Filling of a Circular Crack with Two Non Miscible Fluids

Abstract

Recent works (Bui, 1996) (Bui et al, 2000) on the modelling of a cracked solid, in which the crack is partly filled with a non-wetting fluid are at the origin of our work. This paper studies hydrostatic interaction of the fluid in a crack slowly progressively filled up with two non miscible fluids. We will consider a 3D penny shaped circular crack filled with two non miscible fluids, one of them wetting the solid. In such a situation, the wetting fluid is near the crack tip and the second at the center of the crack. The 3D penny shaped circular crack of radius a is in a linear elastic material. We denote ρ the distance between a crack point and the crack center and r the adimensional radius, such that ρ = a r. We suppose now that the crack is saturated with two incompressible non-miscible fluids. The first fluid is wetting the solid and has a volume V ¹. The second fluid has a volume V ². The first fluid is near the crack tip and has a uniform pressure P ₁; the second fluid is at the center of the crack and has a uniform pressure P ₂. Figure 1 represents the crack filled with the two fluids where τ is the surface tension of the interface between the two fluids and θ the contact angle of the second fluid with the solid. The boundary between the two fluids is a curved surface which is bounded to the crack lips at a dimensionless radius c. In the following analysis, we neglect the consequences of the line force at the triple point (fluid 1, fluid 2, solid). To simplify we suppose that there is no mechanical loading at infinity, then the only loading parameters are V ¹ and V ², the two volumes of the fluids inside the crack. The three mains unknowns are the pressures P ₁ and P ₂ and the dimensionless radius c.