Weakly nonlinear stability of viscous flow past a flexible surface
Weakly nonlinear stability of viscous flow past a flexible surface
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流经柔性表面的粘性流的弱非线性稳定性
DOI：
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发表时间：
2001
影响因子：
3.7
通讯作者：
V. Kumaran
中科院分区：
文献类型：
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作者：
V. Shankar;V. Kumaran
本文分析了零雷诺数下粘性流体绕柔性表面流动的弱非线性稳定性。该系统由牛顿流体通过粘弹性介质的库埃特流组成，粘弹性介质具有无量纲厚度H（壁厚与流体厚度之比）和粘度比μr（壁和流体介质的粘度之比）。壁介质是由流体在一个表面和两种不同类型的边界条件被认为是在壁介质的另一个表面-为“接枝”凝胶零位移条件的应用，而为“吸附”凝胶的位移垂直于表面是零，但表面被允许在横向移动。线性稳定性分析表明，对于接枝凝胶，最不稳定的模式为α → O（1），而对于吸附凝胶，最不稳定的模式为α → 0，其中α是微扰的波数。从弱非线性分析的结果表明，在线性不稳定性的分叉的性质是非常不同的接枝和吸收凝胶。对于流过接枝凝胶的情况，分叉总是亚临界的。然而，它被发现，相对较弱，但有限振幅的扰动不显着降低所需的临界速度，从线性稳定性理论预测的临界速度不稳定的流动。对于吸附态凝胶，在很宽的参数μr和H范围内，在小波数极限下都可以存在超临界平衡态，而在较大波数时分岔变为亚临界分岔，并且随着波数的增大，分岔从超临界向亚临界转变.
The weakly nonlinear stability of viscous fluid flow past a flexible surface is analysed in the limit of zero Reynolds number. The system consists of a Couette flow of a Newtonian fluid past a viscoelastic medium of non-dimensional thickness H (the ratio of wall thickness to the fluid thickness), and viscosity ratio μr (ratio of the viscosities of wall and fluid media). The wall medium is bounded by the fluid at one surface and two different types of boundary conditions are considered at the other surface of the wall medium – for ‘grafted’ gels zero displacement conditions are applied while for ‘adsorbed’ gels the displacement normal to the surface is zero but the surface is permitted to move in the lateral direction. The linear stability analysis reveals that for grafted gels the most unstable modes have α ∼ O(1), while for adsorbed gels the most unstable modes have α → 0, where α is the wavenumber of the perturbations. The results from the weakly nonlinear analysis indicate that the nature of the bifurcation at the linear instability is qualitatively very different for grafted and absorbed gels. The bifurcation is always subcritical for the case of flow past grafted gels. It is found, however, that relatively weak but finite-amplitude disturbances do not significantly reduce the critical velocity required to destabilize the flow from the critical velocity predicted by the linear stability theory. For the case of adsorbed gels, it is found that a supercritical equilibrium state could exist in the limit of small wavenumber for a wide range of parameters μr and H, while the bifurcation becomes subcritical at larger values of the wavenumber and there is a transition from supercritical to subcritical bifurcation as the wavenumber is increased.