A general approach for modeling the motion of rigid and deformable ellipsoids in ductile flows

摘要

A general approach for modeling the motion of rigid or deformable objects in viscous flows is presented. It is shown that the rotation of a 3D object in a viscous fluid, regardless of the mechanical property and shape of the object, is defined by a common and simple differential equation, dQ/dt = -ΘQ., where Q is a matrix defined by the orientation of the object and Θ is the angular velocity tensor of the object. The difference between individual cases lies only in the formulation for the angular velocity. Thus the above equation, together with Jeffery's theory for the angular velocity of rigid ellipsoids, describes the motion of rigid ellipsoids in viscous flows. The same equation, together with Eshelby's theory for the angular velocity of deformable ellipsoids, describes the motion of deformable ellipsoids in viscous flows. Both problems are solved here numerically by a general approach that is much simpler conceptually and more economic computationally, compared to previous approaches that consider the problems separately and require numerical solutions to coupled differential equations about Euler angles or spherical (polar coordinate) angles. A Runge-Kutta approximation is constructed for solving the above general differential equation. Singular cases of Eshelby's equations when the object is spheroidal or spherical are handled in this paper in a much simpler way than in previous work. The computational procedure can be readily implemented in any modern mathematics application that handles matrix operations. Four MathCad Worksheets are provided for modeling the motion of a single rigid or deformable ellipsoid immersed in viscous fluids, as well as the evolution of a system of noninteracting rigid or deformable ellipsoids embedded in viscous flows.