Meshless point collocation for the numerical solution of Navier–Stokes flow equations inside an evaporating sessile droplet

• Published in: Engineering analysis with boundary elements Vol. 36; no. 2; pp. 240 - 247
• Main Authors: Bourantas, G.C., Petsi, A.J., Skouras, E.D., Burganos, V.N.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01-02-2012; Elsevier
• Subjects: Computational methods in fluid dynamics; Droplet; Drops and bubbles; Evaporation; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Mathematics; Meshfree Point Collocation method; Methods of scientific computing (including symbolic computation, algebraic computation); Moving Least Squares method; Navier–Stokes equations; Nonhomogeneous flows; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, boundary value problems; Physics; Radial Basis Function method; Sciences and techniques of general use; Velocity-potential equation; Velocity–vorticity formulation; Velocity–vorticity formulation; Navier–Stokes equations; Meshfree Point Collocation method; Velocity-potential equation; Moving Least Squares method; Radial Basis Function method; Evaporation; Droplet; Gas liquid interface; Shear; Positivity; Boundary condition; Modeling; Model predictive control; Low Reynolds number; Navier Stokes equation; Continuity equation; Least squares method; Convergence rate; Collocation method; Mesh method; Regularity; Velocity-vorticity formulation; Numerical convergence; Mesh generation; Stokes flow; Computational fluid dynamics; Radial basis function; Boundary value problem; Linear system; Meshless method; Navier-Stokes equations
• ISSN: 0955-7997; 1873-197X
• DOI: 10.1016/j.enganabound.2011.07.019

The Navier–Stokes flow inside an evaporating sessile droplet is studied in the present paper, using sophisticated meshfree numerical methods for the computation of the flow field. This problem relates to numerous modern technological applications, and has attracted several analytical and numerical investigations that expanded our knowledge on the internal microflow during droplet evaporation. Two meshless point collocation methods are applied here to this problem and used for flow computations and for comparison with analytical and more traditional numerical solutions. Particular emphasis is placed on the implementation of the velocity-correction method within the meshless procedure, ensuring the continuity equation with increased precision. The Moving Least Squares (MLS) and the Radial Basis Function (RBF) approximations are employed for the construction of the shape functions, in conjunction with the general framework of the Point Collocation Method (MPC). An augmented linear system for imposing the coupled boundary conditions that apply at the liquid–gas interface, especially the zero shear-stress boundary condition at the interface, is presented. Computations are obtained for regular, Type-I embedded nodal distributions, stressing the positivity conditions that make the matrix of the system stable and convergent. Low Reynolds number (Stokes regime), and elevated Reynolds number (Navier–Stokes regime) conditions have been studied and the solutions are compared to those of analytical and traditional CFD methods. The meshless implementation has shown a relative ease of application, compared to traditional mesh-based methods, and high convergence rate and accuracy.
► The flow field inside an evaporating sessile droplet is computed with meshless numerical techniques. ► The Moving Least Squares and the Radial Basis Function approximations are employed. ► An augmented linear system to tackle the zero shear-stress condition at the interface is used. ► The results compare very satisfactorily with analytical and finite volume predictions.