Surface tension and reaction stresses of a linear incompressible second gradient fluid

Abstract

The paper starts with a detailed investigation of the boundary conditions at free and fixed boundaries of any second gradient material and clarifies whether a surface tension is to be expected. The classical approach to the reaction stresses of higher gradient materials leaves a vast indeterminacy in most boundary value problems. An advanced approach is presented that yields much more definite distributions of the reaction stresses and consequently also of the surface tension. It starts from a compressible fluid the stiffness and bulk viscosity of which tend to infinity. Furthermore, the complete set of restrictions on the material parameters of linear incompressible second gradient fluids is derived from the postulate of nonnegative dissipation. Finally, a stirring process with a free surface is studied as an example. The results are based on the numerical solution of the boundary value problems of three partial differential equations of second, fourth and sixth order, respectively.