kelvin–Voigt fluid with a generalized Maxwell–Cattaneo law

We investigate thoroughly a model for thermal convection of a class of viscoelastic fluids in a porous medium of Brinkman–Darcy type. The saturating fluids are of Kelvin–Voigt nature. The equations governing the temperature field arise from Maxwell–Cattaneo theory, although we include Guyer–Krumhansl terms, and we investigate the possibility of employing an objective derivative for the heat flux. The critical Rayleigh number for linear instability is calculated for both stationary and oscillatory convection. In addition a nonlinear stability analysis is carried out exactly

Thermal convection in a higher-gradient Navier–Stokes fluid

We discuss models for flow in a class of generalized Navier–Stokes equations. The work concentrates on producing models for thermal convection, analysing these in detail, and deriving critical Rayleigh and wave numbers for the onset of convective fluid motion. In addition to linear instability theory we present a careful analysis of fully nonlinear stability theory. The theories analysed all possess a bi-Laplacian term in addition to the normal spatial derivative term. The theories discussed are Stokes couple stress theory, dipolar fluid theory, Green–Naghdi theory, Fried–Gurtin–Musesti theory, and a second theory of Fried and Gurtin. We show that the Stokes couple stress theory and the Fried–Gurtin–Musesti theory involve the same partial differential equations while those of Green–Naghdi and dipolar theory are similar. However, we concentrate on boundary conditions which are crucial to understand all five theories and their differences

Stabilization estimates for the Brinkman–Forchheimer–Kelvin–Voigt equation backward in time

The final value value problem for the Brinkman–Forchheimer–Kelvin–Voigt equations is analysed for quadratic and cubic types of Forchheimer nonlinearity. The main term in the Forchheimer equations is allowed to be fully anisotropic. It is shown that the solution depends continuously on the final data provided the solution satisfies an a priori bound in L3. The technique employed avoids the use of a specialist method for an improperly posed problem such as logarithmic convexity

Thermal effects on nonlinear acceleration waves in the Biot theory of porous media

We generalize a theory of Biot for a porous solid based on nonlinear elasticity theory to incorporate temperature effects. Acceleration waves are studied in detail in the fully nonlinear theory. The wavespeeds are found explicitly and the amplitudes are then determined. The possibility of shock formation is discussed

Some Remarks on Functional Analysis

Functional analysis is a key tool in the study of partial differential equations which helps to answer key questions such as existence, well-posedness, and the class in which a solution should belong. We begin these remarks by introducing normed spaces and Banach spaces and then bounded linear operators in normed spaces. Next, we define Hilbert spaces and consider aspects relating to linear operators on Hilbert spaces. With this structure, we are able to consider well-posed of problems and describe the notions of Hölder and Lyapunov stability and Hadamard well-posedness. Finally, we describe how some thermal stress problems can be formulated using abstract operator notation

Global stability for convection when the viscosity has a maximum

Until now, an unconditional nonlinear energy stability analysis for thermal convection according to Navier–Stokes theory had not been developed for the case in which the viscosity depends on the temperature in a quadratic manner such that the viscosity has a maximum. We analyse here a model of non-Newtonian fluid behaviour that allows us to develop an unconditional analysis directly when the quadratic viscosity relation is allowed. By unconditional, we mean that the nonlinear stability so obtained holds for arbitrarily large perturbations of the initial data. The nonlinear stability boundaries derived herein are sharp when compared with the linear instability thresholds

Sharp Instability Estimates for Bidisperse Convection with Local Thermal Non-equilibrium

We analyse a theory for thermal convection in a Darcy porous material where the skeletal structure is one with macropores, but also cracks or fissures, giving rise to a series of micropores. This is thus thermal convection in a bidisperse, or double porosity, porous body. The theory allows for non-equilibrium thermal conditions in that the temperature of the solid skeleton is allowed to be different from that of the fluid in the macro- or micropores. The model does, however, allow for independent velocities and pressures of the fluid in the macro- and micropores. The threshold for linear instability is shown to be the same as that for global nonlinear stability. This is a key result because it shows that one may employ linearized theory to ensure that the key physics of the thermal convection problem has been captured. It is important to realize that this has not been shown for other theories of bidisperse media where the temperatures in the macro- and micropores may be different. An analytical expression is obtained for the critical Rayleigh number and numerical results are presented employing realistic parameters for the physical values which arise

Nonlinear acceleration wave propagation in the DKM theory

We study the evolutionary development of an acceleration wave propagating in a saturated porous material according to a Biot theory proposed by Donskoy, Khashanah and McKee. The theory is fully nonlinear, includes dissipation, and the analysis presented is exact. We derive sufficient conditions to show that two distinct waves propagate, a fast wave and a slower wave. A solution for the wave amplitude is presented for a wave moving into an equilibrium region