Bounded and unbounded patterns of the Benney equation

Journal Article · Mon Jun 01 00:00:00 EDT 1992 · Physics of Fluids A; (United States)

DOI:https://doi.org/10.1063/1.858228· OSTI ID:7044591

Rosenau, P; Oron, A ^[1]; Hyman, J M ^[2]

---

1. Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000 (Israel)
2. Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)

The boundedness of 2-D liquid film flows on an inclined plane in the context of the regularized Benney, {ital u}{sub {tau}}+{lambda}{ital u}{sup 2}{ital u}{sub {ital x}}+(({mu}{ital u}{sup 6}{minus}{nu}{ital u}{sup 3}){ital u}{sub {ital x}}){sub {ital x}} +{sigma}{l brace}{ital u}{sup 3}({ital u}{sub {ital xx}}/(1+{epsilon}{sup 2}{ital u}{sub {ital x}}{sup 2}){sup 3/2}){sub {ital x}}{r brace}{sub {ital x}}=0, and the Benney ({epsilon}=0) equation are studied. Here {ital u}, {ital x}, {tau} are the rescaled film thickness, the longitudinal coordinate, and time, respectively; {lambda}, {mu}, and {nu} are non-negative constants determined at equilibrium; and {epsilon} is the parameter related to the film aspect ratio. For a vertical plane ({nu}=0) a critical curve {lambda}={lambda}{sub {ital c}}({mu}) has been found bifurcating from the point ({lambda},{mu})=(0,1) which divides the {lambda}-{mu} space into two domains. When {lambda}{gt}{lambda}{sub {ital c}}({mu}) the initial data evolves into modulating traveling waves similar to the solutions of the Kuramoto--Sivashinsky equation. However, when {lambda}{lt}{lambda}{sub {ital c}}({mu}), either an infinite spike forms in the solution in finite time and the original Benney model breaks down or the solution of the regularized Benney equation forms an infinite slope when the wavelike solution attempts to become multivalued. In a tilted plane ({nu}{gt}0) the boundedness of the emerging pattern is sensitive to the choice of initial data. It is also found that the Benney equation does not describe wave breaking where the solution develops an infinite slope.

• OSTI ID: 7044591
• Journal Information: Physics of Fluids A; (United States), Vol. 4:6; ISSN 0899-8213
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
FILM FLOW
NAVIER-STOKES EQUATIONS
ASPECT RATIO
LIQUID FLOW
REYNOLDS NUMBER
SURFACE TENSION
THICKNESS
WAVE EQUATIONS
DIFFERENTIAL EQUATIONS
DIMENSIONS
EQUATIONS
FLUID FLOW
PARTIAL DIFFERENTIAL EQUATIONS
SURFACE PROPERTIES
661300* - Other Aspects of Physical Science- (1992-)