Mathematical Formulation of Hele-Shaw Flows
Let, be the directions parallel to the flat plates, and the perpendicular direction, with being the gap between the plates (at ). When the gap between plates is asymptotically small
the velocity profile in the direction is parabolic (i.e. is a quadratic function of the coordinate in this direction). The equation relating the pressure gradient to the velocity is,
where is the velocity, is the local pressure, is the fluid viscosity.
This relation and the uniformity of the pressure in the narrow direction permits us to integrate the velocity with regard to and thus to consider an effective velocity field in only the two dimensions and . When substituting this equation into the continuity equation and integrating over we obtain the governing equation of Hele-Shaw flows,
This equation is supplemented by the no-penetration boundary conditions on the side walls of the geometry,
where is a unit vector perpendicular to the side wall.
Read more about this topic: Hele-Shaw Flow
Famous quotes containing the words mathematical, formulation and/or flows:
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)
“In necessary things, unity; in disputed things, liberty; in all things, charity.”
—Variously Ascribed.
The formulation was used as a motto by the English Nonconformist clergyman Richard Baxter (1615-1691)
“The blood of Abraham, God’s father of the chosen, still flows in the veins of Arab, Jew, and Christian, and too much of it has been spilled in grasping for the inheritance of the revered patriarch in the Middle East. The spilled blood in the Holy Land still cries out to God—an anguished cry for peace.”
—Jimmy Carter (James Earl Carter, Jr.)