Interactions between steady and oscillatory convection in mushy layers

Abstract

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

Nonlinear stability and patterns in granular plane Couette flow: Hopf and pitchfork bifurcations, and evidence for resonance

“…Therefore, the signature of above resonances would appear as a discontinuity in the first Landau coefficient as we shall demonstrate in § 6.3. It is worth pointing out that the mean-flow and 1 : n resonances have been uncovered and are known to play an important role in dynamical transition and pattern formation, via mode interactions, for both Newtonian and non-Newtonian fluids in a variety of flows (Mizushima & Gotoh 1985;Knobloch & Proctor 1988;Proctor & Jones 1988;Manneville 1990;Fujimura 1992;Suslov & Paolucci 1997;Guba & Worster 2010).…”

On nonlinear evolution of convective flow in an active mushy layer

“…Regarding the novelty of the present results, here we make a comparison between our key results and those due to several related studies [7,[16][17][18] that were based essentially on the earlier analytical modeling [6] and led to derivation of the associated amplitude equation. For the steady case considered in the present study, the relevant Landau constant for the associated amplitude equations derived in [7,16,17] is found to be positive for either constant or weakly variable permeability, but larger for the nonconstant permeability case.…”

“…After computing the basic-state solutions, we obtain the marginal stability curve which consists of points given by the critical Rayleigh number as a function of the wavenumber. These points are obtained by solving (16)(17)(18) with boundary conditionsP 0 =θ 0 = 0 at z = 0 and DP 0 =θ 0 =φ 0 = 0 at z = δ using a fourth-order Runge-Kutta shooting method [22, p. 581]. The critical Rayleigh number, designated by R c , is the minimum value of R 0 below which no motion can exist based on the linear system.…”

“…For the constant permeability case, the critical pair is given by (1.33, 15.6162). Once we obtain the critical pair (α c , R c ), we solve the linear system (of order 1 ) given by (16)(17)(18) for the dependent variables numerically using this critical pair. Then we numerically compute the solutions of the adjoint system given by (23), first-order systems (of order 2 ) given by (28-33) using a fourth-order Runge-Kutta method.…”

“…Analysis of convection in a mushy layer with a deformable permeable interface was studied by Roper et al [17]. Guba and Worster [18] studied interactions between steady and oscillatory convection. The problems considered by a number of researchers [7,[16][17][18] were essentially based on the model due to Amberg and Homsy [6], which applies several restrictions including large concentration ratio, small thickness of mushy layer, high far-field temperature, and small deviation of the permeability function from a constant value.…”