We derive a quasi-incompressible hydrodynamic phase field model and consistent physical boundary conditions for flows of binary incompressible fluids of distinct intrinsic densities in porous media subject to an external force using the generalized Onsager principle (GOP). When the external force is conservative, the model not only conserves mass and volume fraction but also dissipates the total mechanical energy with respect to the consistent boundary conditions in a fixed domain. In the case of gravity, an extended family of hydrodynamical phase field models parametrized by a specific volume fraction parameter, $\hat{\varphi }$, is derived via the GOP in the context of the buoyancy force, which reduces to the previous model at $\hat{\varphi }=0$. In the case of a constant mobility and equal intrinsic densities in the binary fluid system, the extended model is identical to the original one for any $\hat{\varphi }$. While $\hat{\varphi }$ is chosen as the spatially averaged volume fraction in the general case, the extended model may not be dissipative and thus different from the model at $\hat{\varphi }=0$. Numerical examples are presented to highlight the difference between the two models and interfacial instability with respect to distinct densities when the two models are distinct.

我们使用广义 Onsager 原理 (GOP) 推导出了一个准不可压缩流体动力学相场模型和一致的物理边界条件，用于多孔介质中不同固有密度的二元不可压缩流体在外力作用下的流动。当外力保守时，模型不仅质量和体积分数守恒，而且相对于固定域中的一致边界条件，还耗散了总机械能。在重力的情况下，由特定体积分数参数参数化的流体动力学相场模型的扩展系列，$\widehat{\phi }$, 是在浮力的背景下通过 GOP 推导出来的，它简化为之前的模型$\widehat{\phi }=0$. 在二元流体系统中流动性恒定且固有密度相等的情况下，扩展模型与原始模型相同$\widehat{\phi }$. 尽管$\widehat{\phi }$在一般情况下选择空间平均体积分数，扩展模型可能不是耗散的，因此与模型不同$\widehat{\phi }=0$. 给出了数值示例以突出两个模型之间的差异以及当两个模型不同时相对于不同密度的界面不稳定性。