The stream function and Gauss' principle of least constraint: Two useful concepts for structural geology

Abstract

To the extent that rock deformation can be approximated by a two-dimensional Newtonian model, a powerful stream-function simulation method is applicable. The significance of stream functions is that velocity, strain, stress and energy derived from the same stream function satisfy automatically three basic conditions of dynamics: 1. (1) the condition of continuity. 2. (2) the Navier-Stokes equations. 3. (3) conservation of energy. Hence we state with Jaeger: “If a stream function can be found which satisfies the boundary conditions of a dynamic model the complete solution follows.” All pertinent bits of dynamic information are implied in the stream function from which they can be directly derived, guaranteed—so to speak—not to violate the basic conditions of dynamics. Stream functions useful in structural geology are solutions of: ∇ 4ϵ= ∂ 4ψ ∂x 4 +2 ∂ 4ψ ∂x 2∂y 2 + ∂ 4ψ ∂y 4 =0 A double-polynomial solution of max. degree 14 is developed, in which the coefficients are related controlled by the ▽ 4 ψ = 0 constraint, and their absolute values are determined by the boundary conditions of specific models and by the condition of maximum rate of energy dissipation or maximum rate of decline of potential energy. The polynomial stream function is applied to a collapsing viscous “nappe” consisting of a thin basal layer with low viscosity on which a thicker layer with high viscosity slides due to gravitational spreading. The velocity of forward movement depends upon absolute and relative values of the following parameters: viscosity, thickness, the aspect ratio and density. The velocity of a variety of nappes with different thicknesses, aspect ratios, viscosities and densities is determined.