We present two second order, structure-preserving numerical schemes for a newly derived thermodynamically consistent, non-isothermal hydrodynamical phase field model for incompressible binary viscous fluid flows. The schemes preserve the volume of each fluid phase, the total energy and the positive entropy production rate. The entropy quadratization approach is employed to devise the two semi-discrete numerical schemes in time, preserving both the total energy and the positive entropy production rate. The first scheme is weakly nonlinear, which is solved using iterative methods aided by fast Fourier algorithms. The second scheme is linear, in which a time-dependent supplementary variable is added to preserve the positive entropy production rate. The semi-discrete schemes are discretized in space by a finite difference method on staggered grids subsequently to yield two fully discrete schemes. Mesh refinement is carried out to confirm the order of the schemes and several numerical examples are provided to show hydrodynamic as well as thermal effects in resolving thermocapillary convection near the fluid interface in the incompressible binary viscous fluid flow.

我们为不可压缩的二元粘性流体流动的新近推导的热力学一致性，非等温流体力学相场模型，提出了两个二阶结构保留数值方案。该方案保留了每个流体相的体积，总能量和正熵生产率。熵平方方法被用来及时设计两个半离散数值方案，同时保留了总能量和正熵生产率。第一种方案是弱非线性的，使用快速傅里叶算法辅助的迭代方法解决。第二种方案是线性的，其中添加了时间相关的补充变量以保持正熵产生率。半离散方案在空间上通过有限差分法在交错网格上离散，随后产生两个完全离散的方案。进行网格细化以确认方案的顺序，并提供了几个数值示例来显示流体动力学和热效应，以解决不可压缩的二元粘性流体流动中流体界面附近的热毛细管对流问题。