Mass Conserving Algorithm for Solving a Solute Advection Diffusion Equation Inside an Evaporating Droplet

A method for simulating and analyzing the evaporation of a solution comprising the steps of:

• (a) discretizing a simulation space as a set of nodes, each particular node associated with a particular cell in the simulation space, wherein each particular node is representative of a particular point in the simulation space;
• (b) identifying a plurality of nodes in the simulation space at a first point in time as environment nodes, wherein the environment nodes are representative of points in the simulation space in which the solution is absent, and the plurality of environment nodes as a group are representative of the environment in which solution is being simulated in;
• (c) identifying a plurality of nodes in the simulation space at the first point in time as droplet nodes, wherein the droplet nodes are representative of points in the simulation space in which the solution is present, and the plurality of droplet nodes as a group are representative of the solution;
• (d) indentifying an interface at the first point in time between the droplet nodes and the environment nodes, wherein the interface represents a surface separating the solution from the environment;
• (e) determining a concentration of solute in the solution at the plurality of droplet nodes at the first point in time;
• and
• (f) determining the concentration of the solute at a plurality of nodes at a second point in time based upon the advection and diffusion of the solution comprises;
• (f.1) identifying a first type of extended node, wherein the first type of extended node is an environment node and is a neighbor of a first type of test node, wherein the first type of test node is a droplet node, only one of the neighbors of the first type of test node is an environment node, and all other neighbors of the first type of test node are droplet nodes;
• (f.2) calculating an extended concentration of solute in the solution at the first type of extended node based upon the concentration of solute at the first type of test node, the concentration of the solute at the droplet nodes neighboring the first type of test node, and the flux across a portion of the interface in a region of simulation space bounded by a cell associated with the first type of test node and a cell associated with the first type of extended node;
• (f.3) identifying a second type of extended node, wherein the second type of extended node is an environment node and half of the second type of extended node's neighboring nodes are droplet nodes, droplet nodes neighboring the second type of extended node are identified as a set of a second type of test nodes associated with the second type of extended node, and a droplet node neighboring at least two nodes in the set the second type of test nodes associated with the second type of extended node is identified as belonging to the set of the second type of test nodes associated with the second type of extended nodes.
• (f.4) calculating an extended concentration of solute in the solution at the second type of extended node based upon the concentration of solute at the set of second type of test nodes associated with the second type of extended node, and the flux across the interface in a region of simulation space bounded by cells associated with the set of second type of test nodes associated with the second type of extended node and a cell associated with the second type of extended node;
• and
• (f.5) calculating the diffusion of the concentration at the droplet nodes at a second point in time based upon the concentration of solute at the droplet nodes and the extended concentration of solute at the first and second type of extended nodes.