scholarly journals A Special Class of Univalent Functions in Hele-Shaw Flow Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Paula Curt ◽  
Denisa Fericean
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Viscous Fluid ◽  
Time Evolution ◽  
Special Class ◽  
Geometric Property ◽  
Univalent Functions ◽  
Property A ◽  
Flow Problems ◽  
Planar Flows

We study the time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection. Applying methods from the theory of univalent functions, we prove the invariance in time ofΦ-likeness property (a geometric property which includes starlikeness and spiral-likeness) for two basic cases: the inner problem and the outer problem. We study both zero and nonzero surface tension models. Certain particular cases are also presented.

Some Problems of Boundary Conditions in the Cavitation

2002 ◽  
Author(s):  
Alexey G. Terentiev
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Gravity Field ◽  
Constant Pressure ◽  
Calculated Data ◽  
Cavitating Flow ◽  
Capillary Surface ◽  
Flow Problems ◽  
Cavity Boundary

Most of cavitation problems are considered by standard conditions i.e. Neumann’s condition on the boundary, constant pressure on the cavity boundary and undisturbed flow in infinty. In this paper cavitating flow problems are considered by following conditions: a) evaporation from free boundary into the cavity, b) capillary surface tension on the boundary of the cavity, c) simulated gravity field in a flow, d) compressibility of the fluid. All these problems have been investigated analytically and obtained a number of calculated data.

scholarly journals On some invariant geometric properties in Hele-Shaw flows with small surface tension

2015 ◽  
Vol 31 (1) ◽  
pp. 53-60
Author(s):  
PAULA CURT ◽  
Keyword(s):  
Surface Tension ◽  
Geometric Property ◽  
Complex Analysis ◽  
Moving Boundary ◽  
Flow Problem ◽  
Unbounded Domains ◽  
Convexity Property ◽  
Bounded Domains ◽  
Small Surface ◽  
Planar Flows

In this paper, by applying methods from complex analysis, we analyse the time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection in the non-zero surface tension case. We study the invariance in time of α-convexity (for α ∈ [0, 1] this is a geometric property which provides a continuous passage from starlikeness to convexity) for bounded domains. In this case we show that the α-convexity property of the moving boundary in a Hele-Shaw flow problem with small surface tension is preserved in time for α ≤ 0. For unbounded domains (with bounded complement) we prove the invariance in time of convexity.

scholarly journals Steady rimming flows with surface tension

2008 ◽  
Vol 597 ◽  
pp. 91-118 ◽  
Author(s):  
E. S. BENILOV ◽  
M. S. BENILOV ◽  
N. KOPTEVA
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Viscous Fluid ◽  
Small Parameter ◽  
Outer Region ◽  
Lubrication Approximation ◽  
Steady Flows ◽  
Matched Asymptotics ◽  
Weak Surface ◽  
Rimming Flows

We examine steady flows of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. If the amount of fluid in the cylinder is sufficiently small, all of it is entrained by rotation and the film is distributed more or less evenly. For medium amounts, the fluid accumulates on the ‘rising’ side of the cylinder and, for large ones, pools at the cylinder's bottom. The paper examines rimming flows with a pool affected by weak surface tension. Using the lubrication approximation and the method of matched asymptotics, we find a solution describing the pool, the ‘outer’ region, and two transitional regions, one of which includes a variable (depending on the small parameter) number of asymptotic zones.

scholarly journals Free boundary problems in shock reflection/diffraction and related transonic flow problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140276 ◽  
Author(s):  
Gui-Qiang Chen ◽  
Mikhail Feldman
Keyword(s):  
Free Boundary ◽  
High Speed ◽  
Free Boundary Problems ◽  
Fluid Flows ◽  
Shock Reflection ◽  
Two Dimensional ◽  
Open Problems ◽  
Boundary Problems ◽  
Flow Problems ◽  
Concave Corner

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection–diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.

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