Weakly nonlinear theory and finite-difference calculations are used to describe steadystate and oscillatory convective heat transport in water-saturated porous media. Two-dimensional rolls in a rectangular region are considered when the imposed temperature difference between the horizontal boundaries is as large as 200 K, corresponding to a viscosity ratio of about 6·5. The lowest-order weakly nonlinear results indicate that the variation of the Nusselt number with the ratio of the actual Rayleigh number to the corresponding critical value R/Rc, is independent of the temperature difference for the range considered. Results for the Nusselt number obtained from finite-difference solutions contain a weak dependence on temperature difference which increases with the magnitude of R/Rc. When R/Rc = 8 the constantviscosity convection pattern is steady, while those with temperature differences of 100 and 200 K are found to oscillate.

Time-dependent flow dynamics within a cylinder with sidewall mass injection are investigated. A time-dependent injection velocity, prescribed along the sidewall boundary of a long, narrow, half-open cylinder, induces a low Mach number, high Reynolds number flow. The injection is the source of planar acoustic disturbances which interact with the injected fluid to produce vorticity on the sidewall in an inviscid manner. The analysis of these flow processes is based on the Navier–Stokes equations, which are reduced to simpler forms using a multiple-scale asymptotic analysis. The equations that arise from the analysis describe the leading-order vorticity dynamics. These nonlinear equations possess both wave and diffusion properties and are solved in an initial value sense. The results show that the vorticity produced at the sidewall convects toward the center of the cylinder, diffuses radially, and convects downstream.

The spatially homogeneous model of a high activation energy thermal explosion is studied when the heat loss parameter a is close to the classically-defined critical value a = e. Asymptotic solutions are developed which describe the time-history of the temperature and reactant depletion. It is shown that there is a critical time period, large with respect to the characteristic conduction time, in which the temperature variation is described by a Riccati equation. The solution properties of this nonlinear equation permit one to define a value of A = a-e which separates subsequent subcritical and supercritical behaviour.

The onset of convection in a horizontal, isotropic, water-saturated porous medium is considered. The temperature difference between the top and bottom is as large as 250 °C. The effects of an eightfold variation in kinematic viscosity are included. The critical Rayleigh number is found to be substantially reduced from the classical value although the associated wavenumber is nearly the same. Neutral mode streamline and isotherm patterns are considerably distorted in the vertical direction in distinction to the symmetric patterns found in the constant viscosity classical calculation.

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A mathematical model is formulated to describe the initiation and evolution of intense unsteady vorticity in a low Mach number (M), weakly viscous internal flow sustained by mass addition through the sidewall of a long, narrow cylinder. An O(M) axial acoustic velocity disturbance, generated by a prescribed harmonic transient endwall velocity, interacts with the basically inviscid rotational steady injected flow to generate time-dependent vorticity at the sidewall. The steady radial velocity component convects the vorticity into the flow. The axial velocity associated with the vorticity field varies across the cylinder radius and in particular has an instantaneous oscillatory spatial distribution with a characteristic wavelength O(M) smaller than the radius. Weak viscous effects cause the vorticity to diffuse on the small radial length scale as it is convected from the wall toward the axis. The magnitude of the transient vorticity field is larger by O(M−1) than that in the steady flow.An initial-boundary-value formulation is employed to find nonlinear unsteady solutions when a pressure node exists at the downstream exit of the cylinder. The complete velocity consists of a superposition of the steady flow, an acoustic (irrotational) field and the rotational component, all of the same magnitude.

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Weakly nonlinear analysis is used to calculate the possible two- and three-dimensional convection patterns in a rectangular parallelepiped of saturated porous media when the horizontal dimensions are integral multiples of the vertical dimension. A two-term expansion for the Nusselt number is found for values of the Rayleigh number close to the critical values. It is shown that the two-dimensional roll configurations transfer heat more effectively than does the three-dimensional pattern of motion when the Rayleigh number is just above the critical value.