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A finite element analysis of a free surface drainage problem of two immiscible fluids
Author(s) -
Mori Masatake,
Natori Makoto,
GuoFeng Zhang
Publication year - 1989
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650090507
Subject(s) - free surface , discretization , laplace transform , finite element method , mathematics , partial differential equation , laplace's equation , boundary value problem , flow (mathematics) , mathematical analysis , ordinary differential equation , system of linear equations , free boundary problem , mechanics , differential equation , geometry , physics , thermodynamics
A sharp interface problem arising in the flow of two immiscible fluids, slag and molten metal in a blast furnace, is formulated using a two‐dimensional model and solved numerically. This problem is a transient two‐phase free or moving boundary problem, the slag surface and the slag–metal interface being the free boundaries. At each time step the hydraulic potential of each fluid satisfies the Laplace equation which is solved by the finite element method. The ordinary differential equations determining the motion of the free boundaries are treated using an implicit time‐stepping scheme. The systems of linear equations obtained by discretization of the Laplace equations and the equations of motion of the free boundaries are incorporated into a large system of linear equations. At each time step the hydraulic potential in the interior domain and its derivatives on the free boundaries are obtained simultaneously by solving this linear system of equations. In addition, this solution directly gives the shape of the free boundaries at the next time step. The implicit scheme mentioned above enables us to get the solution without handling normal derivatives, which results in a good numerical solution of the present problem. A numerical example that simulates the flow in a blast furnace is given.