Hadamard instability

Loss of evolution Hadamard instabilities or instabilities to short waves... 

This section concerns the Cauchy problem or initial value problem, where initial data at time t = 0 are given. It was noticed by Rutkevitch [6,7], and systematized by Joseph et al. [8], Joseph and Saut [9], and Dupret and Marchal [10] that Maxwell type models can present Hadamard instabilities, that is, instabilities to short waves. (See [11] for a recent discussion of more general models.) Then, the Cauchy problem is not well-posed in any good class but analytic. Highly oscillatory initial data will grow exponentially in space at any prescribed time. An ill-posed problem leads to catastrophic instabilities in numerical simulations. For example, even if one initiates the solution in a stable region, one could get arbitrarily close to an unstable one. 

It can be shown [8] that Hadamard instabilities are possible for admissible motions if a is in the interval (—1,1), e.g., in extensional flows. On the other hand, restrictions on the eigenvalues of r prevent Hadamard instabilities for a = 1. This is immediately seen from the integral forms qf (4)-(5) for the upper- and lower-convected Maxwell models, which imply constraints on the eigenvalues of the Cauchy-Green tensors. (See, for instance, [12].)... 

We now turn to local existence of solutions for Maxwell-type models. The situation is much trickier here since these models can display Hadamard instabilities (see Section 2.1), and no general results seem to be known so far. One has, in any case, to restrict initial data to Hadamard stable ones. A possible way to overcome the difficulty is to consider models satisfying an eUipticity condition, which will imply well-posedness. This approach was followed by Renardy [41], whose results are briefly described below. 

Note that, for Maxwell models with —1 < a < 1, relation (15) is satisfied locally in time provided it is satisfied at time < = 0. For a = 1, relation (15) is equivalent to relation (5), which insures that the initial value problem is well-posed this is a natural condition to impose on the stress. But, for a 1, condition (15) reveals that the model is not always of evolution type, which means that Hadamard instabilities can occur. (See Section 2.1)... 

One point that has not been emphasized is that all of the preceding analysis and discussion pertains only to the steady-state problem. From this type of analysis, we cannot deduce anything about the stability of the spherical (Hadamard Rybczynski) shape. In particular, if a drop or bubble is initially nonspherical or is perturbed to a nonspherical shape, we cannot ascertain whether the drop will evolve toward a steady, spherical shape. The answer to this question requires additional analysis that is not given here. The result of this analysis26 is that the spherical shape is stable to infinitesimal perturbations of shape for all finite capillary numbers but is unstable in the limit Ca = oo (y = 0). In the latter case, a drop that is initially elongated in the direction of motion is predicted to develop a tail. A drop that is initially flattened in the direction of motion, on the other hand, is predicted to develop an indentation at the rear. Further analysis is required to determine whether the magnitude of the shape perturbation is a factor in the stability of the spherical shape for arbitrary, finite Ca.21 Again, the details are not presented here. The result is that finite deformation can lead to instability even for finite Ca. Once unstable, the drop behavior for finite Ca is qualitatively similar to that predicted for infinitesimal perturbations of shape at Ca = oo that is, oblate drops form an indentation at the rear, and prolate drops form a tail. 

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