Determination of the average electro-thermophoretic force acting on a system of polarizable particles in an inhomogeneously heated fluid

The average force acting on the system of polarizing particles from the electric field in a non-uniformly heated dielectric liquid is determined. The case of pair interactions in the system is examined. To find the force acting on the particles, the interaction of two particles in a liquid is modelled in the presence of a given temperature gradient and the electric field strength far from the particles. The dependence of the particle permittivity on temperature is taken into account. The resulting expression for the force acting on two particles has such a power-law dependence on the distance between the particles, that allows to carry out the direct averaging procedure for a system of particles located in an infinite volume of liquid. When determining the average force, the probability density function of a continuous random variable is used, and the vector connecting the centers of particles plays the role of this variable. The differential equation for finding the probability density function is derived from two conditions. First, the pairs of particles are preserved in the space of all their possible configurations. Second, each pair of particles moves like a point with a speed equal to the speed of their relative motion. The resulting equation in the case under consideration has a set of solutions. Basing on the physical analysis of the problem, the choice of the probability density function is proposed, which allows one to determine the average electro-thermophoretic force acting in such a system with an accuracy up to the second degree of the volume concentration of particles.