两个同心旋转圆柱之间的两种流体的交界面几何形状问题

Geometric Shape of Interface Surface of Bicomponent Flows Between Two Concentric Rotating Cylinders

研究两个同心旋转圆柱之间的两种流体的交界面几何形状问题.利用张量分析工具,给出了忽略耗散能量影响下交界面几何形状是一种能量泛函的临界点,其对应的Euler-Lagrange方程是1个非线性椭圆边值问题.对于粘性引起的耗散能量不能忽略的情况下,同样给出了1个带有耗散能量的能量泛函,其临界点是交界面几何形状,相应的Euler-Lagrange方程也是1个二阶的非线性椭圆边值问题.这样,交界面几何形状问题转化为求解非线性椭圆边值问题.

The shape problem of interface surface of bicomponent flows between two concentric rotating cylinders is investigated. By the tool of tenor analysis, this problem can be reduced to an isoperimetric problem of energy functional when neglecting the effects of dissipative energy caused by viscosity. The associated Euler-Lagrangian equation, which is a nonlinear elliptic boundary value problem of second order was derived.Moreover,in the case of considering the effects of dissipative energy,another total energy functional with dissipative energy to characterize the geometric shape of interface surface was proposed, and the corresponding Euler-Lagrangian equation which is also a nonlinear elliptic boundary value problem of second order was obtained. Thus, the problerm of geometric shape is transfomed into the nonlinear boundary value problem of second order in both cases.