Influence of Motion of Environment on Thermo-, Photo- and Diffusiophoresis of a Solid Aerosol Particle of the Spheroidal Form

In Stokes approach, the theoretical description of stationary motion of a large solid aerosol particle of the spheroidal form in external fields of temperature and concentration of gradients, on which powerful electromagnetic radiation in a binary gas mixture falls down, is carried out. At motion consideration it was supposed, that the average temperature of a surface of a particle slightly differs from the temperature of the gaseous environment surrounding it. In the course of the gasdynamics equations solution analytical expressions for force and speed of thermo -, photo-, and diffusion-phoresis taking into account influence of movement of environment are obtained.


Introduction
In a modern science and the technique, in areas of chemical technologies, hydrometeorology, preservations of the environment etc. multiphase mixes are widely applied. The disperse mixtures, consisting of two phases one of which is a particle and the other one is the viscous environment (gas or a liquid), hold the greatest interest today. Gas (liquid), with the particles suspended in it is called aerosol and the particles are called aerosol particles. Hydro -and aerosol particles can make considerable impact on a course of physical and physicalchemical processes of a various types in disperse systems (for example, processes of mass -and heat exchange). The size of particles of a disperse phase lies in a very wide limits: from macroscopic (∼ 500mkm) to molecular (∼ 10nm) values; a concentration of particles varies accordingly -from one particle to highly concentrated systems (> 10 10 −3 ). Nowadays, with development of nano-technologies and nano-materials, the essential prospect is represented by application of ultradisperse (nano-) particles, for example, in nano-electronics and nano-mechanics, etc.
The forces of the various nature can effect on the particles of disperse systems causing their ordered movement concerning the centre of inertia of the viscous environment. For example, the sedimentation occurs in the field of the gravitational force. In gaseous environments with non-uniform distribution of temperature there can be ordered movement of the particles which is caused, for example, by externally set gradients of temperature and concentration that is called thermophoresis and diffusiophoresis [1]. If movement is caused at the expense of internal sources of heat non-uniformly distributed in particle volume such movement is called photophoretic [2][3][4].
Average distance between aerosol particles of a considerable part of aerodisperse systems meeting in practice is much greater than a characteristic size of particles. In such systems the account of influence of an aerosol on development of physical process can be carried out, basing on knowledge of laws of dynamics of movement and heat -and mass exchange with infinite environment of separate aerosol particles. Mathematical modeling of evolution of aerosol systems and the solution of such important question as purposeful influence on aerosols is impossible without knowledge of laws of this type of behavior.
Many particles that can be found in industrial devices and nature, have the form of a surface different from spherical, for example, spheroidal (ellipsoid of rotation). Therefore studying of laws of motion of separate particles in gaseous (liquid) both homogeneous, and non-homogeneous environments is the important actual problem representing considerable theoretical and practical interest. means of external sources a small gradient of temperature ∇T and concentration ∇C 1 is sustained. Here C 1 + C 2 = 1, C 1 = n 1 /n g , C 2 = n 2 /n g , n g = n 1 + n 2 , ρ g = ρ 1 + ρ 2 , ρ 1 = m 1 n 1 , ρ 2 = m 2 n 2 , m 1 , n 1 and m 2 , n 2 -mass and concentration of the first and second components of the binary gas mixture. In the researches on the theory of thermo -, the photo -, and diffusiophoresis of large solid aerosol particles of the spheroidal form at small relative temperature drops, which were [5] published until the present time, the simultaneous influence of the environment movement (i.e. convective terms in the heat conductivity and diffusion equations) and surface heating on thermo -, photo-, and diffusiophoresis, and which represents theoretical and practical interest, was not considered. In the given article the estimation of this influence is evaluated.
At the theoretical description of the thermo -, photo-, and the diffusiophoretic particle movement process we will assume, that due to a small time of thermal and diffusion relaxation process, the thermal and mass transfer process in the system of a particle, the gaseous environment takes place quasi-stationary. The particle movement occurs at small Peclet and Reynolds numbers, and at small relative temperature drops in its vicinity, i.e. when (T s −T ∞ )/T ∞ 1, T s -is an average temperature of the particle surface, T ∞ -is the temperature of the gaseous environment far from it. At fulfillment of this condition it is possible to consider coefficients of heat conductivity, dynamic and kinematic viscosity as constant [13], and the gas as the continuous environment. The problem is solved by a hydrodynamic method, the equations of hydrodynamics with corresponding boundary conditions are solved and it is considered, that a phase transition is absent, and the particle is homogeneous over its composition.
Let's assume also, that at some moment of time the flat monochromatic wave of intensity I 0 falls on the particle. Energy of electromagnetic radiation, being absorbed in the particle volume, will be transformed to thermal energy. Non-uniformly in its volume is the local distribution of the heat sources arisen by this way, which can be described by some function q p called the volume density of internal sources of heat [12].
The description of thermo -, photo-, and diffusiophoretic particle motions we will carry out in spheroidal system of co-ordinates (ε, η, ϕ) with the beginning in the spheroid center, i.e. we chose the beginning of the motionless system of co-ordinates in an instant position of the center of the particle. Curvilinear co-ordinates ε, η, ϕ are connected with the Cartesian co-ordinates by the following relations [6]: x = c sinh ε sin η cos ϕ, y = c sinh ε sin η sin ϕ, where c = √ a 2 − b 2 in case of the oblate spheroid (a > b, formulae (1)), and c = √ b 2 − a 2 -in case of the prolate (a < b, formulae (2)); and b -semi-axes of the spheroid. At that the position of the Cartesian system of co-ordinates is fixed concerning the particle so that the axis coincides with the axis of symmetry of the spheroid.
Within the limits of the formulated assumptions the distribution of speed U g , pressure P g , temperatures T g and T p , and a relative concentration of the first component of the binary gas mixture C 1 are described by the following system of the equations [7]: The system of equations (3) was solved with the following boundary conditions [5]: here e η , e ε are unit vectors of spheroidal system of coordinates; U ε , U η -components of the mass speed of the gas U g , U = |U| -characteristic speed of the particle movement; λ g , λ p -coefficients of the heat conductivity of the gas and the particle respectively; ν g , µ g -kinematic and dynamic viscosity; H ε = c cosh 2 ε − sin 2 η -Lamé coefficient; K T S and K DS -coefficients of thermal and diffusion slips which are determined by methods of the kinetic theory of gases. For example, at accommodation coefficients of a tangential impulse and the energy, equal to unity, the gaskinetic coefficient (in case of a spherical particle) K T S ≈ 1, 152, K DS ≈ 0, 3 [1,8]. ε = ε 0 -the co-ordinate surface corresponding to a surface of the particle. 2.2. Distribution of temperature and relative concentration out of and in the particle. Let us make equations (3) and boundary conditions (4) -(6) dimensionless, having entered dimensionless temperature and speed as follows: In the problem except Reynolds's and the Peclet dimensionless numbers there are two more controllable small parameters ξ 1 = a|∇T g |/T ∞ 1, ξ 2 = a|∇C 1 | 1, characterizing relative temperature drop and concentration over the size of a particle. For pure thermophoresis the characteristic speed U is of the order of magnitude U ∼ (µ g /ρ g T ∞ )|∇T g |, and for pure diffusion phoresis -U ∼ D 12 |∇C 1 |. If we take this into account, the solution of the boundary problem (3) -(6) will be found in the form of expansion of corresponding physical sizes over powers of ξ 1 . The small parameter ξ 2 is expressed through ξ 1 , and also the Reynolds number (ξ = Re = ρ g U a/µ g 1) calculated by the characteristical speed of pure thermophoresis coincides with ξ 1 , which proves the expansion over the given small parameter.
Searching the force and the speed of thermo-, photoand diffusiophoresis, we will be limited by the corrections of the first order smallness over ξ 1 . In order to find them it is necessary to know the distribution of speed, pressure, temperatures and concentration in the spheroid vicinity. Substituting (7) in (3), leaving terms of the order ξ 1 , solving the obtained systems of the equations by the method of separation of variables, finally we obtain for zero and the first approximations t g0 (λ) = 1 + γλ 0 arcctgλ, where λ = sinh ε, λ 0 = sinh ε 0 , δ = λ g /λ p , γ = t s − 1 -dimensionless parameter characterizing heating of the spheroid; t s = T s /T ∞ , T s -average temperature of the spheroid surface defined by the formula Integration in the formulae (14) is carried out over the whole particle's volume.
x = cos η, P n (x) -Legendre polynomials [9]. The constants entering into expressions for fields of temperatures out of and in the particle (11), (12) are determined from corresponding boundary conditions on the spheroid surface. Considering, that later the expression for the coefficient is required, let us represent its explicit form Determination of the force and the speed of thermo-, photo-, and diffusiophoresis. A general solution of hydrodynamic equations in the spheroid coordinate system has a form [5]: Constants of integration A 1 , A 2 , are determined from the boundary conditions of the spheroid surface, in particular, Main force acting on the spheroid is determined by integration of the stress tensor over the surface of the aerosol particle [5,6] and has a form: Granting the explicit form of the coefficient A 2 , we obtain the general expression for the force acting on the spheroidal particle, which additively consists of the force of the environment viscous resistance F µ , thermophoretic force F th , photophoretic force F ph proportional to the dipole moment of thermal sources density, non-uniformly distributed in the volume of the particle, force considering the influence of the environment movement, and the diffusion phoretic force F dh 0 )arcctgλ 0 Equating the general force to zero, we obtain the general expression for the speed of the ordered movement of the oblate spheroidal particle in the external set field of the gradient of temperature and concentration [arcctg 2 λ 0 + +4(λ 0 arcctgλ 0 −1)+(λ 0 arcctgλ 0 −1) 2 ])−K DS D 12 |∇C 1 | (19) In order to obtain an expression for the force and the speed of thermo -, photo -, and diffusiophoresis of the prolate spheroid, it is necessary to replace λ for iλ in (18), (19) and c for -ic (i -imaginary unit).
Thus, formulas (18), (19) have the most the general character and allow to estimate the general force acting the solid aerosol particle of the spheroidal form and speed of its ordered movement in the external set field of the gradient of temperature and concentration when the non-uniformly distributed sources of heat act inside the particle. Thus, the influence of the environment movement is calculated at small relative temperature drops in its vicinity.

Analysis of obtained results
One can see from formulas (18), (19), that the movement of environment does not bring the contribution into the diffusion phoresis . Convective terms in the diffusion equation do not influence on the movement of a solid aerosol particle in the field of the concentration gradient. It can be of great importance in practical applications, for example, at the description of aerosol particles movement in the diffusion fields. In order the convective terms to influence on the diffusion phoresis it is necessary to change boundary conditions for the diffusion part. It can be made in variety of ways. For example, to consider the movement not of the solid particle, but of an evaporating drop.
Movement of the environment brings the contribution both in the thermophoresis, and in the photophoresis. This contribution is proportional to the product of Prandtl number and an average temperature of the particle surface. Prandtl number for the majority of gases is of the order of unity. As the problem was solved at small relative temperature drops the movement contribution can give no more than 10 -12 percent.
Consideration of special cases of movement of particles of the spheroidal form is of interest.
If not to consider the influence of the environment movement and internal sources of heat, then (19) transforms into expression for pure thermophoresis speed of the spheroidal particle.
In case of a sphere the formula (19) transforms in expression for thermo -, photo -, diffusiophoretic speed of a solid spherical particle of radius R, considering influence of the environment movement and internal sources of heat: Without influence of the environment movement and internal sources of heat we have the classical formula for the thermophoresis speed of a large spherical particle [10,11].
To estimate, what influence renders the environment movement on the speed of thermo-and photophoresis of a spheroidal particle, it is necessary to concretize the nature of thermal sources non-uniformly distributed in its volume. As an example we will consider the most simple case when the particle absorbs radiation as a black body, i.e. the radiation absorption occurs in a thin layer in the thickness δε ε 0 , adjoining to a heated up part of the particle surface. At that the density of thermal sources in a layer in the thickness δε is equal [12,13] to cosh ε cos η c(cosh 2 ε − sin 2 η)δε I 0 , π 2 ≤ η ≤ π, The expression for the speed of thermo-and photophoresis includes integrals V q p dV , V q p zdV . Substituting (23) in these integrals and accounting that δε ε 0 after integration one obtains: With accounting of (24), (25) the expression (19) has a form: 0 ∆ In case of the sphere the expression (26) has a form: To illustrate the contribution of the form-factor (the relation of semiaxes of the spheroid), the influence of the environment movement and an internal thermal release for the speed thermo-and photophoresis (26), in the picture curves connecting values f = f * t.ph /f * * t.ph | T∞=300K with the intensity of falling radiation for particles of boride graphite (λ p = 55W/(mK)) with spheroidal (a curve 1 -taking into account the environment movement, a curve 2 -without movement) and spherical (a curve 3) forms of the surfaces suspended in air at T ∞ = 300K and P g = 10 5 P a, for various relations of semiaxes of a spheroid are represented.
The numerical analysis has shown, that at the fixed relation of semi-axes with increase in intensity of the falling radiation the total contribution of the motion of environment and of internal thermal release leads to monotonous reduction of the thermo-, and photophoresis speed, and this reduction essentially depends on equatorial radius of a spheroid a.
Quantitative research of the discussed phenomenon for solid heated particles represents quite real experimental problem.