B. Straughan

Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems

The Chebyshev tau method is examined in detail for a variety of eigenvalue problems arising in hydrodynamic stability studies, particularly those of Orr-Sommerfeld type. We concentrate on determining the whole of the top end of the spectrum in parameter ranges beyond those often explored. The method employing a Chebyshev representation of the fourth derivative operator, D^4, is compared with those involving the second and first derivative operators, D^2, D, respectively; the latter two representations require use of the QZ algorithm in the resolution of the singular generalised matrix eigenvalue problem which arises. The D^2 method is shown to be different from the stream function - vorticity scheme in certain (important and practical) cases. Physical problems explored are those of Posieuille, Couette, and pressure gradient driven circular pipe flow. Also investigated are the three-dimensional problem of Posieuille flow arising from a normal velocity - normal vorticity interaction, and finally Couette and Posieuille problems for two viscous, immiscible fluids, one overlying the other are studied.

A sharp nonlinear stability threshold in rotating porous convection.

A nonlinear stability analysis is performed for the Darcy equations of thermal convection in a fluid‐saturated porous medium when the medium is rotating about an axis orthogonal to the layer in the direction of gravity. A best possible result is established in that we show that the global nonlinear stability Rayleigh number is exactly the same as that for linear instability. It is important to realize that the nonlinear stability boundary holds unconditionally, i.e. for all initial data, and thus for the rotating porous convection problem governed by the Darcy equations, subcritical instabilities are not possible.

Growth and uniqueness in thermoelasticity

A uniqueness theorem is proved for two theories of thermoelasticity capable of admitting finite speed thermal waves, the theories having been proposed by Green & Naghdi. Uniqueness is proved under the weak assumption of requiring only major symmetry of the elasticity tensor; no definiteness whatsoever is postulated. It is shown how to demonstrate uniqueness by a Lagrange identity method and also by producing a novel functional to which to apply the technique of logarithmic convexity. It is remarked on how to extend the result to an unbounded spatial domain without requiring decay restrictions on the solution at infinity. Finally, conditions are derived which show how a suitable measure of the solution will grow at least exponentially in time if the initial ‘energy’ satisfies appropriate conditions. This complements the fundamental work of Knops & Payne, who produced corresponding growth results in the isothermal elasticity case.

Convergence and Continuous Dependence for the Brinkman–Forchheimer Equations

The Brinkman–Forchheimer equations for non-slow flow in a saturated porous medium are analyzed. It is shown that the solution depends continuously on changes in the Forchheimer coefficient, and convergence of the solution of the Brinkman–Forchheimer equations to that of the Brinkman equations is deduced, as the Forchheimer coefficient tends to zero. The next result establishes continuous dependence on changes in the Brinkman coefficient. Following this, a result is proved establishing convergence of a solution of the Brinkman–Forchheimer equations to a solution of the Darcy–Forchheimer equations, as the Brinkman coefficient (effective viscosity) tends to zero. Finally, upper and lower bounds are derived for the energy decay rate which establish that the energy decays exponentially, but not faster than this.

Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity

The equations for convective fluid motion in a porous medium of Brinkman or Forchheimer type are analysed when the viscosity varies with either temperature or a salt concentration. Mundane situations such as salinization require models which incorporate strong viscosity variation. Therefore, we establish rigorous a priori bounds with coefficients which depend only on boundary data, initial data and the geometry of the problem, which demonstrate continuous dependence of the solution on changes in the viscosity. A convergence result is established for the Darcy equations when the variable viscosity is allowed to tend to a constant viscosity.

A note on discontinuity waves in type III thermoelasticity

Two recent nonlinear theories of thermoelasticity, developed by Green and Naghdi, are examined. It is shown that in type II theory, second sound is permissible and both mechanical and temperature waves may propagate. In type III theory we show that the situation is more analogous to that in classical nonlinear thermoelasticity: one wave propagates and a homothermal temperature wave is allowed.

Global nonlinear stability in porous convection with a thermal non-equilibrium model

We show that the global nonlinear stability threshold for convection with a thermal non-equilibrium model is exactly the same as the linear instability boundary. This result is shown to hold for the porous medium equations of Darcy, Forchheimer or Brinkman. This optimal result is important because it shows that linearized instability theory has captured completely the physics of the onset of convection. The equivalence of the linear instability and nonlinear stability boundaries is also demonstrated for thermal convection in a non-equilibrium model with the Darcy law, when the layer rotates with a constant angular velocity about an axis in the same direction as gravity.

Thermoelasticity at cryogenic temperatures

Abstract A theory of thermoelasticity is developed which is suitable for application at cryogenic temperatures. The thermodynamic functions are so chosen as to give correct predictions with experimental findings of second-sound wave speeds in NaF, in the temperature range 8–20 K and in Bi, in the temperature range 1–4 K. After the non-linear theory is presented, a linearized theory is developed. A uniqueness theorem is provided for the linearized theory on an unbounded domain. The question of a half-space heated on its boundary is addressed and, in particular, the question of transverse elastic wave propagation is studied. Finally, a simplified theory in which only unidirectional solutions are allowed is examined.

A note on heroin epidemics

We show that the steady states of the White and Comiskey [E. White, C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci. 208 (2007) 312-324.] model of heroin epidemics are stable.

Anisotropic porous penetrative convection

A linear instability analysis and a nonlinear energy stability analysis is developed for convection in an anisotropic porous medium. The nonlinear analysis is very important since a standard energy method does not in the present situation yield unconditional stability and a weighted analysis must be employed to yield global results, and in addition the nonlinear energy results yield a valuable threshold indicating where possible subcritical instabilities may form. Significantly, we find that when a quadratic density temperature law is used in the anisotropic convection model of Tyvand & Storesletten (1991), then the growth rate σ is always complex provided we are in the anisotropic situation. Thus, the nature of bifurcation into convection is very different from the Boussinesq situation and is always via an oscillatory instability.