ALBERT

Description: The finite wavelength instability of viscosity-stratified three-layer flow down an inclined wall is examined for small but finite Reynolds numbers. It has previously been demonstrated using linear theory that three-layer zero-Reynolds-number instabilities can have growth rates that are orders of magnitude larger than those that arise in two-layer structures. Although the layer configurations yielding large growth instabilities have been well characterized, the physical origin of the three-layer inertialess instability remains unclear. Using analytic, numerical and experimental techniques, we investigate the origin and evolution of these instabilities. Results from an energy equation derived from linear theory reveal that interfacial shear and Reynolds stresses contribute to the energy growth of the instability at finite Reynolds numbers, and that this remains true in the limit of zero Reynolds number. This is thus a rare example that demonstrates how the Reynolds stress can play an important role in flow instability, even when the Reynolds number is vanishingly small. Numerical solutions of the Navier-Stokes equations are used to simulate the nonlinear evolution of the interfacial deformation, and for small amplitudes the predicted wave shapes are in excellent agreement with those obtained from linear theory. Further comparisons between simulated interfacial deformations and linear theory reveal that the linear evolution equations are surprisingly accurate even when the interfaces are highly deformed and nonlinear effects are important. Experimental results obtained using aqueous gelatin systems exhibit large wave growth and are in agreement with both the theoretical predictions of small-amplitude behaviour and the nonlinear simulations of the large-amplitude behaviour. Quantitative agreement is confounded owing to water diffusion driven by differences in gelatin concentration between the layers in experiments. However, the qualitative agreement is sufficient to confirm that the correct mechanism for the experimental instability has been determined. © 2005 Cambridge University Press.

Description: The complete integral solution is found for the convectively unstable and oscillatory-forced linear Klein-Gordon equation as a function of spatial variable, x, and time,t. A comparison of the integral solution with series solutions of the Klein-Gordon equation elucidates salient features of both the transient and long-time spatially growing solutions. A rigorous method is developed for identifying the key x/t rays associated with saddle points that can be used to characterize the transition between transient temporally growing and long-term spatially growing waves. This method effectively combines the procedure given by Gordillo & Pérez-Saborid(Phys. Fluids, vol. 14, 2002, pp. 4329-4343) for determining the x/t ray at which the forced spatial growth response affects the observed waveform and competes with the transient response, with an established methodology for identifying the leading and trailing edge rays of an impulse response. The method is applied to a linearized system describing an oscillatory-forced liquid sheet and asymptotic predictions are obtained. Series solutions are used to validate these predictions. We establish that the portion of the solution responsible for spatial growth in the signalling problem is correctly identified byGordillo & Pérez-Saborid(Phys. Fluids, vol. 14, 2002, pp. 4329-4343), and that this interpretation is in contrast with the classical literature. The approach provided here can be applied in multiple ways to study a convectively unstable oscillatory-forced medium. In cases where numerical or series solutions are readily available, the proposed method is used to extract key features of the solution. In cases where only the forced long time behaviour is needed, the dispersion relation is used to extract: (i) the time required to see the forced solution; (ii) the amplitude, phase and spatial growth of the forced solution; and (iii) the breadth of the transient. © 2012 Cambridge University Press.