![]() This depends on pH, and in our experimental setup, the distribution of the pH (H +) has a sigmoidal shape because an acidic source continuously releases H + ions into an alkaline solution (where the concentration of H + is low). We can use this approximation, because the surface tension is determined by the concentration of deprotonated 2-HDA. We assume that the surface tension at the liquid–air interface can be approximated by the following functional form (Figure 2a) (6)where a and x 0 are constants. The numerical solutions to the nonlinear problem (eqs 4 and 5) are found by using a simple second-order, finite-difference discretization on a structured, equidistant grid which is combined with the Newton–Raphson iteration in order to find the solution the biharmonic term, ∇ 4ψ, in eq 5 is discretized using the 13-point stencil, and the Laplacian in eq 5 is discretized using the standard 9-point stencil. In the stream-function representation, the boundary conditions become (4)We look for a fully developed steady solution of eq 3 which satisfies (5)with the same boundary conditions as in eq 4. By taking advantage of the 2D approximation, we can represent eq 1 as (3)where the scalar stream function, ψ, is related to the velocity via u = (∂ yψ, −∂ xψ), ∂(∇ 2ψ, ψ)/∂( x, y) ≡ ∂ x∇ 2ψ∂ yψ – ∂ y∇ 2ψ∂ xψ with ∂ x ≡ ∂/∂ x, ∂ y ≡ ∂/∂ y. The Marangoni-type flow considered here is driven by the nonuniform surface tension on the top (liquid–air) boundary (assumed rigid) which leads to the following constraints on the tangential stresses (23) (2)where γ̂ is the dimensionless surface tension and the boundary is decomposed as ∂Ω ≡ ∂Ω T ∪ ∂Ω̃, where ∂Ω T denotes the top surface and ∂Ω̃ = ∂Ω/∂Ω T the remaining boundary conditions are no-slip, i.e., u( x, y) = 0 on ∂Ω̃. The Reynolds number can be interpreted as a measure of the importance of advection against the diffusion of the (linear) momentum of the fluid in the dynamics. The fluid flow within the domain is described by the nondimensionalized Navier–Stokes equation (22) (1)where u = ( u x( x, y), u y( x, y)) is the 2D velocity field and Re = UL/ν is the Reynolds number with a characteristic horizontal length L and velocity U and a dynamic viscosity ν. In order to model this phenomenon, namely, the fluid flow in a 2D cross section in a channel, we considered the fully developed, steady flow within the 2D rectangular domain Ω ⊂ 2 driven by a spatially nonuniform surface tension, γ, at the liquid–air interface. (10, 11) In the latter example, it has been shown that a small fatty acid droplet suspended at the liquid–air interface in an alkaline solution can solve the maze and find the shortest path if a pH gradient is established in the maze. They rely either solely on maze topology (examples include applying a pressure difference in fluid through the maze, (3) processing the path of propagating chemical waves (4) or plasma, (5) and using a network of memristors (6) and the reconfiguration of an organism between two food sources within the maze (7-9)) or on a path-finding entity (a solid hydrogel in a microfluidic network or a droplet suspended at the liquid–air interface) that can react to environmental stimuli. (1, 2) However, there are other possibilities such as analog computational methods that have been successfully used to solve these problems. These problems can be solved using inherently slow computational methods where the solving time, t, scales with the size of the system, n, raised to some power x, i.e., t ≈ n x. Efficient maze solving, i.e., finding either the shortest path or all possible paths between two points in a maze, is a challenging mathematical problem especially if computational constraints are imposed.
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