Traveling wave solutions are studied numerically and theoretically for models of viscous core–annular flows and falling film flows inside a tube. The models studied fall into one of two classes, referred to here as ‘thin-film’ or ‘long-wave’. One model of each type is studied for three problems: a falling viscous film lining the inside of a tube, and core–annular flow with either equal- or unequal-density fluids. In recent work, traveling wave solutions for some of these equations were found using a smoothing technique that removes a degeneracy and allows for continuation onto a periodic family of solutions from a Hopf bifurcation. This paper has three goals. First, the smoothing technique used in earlier studies is justified for these models using asymptotics. Second, this technique is used to find numerically families of traveling wave solutions not previously explored in detail, including some which have multiple turning points due to the interaction between gravity, viscous forces, surface tension, and pressure-driven flow. Third, the stability of these solutions is studied using asymptotics near the Hopf bifurcation point, and numerically far from this point. In particular, a simple theory using the constant solution at the Hopf bifurcation point produces estimates for the eigenvalues in good agreement with numerics, with the exception of the eigenvalues closest to zero; higher-order asymptotics are used to predict these eigenvalues. Far from the Hopf bifurcation point, the stabilizing role of increasing surface tension is quantified numerically for the thin-film models, while multiple changes in stability occur along families of solutions for some of the long-wave models.

对行波解在数值上和理论上都研究了粘性的核-环状流和管内降膜流模型。研究的模型属于两类之一，在这里被称为“薄膜”或“长波”。研究了每种类型的模型中的三个问题：管道内部衬里的粘性膜下降，以及密度相等或不相等的流体的核环流。在最近的工作中，使用平滑技术找到了其中一些方程的行波解，该技术消除了简并并允许从Hopf分支延续到周期解的周期族上。本文有三个目标。首先，在早期研究中使用的平滑技术对于使用渐近线的这些模型是合理的。第二，该技术用于查找以前未详细探讨的行波解的数值族，其中包括一些由于重力，粘性力，表面张力和压力驱动的流动之间的相互作用而具有多个转折点的解。第三，使用渐近点在Hopf分叉点附近并且在数值上远离该点的渐近性来研究这些解决方案的稳定性。特别是，一个简单的理论在Hopf分叉点使用常数解，可以得出特征值的估计值与数字具有很好的一致性，但特征值最接近于零。高阶渐近线用于预测这些特征值。远离霍普夫分叉点，对于薄膜模型，通过数值量化了增加表面张力的稳定作用，