A finite difference scheme is used to develop a numerical method to solve the flow of an unbounded viscoelastic fluid with zero to moderate inertia around a prolate spheroidal particle. The equations are written in prolate spheroidal coordinates, and the shape of the particle is exactly resolved as one of the coordinate surfaces representing the inner boundary of the computational domain. As the prolate spheroidal grid is naturally clustered near the particle surface, good resolution is obtained in the regions where the gradients of relevant flow variables are most significant. This coordinate system also allows large domain sizes with a reasonable number of mesh points to simulate unbounded fluid around a particle. Changing the aspect ratio of the inner computational boundary enables simulations of different particle shapes ranging from a sphere to a slender fiber. Numerical studies of the latter particle shape allow testing of slender body theories. The mass and momentum equations are solved with a Schur complement approach allowing us to solve the zero inertia case necessary to isolate the viscoelastic effects. The singularities associated with the coordinate system are overcome using L'Hopital's rule. A straightforward imposition of conditions representing a time-varying combination of linear flows on the outer boundary allows us to study various flows with the same computational domain geometry. {For the special but important case of zero fluid and particle inertia we obtain a novel formulation that satisfies the force- and torque-free constraint in an iteration-free manner.} The numerical method is demonstrated for various flows of Newtonian and viscoelastic fluids around spheres and spheroids (including those with large aspect ratio). Good agreement is demonstrated with existing theoretical and numerical results.

翻译：采用有限差分格式发展了一种数值方法，用于求解零至中等惯性范围内的无界粘弹性流体绕长球颗粒的流动。方程在长球坐标系中建立，颗粒形状被精确解析为计算域内边界的坐标曲面之一。由于长球坐标网格在颗粒表面附近自然聚集，在相关流动变量梯度最显著的区域可获得良好的分辨率。该坐标系还允许在合理网格点数下采用大计算域尺寸，以模拟颗粒周围的无界流体。通过改变内计算边界的纵横比，可模拟从球体到细长纤维的不同颗粒形状。针对后者的数值研究可用于检验细长体理论。采用舒尔补方法求解质量与动量方程，从而能够求解为隔离粘弹性效应所需的零惯性情形。利用洛必达法则克服了坐标系相关的奇异性。在外边界上直接施加表示线性流动时变组合的条件，使得我们能在相同计算域几何下研究多种流动。特别针对流体与颗粒惯性均为零的重要情形，我们提出了一种无需迭代即可满足无力和无力矩约束的新公式。该方法已在牛顿流体和粘弹性流体绕球体及长球体（包括大纵横比情形）的各种流动中得到验证，并与现有理论和数值结果吻合良好。