Orientation of Rigid Bodies Freefalling in Newtonian and Non-Newtonian Fluids
This thesis deals with the subject of terminal orientations of rigid bodies, sedimenting in Newtonian and non-Newtonian liquids. It is a well established fact that homogeneous bodies of revolution around an axis ('a') with fore-aft symmetry will orient themselves with respect to the direction of gravity ('g') depending upon their shape and upon the nature of the fluid in which they are immersed. If, for instance, we are considering an ellipsoidal object falling in a Newtonian fluid such as water, then the body falls with 'a' eventually becoming perpendicular to the direction of 'g'. However, if the same body falls in a viscoelastic fluid where the inertial effects can be disregarded, then 'a' will eventually become parallel to 'g'. It has also been noted that long bodies falling in fluids with certain polymeric concentrations can take on angles between the horizontal and vertical orientations. These intermediate angles are referred to as tilt angles. The objective of this thesis is the explanation of this orientation phenomenon in different liquids.
Our approach to the problem has been three-fold, experimental, mathematical and also numerical. We perform several experiments on sedimentation of particles in a variety of viscoelastic and Newtonian liquids to verify and fill gaps in the previous observations. A second set of experiments that we perform involves a modified flow chamber setup where the particle is fixed at the center of the chamber while water flows past it. We are able able to replicate previous experiments at low and intermediate Re, with both these experiments.
The equations to describe the problem of freefall of a rigid body of arbitrary shape, in a liquid, are obtained from a frame attached to the body and is formulated for any general fluid model. In addition, we also obtain the equations for the body, since the problem we are dealing with is one of fluid-structure interaction. We establish well-posedness of the equations by showing the exitence and uniqueness of steady solutions to the problem of sedimentation in a Second order fluid, with Re=0 and arbitrary material parameters using the Banach fixed point theorem.
In order to explain the terminal orientation assumed by the body, we consider the effect of torques imposed by different components of the liquid such as inertia, viscoelasticiy and shear-thinning. The equilibrium resulting from the competition of the different torques should reveal the terminal angle. Guided by the fact that the orientation phnomenon is observed at very small Re and We, we formulate the torque equations at first order in these material parameters. The calculation is performed for four different liquid models, Newtownian, Power-law, Second order fluid and a modified Second order model which we introduce here for the first time. The different orientation observations seen in experiments is well explained by these models. Finally, a simple quasi-steady stability argument is used to establish stability of the equilibrium states. For this final argument, we numerically evaluate the torque imposed by the individual components of the liquid upon a sedimenting prolate ellipsoid in an unbounded three dimensional fluid domain surrounding the body.
Advisor:Dr. Walter Goldburg; Dr. Anne M. Robertson; Dr. Peyman Givi; Dr. Giovanni P. Galdi
School:University of Pittsburgh
School Location:USA - Pennsylvania
Source Type:Master's Thesis
Date of Publication:06/09/2004