Thermophoresis of an aerosol spheroid along its axis of revolution

Abstract

The thermophoretic motion of a spheroidal particle freely suspended in a gaseous medium prescribed with a uniform temperature gradient along the axis of revolution of the particle is studied theoretically in the steady limit of small Peclet and Reynolds numbers. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the particle surface. The general solutions in prolate and oblate spheroidal coordinates can be expressed in infinite-series forms of separation of variables for the temperature distribution and of semiseparation of variables for the stream function. The jump/slip boundary conditions on the particle surface are applied to these general solutions to determine the unknown coefficients of the leading orders, which can be numerical results obtained from a boundary-collocation method or explicit formulas derived analytically. Numerical results for the thermophoretic velocity of the spheroidal particle are obtained in a broad range of its aspect ratio with good convergence behavior for various cases. For the axisymmetric thermophoresis of an aerosol spheroid, prolate, or oblate, with no temperature jump and frictional slip at its surface, our results agree excellently with the analytical solution obtained previously. The agreement between our results and the available numerical solutions obtained by using a singularity method is also very good. For most practical cases of a spheroid with a specified aspect ratio, the thermophoretic mobility of the particle is not a monotonic function of its relative jump/slip coefficients and thermal conductivity. The axisymmetric thermophoretic mobilities of a prolate spheroid and of an oblate spheroid with large aspect ratios can be much greater and smaller, respectively, than that of a sphere with the same equatorial radius.

Thermophoretic motion of slightly deformed aerosol spheres

“…The exact analytical solutions for the thermophoretic motion of a spheroid without temperature jump and frictional slip at the surface ðC Ã t ¼ C Ã m ¼ 0Þ in an arbitrary direction summarized in Eq. (53) and the numerical solutions for the axisymmetric thermophoresis (relevant to the axial mobility parameter m z ) of a spheroid obtained by Keh and Chang (2009) using a boundary collocation method are also given in these tables for comparison. It can be seen in Tables 1 and 2 that our asymptotic results obtained from Eq.…”

“…(48) or (56) for their expansions in powers of the small deformation parameter e up to the first and second orders as functions of Table 1 Leading-order asymptotic results of the thermophoretic mobility parameter mz calculated from Eq. (48a) in powers of the deformation parameter e for various values of the parameters k à , C à m and b/a of a spheroid with Exact solutions and numerical solutions are obtained from Keh and Ou (2004) and Keh and Chang (2009), respectively.…”

“…For a prolate spheroid with a fixed value of k à , one always has m z 414m x , whereas for an oblate spheroid the reverse is true. Keh and Chang (2009) are also shown in Fig. 4 for comparison.…”

Thermophoresis and Photophoresis of an Aerosol Cylinder with Thermal Stress Slip

“…The thermophoretic and photophoretic forces exerted on the particle obtained from these analyses were found to agree better with experimental data in comparison with those predicted by Brock (1962) and Mackowski (1989) excluding the effect of thermal stress slip. It is widely noticed that aerosol particles are generally non-spherical and exploration of effects of particle shape on thermophoresis is needed (Leong 1984;Williams 1986;Keh and Ou 2004;Keh and Chang 2009;Wang and Keh 2010;Chen and Keh 2014). Also, the photophoresis of a long circular cylinder, such as asbestos fiber or tobacco mosaic virus, with the thermal stress slip at the particle surface has been studied to some extent (Tzeng et al 2012).…”

Thermophoretic Motion of a Cylindrical Particle with Chemical Reactions

“…Most aerosol particles are nonspherical and thus it is interesting to explore the effect of particle shape on thermophoresis (Williams 1986;Keh and Ou 2004;Keh and Chang 2009;Keh 2009, 2010a,b). The thermophoretic velocity of a long circular cylindrical particle, such as tobacco mosaic virus or asbestos fiber, of radius a under a temperature gradient ∇T ∞ in the direction normal to its axis corresponding to Equation (1) for a spherical particle can be obtained as…”