Bernoulli Problem Research Articles

Wicking of liquids into sagged fabrics

The classical Bernoulli problem of a freely hanging fabric sagged between two posts is used for the analysis of wicking phenomena. We show that wicking of a wetting liquid into a Bernoulli catenary is an instructive nontrivial experiment illustrating an unusual coupling between mechanical and capillary forces. When the liquid wicks into the material, it causes the catenary to sway back and forth. We studied theoretically and experimentally the kinetics of wicking into sagged nonwoven materials. The proposed experiment can be used for the analysis of transport and tensile properties of thin porous films.

On a Bernoulli problem with geometric constraints

A Bernoulli free boundary problem with geometrical constraints is studied. The domain $\Om$ is constrained to lie in the half space determined by $x_1\geq 0$ and its boundary to contain a segment of the hyperplane $\{x_1=0\}$ where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

An isoperimetric inequality related to a Bernoulli problem

Given a bounded domain Ω we look at the minimal parameter Λ(Ω) for which a Bernoulli free boundary value problem for the p-Laplacian has a solution minimising an energy functional. We show that amongst all domains of equal volume Λ(Ω) is minimal for the ball. Moreover, we show that the inequality is sharp with essentially only the ball minimising Λ(Ω). This resolves a problem related to a question asked in Flucher et al. (Reine Angew Math 486:165–204, 1997).

On the convexity of some free boundaries

We demonstrate that a method of Colesanti and Salani, which compares solutions of elliptic differential equations to their quasiconcave envelopes, can be extended to derive convexity of free boundaries. As examples we present the so-called dam problem, a free boundary problem modelling pollution and a Bernoulli problem. Moreover, we prove strict convexity of the wet region in the dam problem in arbitrary dimensions.

Minimal advective travel time along arbitrary streamlines of porous media flows: The Fermat–Leibnitz–Bernoulli problem revisited

Travel time of marked fluid particles along arbitrary streamlines in arbitrary porous streamtubes is estimated from below based on the Cauchy–Bunyakovskii (Schwartz) and Jensen inequalities. In homogeneous media the estimate is strict and expressed through the length of the streamline, hydraulic conductivity, porosity and the head fall. The minimum is attained at streamlines of unidirectional flow. The bounds for heterogeneous soils, non-Darcian flows and unsaturated media are also written. If such bounds are attained the corresponding trajectories become brachistochrones . For example, in a two-layered aquifer and seepage perpendicular to the layers there is a unique conductivity–porosity ratio which makes a broken streamline brachistocronic . Similarly, if conductivities of two layers are fixed there is a unique incident angle between flow in one medium and the interface which makes a refracted streamline brachistocronic .

Some geometric and analytic properties of solutions of Bernoulli free-boundary problems

A Bernoulli free-boundary problem is one of finding domains in the plane on which a harmonic function simultaneously satisfies linear homogeneous Dirichlet and inhomogeneous Neumann boundary conditions. For a general class of Bernoulli problems, we prove that any free boundary, possibly with many singularities, is necessarily the graph of a function. Also investigated are convexity and monotonicity properties of free boundaries. In addition, we obtain some optimal estimates on the gradient of the harmonic function in question.

On the curvature of free boundaries with a Bernoulli-type condition

In this paper we study the classical external Bernoulli problem set in an annular domain Ω of the plane. We focus on the curvature of the free boundary Γ (outer component of the boundary of our domain) and establish a one-to-one correspondence between positive/negative curvature arcs of Γ and of the curve γ representing the data, extending a method put forward by A. Acker. Moreover we show that the positive curvature arcs on the free boundary bend less than the corresponding arcs on the inner curve, i.e. the maximum attained by the curvature is greater on γ than on Γ . Thus we can draw the following conclusions: the geometry of Γ is simpler than that of γ (an already known result); the shape of Γ is alleviated with respect to that of γ .

Fast Numerical Methods for Bernoulli Free Boundary Problems

The numerical solution of the free boundary Bernoulli problem is addressed. An iterative method based on a level-set formulation and boundary element method is proposed. Issues related to the implementation, the accuracy, and the generality of the method are discussed. The efficiency of the approach is illustrated by numerical results.

Singularities of Bernoulli Free Boundaries

In this article we show that, for a large class of Bernoulli problems, a free boundary which is symmetric with respect to a vertical line through an isolated singular point must necessarily have a corner at that point, and we give a formula for the contained angle. The assumptions used admit the possibility of other singular points, even uncountably many, on the free boundary. This result is an extension of the first Stokes conjecture in the theory of hydrodynamic waves. We also show that, even in the presence of singularities, a geometrically simple Bernoulli free boundary is necessarily symmetric.

Variational approach to shape derivatives for a class of Bernoulli problems

The shape derivative of a functional related to a Bernoulli problem is derived without using the shape derivative of the state. The gradient information is combined with level set ideas in a steepest descent algorithm. Numerical examples show the feasibility of the approach.

Numerical solution of the free boundary Bernoulli problem using a level set formulation

We present a numerical method based on a level set formulation to solve the Bernoulli problem. The formulation uses time as a parameter of boundary evolution. The level set formulation enables to consider nonconnected domains. Numerical experiments show the efficiency of the method if boundary conditions are handled accurately. In particular, the case of multiple solutions is treated.

The nonstationary Bernoulli problem with fixed boundary

We consider a two dimensional free boundary problem for an inviscid, incompressible and irrotational fluid. Imbedded in the fluid is a gas bubble of prescribed shape. Given some initial configuration we investigate the long time evolution of the potential flow. We prove existence, uniqueness and regularity of a solution.

Exact Controllability of the Euler–Bernoulli Equation with Controls in the Dirichlet and Neumann Boundary Conditions: A Nonconservative Case

This paper considers the Euler–Bernoulli problem (1.1a–d) with boundary controls $g_1 $, $g_2 $ in the Dirichiet and Neumann boundary conditions, respectively. Several exact controllability results are shown, including the following. Problem (1.1a–d) is exactly controllable in an arbitrarily short time $T > 0$ in the space (of maximal regularity) $H^{ - 1} (\Omega ) \times V'$, V as in (1.4), (i) with boundary controls $g_1 \in L^2 (\Sigma )$, $g_2 \equiv 0$ under some geometrical conditions on $\Omega $; (ii) with boundary controls $g_1 \in L^2 (\Sigma )$ and $g_2 \in L^2 (0,T;H^{ - 1} (\Gamma ))$ without geometrical conditions on $\Omega $. A direct approach is given, based on an operator model for problem (1.la–d) and on multiplier techniques. An additional difficulty of the particular boundary conditions is due to the fact that, in the natural norms for the solution arising in the application of multiplier techniques, the corresponding homogeneous problem is not conservative. This difficulty is overco...

The stochastic geometry of invasion percolation

Invasion percolation, a recently introduced stochastic growth model, is analyzed and compared to the critical behavior of standardd-dimensional Bernoulli percolation. Various functions which measure the distribution of values accepted into the dynamically growing invaded region are studied. The empirical distribution of values accepted is shown to be asymptotically unity above the half-space threshold and linear below the point at which the expected cluster size diverges for the associated Bernoulli problem. An acceptance profile is defined and shown to have corresponding behavior. Quantities related to the geometry of the invaded region are studied, including the surface to volume ratio and the volume fraction. The former is shown to have upper and lower bounds in terms of the above defined critical points, and the latter is bounded above by the probability of connection to infinity at the half-space threshold. Provided that the critical regimes of Bernoulli percolation possess their anticipated properties, as is known to be the case in two dimensions, these results verify numerical predictions on the acceptance profile, establish the existence of a sharp surface to volume ratio and show that the invaded region has zero volume fraction. Large-time asymptotics are analyzed in terms of the probability that the invaded region accepts a value greater thanx at timen. This quantity is shown to be bounded below byh(x)exp[−c(x)n (d-1)/d ] forx above threshold, and to have an upper bound of the same form forx larger than a particular value (which coincides with the threshold ind=2). For two dimensions, it is also established that the infinite-time invaded region is essentially independent of initial conditions.

Finite memory multiple hypothesis testing: Close-to-optimal schemes for Bernoulli problems

The design of optimal, time-invariant, randomized, finite-state automata for K -hypothesis testing is an open problem for K > 2 . A lower bound is constructed on the smallest probability of error achievable by m -state automata solving a three-hypothesis Bernoulli problem. A class of close-to-optimal automata is exhibited that requires at most one extra bit of memory to match the performnnce of an optimal automaton.