Universal Shapes and Bifurcation for Rotating Incompressible Fluid Drops

Axisymmetric shapes of Beer, Rosenthal, D. Ross, Gulliver, and others provide universal solutions for a problem related to Plateau's rotating drops, for the material class of Noll's simple incompressible fluid subject to Laplace's frequently used linear constitutive relation between pressure jump and mean curvature at the free boundary. A bifurcation from spheroidal to toroidal shapes occurs within this family, leading initially to a decrease in angular momentum. It is shown that the spheroidal and toroidal axisymmetric families connect tangentially in the angular speed-angular momentum plane, and the corresponding bifurcation slope is calculated. It is also shown for a general (possibly nonaxisymmetric) rigidly rotating drop without body force that the center of mass of the drop is necessarily on the axis of rotation, complementing a result of Wente. 1. Preliminary results The steady rigid rotation of a homogeneous incompressible fluid drop which is surrounded by a rigidly rotating incompressible fluid is considered; see Ross and Smith [12] for a discussion of relevant experiments of Plateau [7], Wang, Trinh, Croonquist, and Elleman [16], and others. An interfacial surface separates the drop from the surrounding fluid, and interfacial surface tension balances the pressure gradients due to the rotational acceleration. The rotating drop occupies a domain Vi = Vi(t) at time t and the surrounding fluid occupies an outer domain V2 = ^(t) enclosing Vi. The fluids are incompressible homogeneous simple fluids of respective mass densities pi and p2, and they undergo separate steady rigid rotations about a common axis with respective angular rotation speeds UJI and UJ2. (A simple fluid is a simple material (in the sense of Noll) that has maximal symmetry, with symmetry group consisting of the entire unimodular group; cf. Smith [13].) A steady rigid rotation about an axis takes initial points X = (Xi,X2,X3) into points x = (xi,X2,X3) at time t with x = x(X,t) = F(t)X, F(t) = where UJ is the angular speed and Cartesian coordinates are used with the xs-axis taken to coincide with the axis of rotation, and the origin is placed on the axis of rotation. For brevity, the notations x and X are used for points and also for the corresponding vectors that translate the origin to the given points. A routine calculation based on the balance of linear momentum shows that any such steady rigid rotation with constant coswt smu>t 0 sin ut coswt 0 0 0 1 Received June 21, 1993. 1991 Mathematics Subject Classification: 34A47.