On the motion of several rigid bodies in an incompressible non-Newtonian and heat-conducting fluid

Over the past years, there has been a great impulse in studying the motion of a rigid body. We would like to quote the work of Brenner [2] concerning the steady motion of one or more bodies in a linear viscous liquid in the Stokes approximation as well as Weinberger [18], [19], Serre [16] regarding the fall of a body in an incompressible Navier-Stokes fluid under the action of gravity and Borchers [1] for the existence of weak solutions. We would like to distinguish that there are two different problems. First one which is modelling the motion of fluid around a moving body with prescribed velocity see work of Galdi, Hishida, Farwig, [6, 7, 8, 9, 12]. The another problem is the motion of rigid bodies in a viscous fluids, which satisfies the second Newton’s law and the conservation of momentum of momentum. In our paper we will deal with the second problem. We consider the motion of one or several rigid bodies in a viscous fluid occupying a bounded domain Ω ⊂ R represents an interesting theoretical problem featuring, among others, possible contacts of two or more solid objects. We will consider the problem of rigid bodies in viscous non-Newtonian fluids where both are heat conducting materials. Using penalization method developed by Conca, San Martin, Tucsnak and