The formalism of the internal variable theory is applied to extend Navier-Stokes equations. The internal variable theory provides a thermodynamically consistent derivation of constitutive relations and equations of motion without a priori specifying the nature of internal variables. Both single and dual internal variables cases are thoroughly examined. The similarities and differences of the approaches are emphasized. In the single internal variable framework, the elimination of the internal variable results in Maxwell-type constitutive relations and hyperbolic equations of motion. The dual internal variable technique enables us to create even more sophisticated fluid flow models with coupled equations for fluid motion and internal variable evolution.

中文翻译：

---

内部变量作为扩展 Navier-Stokes 方程的工具

应用内变量理论的形式来扩展 Navier-Stokes 方程。内部变量理论提供了本构关系和运动方程的热力学一致推导，而无需先验指定内部变量的性质。对单内变量和双内变量情况都进行了彻底检查。强调了这些方法的异同。在单内变量框架中，内变量的消除导致麦克斯韦型本构关系和双曲运动方程。双内变量技术使我们能够创建更复杂的流体流动模型，其中包含流体运动和内变量演化的耦合方程。