Nonlinear waves in gas-fluidized beds

A mathematical model for gas-fluidized beds is examined which treats both the particles and gas as continua by volume averaging. The system is then considered as two interlocking one-phase fluids. For small perturbations to the uniform state, these equations have been shown by Crighton (1991) to reduce to the Burgers-KdV equation and under certain criteria, we have instability. We consider the unstable situation when the amplification effects are a perturbation to the KdV equation and take an initial condition of a single KdV soliton. The growth of this soliton is followed through several regions in which the unstable Burgers-KdV equation is no longer appropriate, but KdV remains the leading order equation. Eventually, there is a fundamental change in the solution and the new governing equations are fully nonlinear and O(1). These admit a solitary wave solution which matches back onto the KdV soliton. Thus, we can follow the formation of a bubble from a small amplitude perturbation to the uniform state.