In this study, we propose an efficient mathematical equation to describe incompressible fluid flow-coupled crystallization. By using the $L^2$-gradient flow, the evolutional equation is derived from an energy functional and a nonlocal Lagrange multiplier is introduced to preserve the total mass. Various physical phenomena of colloidal crystals can be modeled using a generalized nonlinearity and the fluid dynamics are described by incompressible Navier–Stokes (NS) equations. To efficiently manage the nonlinear and coupled terms in this complex fluid system and maintain the discrete version of the energy stability, we develop a linear, second-order time-accurate, and energy-stable numerical method for the fluid flow-coupled phase-field crystal model. The energy stability of the discrete method is estimated. In each time step, all variables are completely decoupled, and we only need to solve several linear elliptic-type equations. We adopt a relaxation method to correct the energy. The main merits of this study are as follows: (i) a simple fluid flow-coupled crystal equation with generalized nonlinearity is derived; (ii) the proposed scheme not only satisfies the energy stability but also achieves improved consistency; and (iii) the calculation in each time step is highly efficient in implementing the proposed scheme. Numerical experiments, including flow-coupled phase transition, crystallization in shear flow, and sedimentation of crystals, are performed to validate the accuracy, stability, and superior performance of the proposed method.

在这项研究中，我们提出了一个有效的数学方程来描述不可压缩流体流动耦合结晶。通过使用${\mathrm{大号}}^{\mathrm{2个}}$-梯度流，进化方程是从能量泛函导出的，并且引入了非局部拉格朗日乘数以保持总质量。可以使用广义非线性对胶体晶体的各种物理现象进行建模，并且流体动力学由不可压缩的 Navier-Stokes (NS) 方程描述。为了有效地管理这个复杂流体系统中的非线性和耦合项并保持能量稳定性的离散版本，我们为流体流动耦合相场开发了一种线性、二阶时间精确和能量稳定的数值方法水晶模型。估计了离散方法的能量稳定性。在每个时间步中，所有变量完全解耦，我们只需要求解几个线性椭圆型方程。我们采用放松法来修正能量。本研究的主要优点如下： (i) 推导了具有广义非线性的简单流体流动耦合晶体方程；(ii) 所提出的方案不仅满足能量稳定性，而且提高了一致性；(iii) 每个时间步长的计算对于实施所提出的方案都是非常有效的。进行了包括流动耦合相变、剪切流结晶和晶体沉降在内的数值实验，以验证所提出方法的准确性、稳定性和优越性能。(iii) 每个时间步长的计算对于实施所提出的方案都是非常有效的。进行了包括流动耦合相变、剪切流结晶和晶体沉降在内的数值实验，以验证所提出方法的准确性、稳定性和优越性能。(iii) 每个时间步长的计算对于实施所提出的方案都是非常有效的。进行了包括流动耦合相变、剪切流结晶和晶体沉降在内的数值实验，以验证所提出方法的准确性、稳定性和优越性能。