Symbolic computation of the phoretic acceleration of convex particles suspended in a non-uniform gas

A package has been developed for calculating analytic expressions for forces and torques onto an arbitrarily shaped convex tracer (aerosol) particle small compared to the mean free path of the surrounding nonequilibrium gas. The package Phoretic allows to compute analytical (and also numerical) expressions for forces and torques stemming from elastic and diffusive scattering processes parameterized by an accommodation coefficient. The method is based on calculating half-sphere integral tensors of arbitrary rank and on integrating forces and torques acting on surface elements. The surrounding gas is completely specified by an arbitrarily shaped velocity distribution function. Accordingly, Phoretic requires two inputs: A particle (surface) geometry and a velocity distribution function. For example, the particle may be a cylinder with flat end caps, and the distribution function the one of Maxwell (isotropic) or Grad (13th moment approximation). The package reproduces analytic results for spheres which were available in the literature, and the ones for other geometries (cylinders, cuboids, ellipsoids) which were, however, only partially available (some works considered only elastic collisions, others temperature, or pressure, or only velocity gradients, etc.). In addition, Phoretic takes into account angular velocities which have been usually neglected and become relevant for non-spherical particles. The package is geared towards the implementation of dynamical equations for aerosol particles suspended in dilute or semidilute gases and as such helps to obtain concentration profiles and mobilities of aerosol particles depending on their shape (distribution) and environmental conditions.Program summaryTitle of program:PhoreticCatalogue identifier:ADYI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYI_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: Persons requesting the program must sign the standard CPC-non-profit use license (see license agreement printed in every issue)Computer for which the program is designed and others on which it has been tested: All platforms with a monitorOperating systems or monitors under which the program has been tested: Linux, Windows XP, Unix, Mac-OSProgram language used: Mathematica®, version 5.2 or later. Phoretic makes use of the DiscreteMath‘Combinatorica’ Mathematica® packageMemory required to execute with typical data: 10 MByteNo. of lines in distributed program, including test data, etc.: 22 410No. of bytes in distributed program, including test data, etc.: 114 657Distribution format:tar.gzNature of physical problem:   Starting from a non-uniform velocity distribution function of a gas in terms of its moments, i.e. field variables, and field gradients such as temperature, pressure, or velocity field, the problem is to analytically calculate forces and torques acting onto arbitrarily shaped convex tracer (aerosol) particles small in size compared to the mean free path of the gas. The collision process is modeled as a superposition of elastic and diffusive scattering processes (parameterized by 0⩽α⩽10⩽α⩽1).Method of solution: We implemented the solution to this problem in the symbolic programming language Mathematica®. The program allows to specify an arbitrary shape of the tracer particle and an arbitrary distribution function of the gas and returns symbolic or numerical expressions for forces and torques. The solution requires the calculation of half-sphere and base surface integrals and subsequent symbolic algebraic and tensorial manipulations.Restrictions on the complexity of the problem: Not known. In case the software cannot calculate surface integrals analytically it offers the possibility to proceed with a numerical evaluation of the corresponding terms.Typical running time: Typical running times mostly depend on the shape of the tracer particle. For all examples coming together with the software distribution run times are below 5 minutes on a modern single-processor platform.