Interaction of spheres in a viscoelastic fluid

Abstract

The interaction between two identical spheres of radiusa in a second-order fluid is studied, if the undisturbed flow is a general homogeneous flow. WithR the (instantaneous) distance between the sphere centers only the situationa/R ≪ 1 is considered. It turns out that it is not sufficient to know thea/R-term of the perturbation velocity, since certain contributions of the (a/R)2-terms are also needed. For two spheres sedimenting in a quiescent fluid a change of the relative position vector is predicted: the distance decreases and so does the orientation, i.e. the spheres tend to fall along their line of centers. If the motion of the individual sphere is restrained via a rigid connection (rigid dumbbell) this change of orientation implies that the dumbbell rotates until its axis is parallel to the direction of the applied force (stable orientation). In simple shear the “first-order” dumbbell (a/R-terms due to interaction) ultimately ends up in the plane normal to the gradient direction, independent of the rate of shear. This contrasts the behavior of a “second-order” dumbbell: if the symmetry axis lies in the plane of flow it will rotate around the vorticity axis at small rates of shear. Increasing the shear rate this dumbbell reaches a spinfree terminal state in which the angle between the symmetry axis and the flow direction is non-zero (although it is small). It is conjectured that for arbitrary initial orientations (not in the flow plane) the axis of the “second-order” dumbbell will not rotate in the Jeffrey orbits but rather show a systematic drift to become oriented parallel to the vorticity axis.