Axisymmetric viscous interfacial oscillations – theory and simulations
Axisymmetric viscous interfacial oscillations – theory and simulations
复制标题
轴对称粘性界面振荡 – 理论与模拟
DOI：
10.1017/jfm.2017.443
复制
发表时间：
2017
影响因子：
3.7
通讯作者：
R. Dasgupta
中科院分区：
文献类型：
--
作者：
P. K. Farsoiya;Y. S. Mayya;R. Dasgupta
我们研究轴对称，自由振荡驱动的重力和表面张力在两个粘性，不混溶，径向无界流体的界面，分析和数值。界面作为零阶贝塞尔函数（在空间中）扰动，其演变作为时间的函数获得。在线性化近似中，我们求解初值问题（IVP），得到波振幅随时间演化的解析表达式。结果表明，线性贝塞尔模式在时间上完全相同的方式在平面几何形状中的傅立叶模式的演变。我们得到新的解析表达式的时变涡量和压力场在这两种流体。对于小的初始振幅，我们的分析结果显示出良好的协议从解决轴对称Navier-Stokes方程数值。我们还比较了我们的结果与正常模式近似，并发现后者是一个准确的表示在非常早期和晚期的时间。简正模近似与IVP解之间的偏差随粘度比的增大而增大。涡量场在界面处有一个跳跃的不连续性，我们发现这种跳跃依赖于两种流体的粘度和密度比。在模拟中增加初始扰动幅度后，非线性产生了分析IVP解决方案中不存在的定性新功能。值得注意的是，发现射流出现在对称轴上升到大于初始扰动幅度的高度。增加扰动幅度进一步导致射流经历末端夹断，产生子液滴。这可能发生在前进或后退射流中，取决于粘度比。最大射流高度和扰动振幅之间的关系。界面的汉克尔变换表明，在大的扰动振幅更高的波数出现，共享一些最低模式的能量。当这些额外的更高的模式存在时，界面具有尖的波峰和圆的波谷。
We study axisymmetric, free oscillations driven by gravity and surface tension at the interface of two viscous, immiscible, radially unbounded fluids, analytically and numerically. The interface is perturbed as a zeroth-order Bessel function (in space) and its evolution is obtained as a function of time. In the linearised approximation, we solve the initial value problem (IVP) to obtain an analytic expression for the time evolution of wave amplitude. It is shown that a linearised Bessel mode temporally evolves in exactly the same manner as a Fourier mode in planar geometry. We obtain novel analytical expressions for the time varying vorticity and pressure fields in both fluids. For small initial amplitudes, our analytical results show excellent agreement with those obtained from solving the axisymmetric Navier–Stokes equations numerically. We also compare our results with the normal mode approximation and find the latter to be an accurate representation at very early and late times. The deviation between the normal mode approximation and the IVP solution is found to increase as a function of viscosity ratio. The vorticity field has a jump discontinuity at the interface and we find that this jump depends on the viscosity and the density ratio of the two fluids. Upon increasing the initial perturbation amplitude in the simulations, nonlinearity produces qualitatively new features not present in the analytical IVP solution. Notably, a jet is found to emerge at the axis of symmetry rising to a height greater than the initial perturbation amplitude. Increasing the perturbation amplitude further causes the jet to undergo end pinch off, giving birth to a daughter droplet. This can happen either for an advancing or a receding jet, depending on the viscosity ratio. A relation is found between the maximum jet height and the perturbation amplitude. Hankel transform of the interface demonstrates that at large perturbation amplitudes higher wavenumbers emerge, sharing some of the energy of the lowest mode. When these additional higher modes are present, the interface has pointed crests and rounded troughs.