On the motion of several disks in an unbounded viscous incompressible fluid

Abstract

In this paper, we study the time evolution of a finite number of homogeneous rigid disks within a viscous homogeneous incompressible fluid in the whole domain R 2 . The motion of the fluid is governed by Navier-Stokes equations, whereas the movement of each rigid body is described by the standard conservation laws of linear and angular momentum. The motion of the rigid bodies inside the fluid makes the fluid domain time dependent and unknown a priori. At first, we prove the existence and uniqueness of strong solutions of the considered problem up to collision. Then, we prove that contact between rigid bodies cannot occur for almost arbitrary configurations.

Long-time behavior of several point particles in a 1D viscous compressible fluid

“…For example, to show (27), it suffices to check that 𝐼 = − 𝐺 0+ (𝑥, 𝑡 − 𝑠) + 𝐺 0+ (𝑥, 𝑡 − 𝑠) 0 𝑁 (0 + , 𝑠)+𝐺 −+ (𝑥, 𝑡−𝑠) 0 𝑁 (0 − , 𝑠)+𝐺 + 0𝑏 (𝑥, 𝑡−𝑠) 0 𝑁 (0, 𝑠)and𝐼 𝐼 = − 𝐺 ++ (𝑥 − 1, 𝑡 − 𝑠) + 𝐺 ++ (𝑥 + 1, 𝑡 − 𝑠) , 𝑠) + 𝐺 0+ (𝑥 − 1, 𝑡 − 𝑠) + 𝐺 0+ (𝑥 + 1, 𝑡 − 𝑠) 0 𝑁 (1 − , 𝑠) + 𝐺 + 1𝑏 (𝑥, 𝑡 − 𝑠) 0 𝑠) vanish.For brevity, we shall only show that 𝐼 is zero. First, note that by(23), we have𝐺 0+ (𝑥, 𝑡 − 𝑠) + 𝐺 0+ (𝑥, 𝑡 − 𝑠) 2𝑖 (𝑥 + 2𝑖, 𝑡 − 𝑠) + 𝐺 1,2𝑖+1 (𝑥 + 2𝑖, 𝑡 − 𝑠) 𝐺 −+ (𝑥, 𝑡 − 𝑠) 0 𝑁 (0 + , 𝑠).Note also that 𝐺+ 0𝑏 = 𝐺 −+ by definition. From these, it follows that 𝐼 = −𝐺 −+ (𝑥, 𝑡 − 𝑠) 0 𝑁 (0, 𝑠) + 𝐺 + 0𝑏 (𝑥, 𝑡 − 𝑠) 0 𝑁 (0, 𝑠) = 0.…”

“…We also refer to [8] for a related result. These works then prompted investigations on the possibility of collisions between solids in multi-dimensions: see, e.g., [6,7,9,10,11,21,22,23] for such results. Now what happens for the motion of several point particles in the 1D viscous compressible fluid considered in [12,13]?…”