Thermocapillary migration of a bidisperse suspension of bubbles

We consider the thermocapillary motion of a well-mixed suspension of non-conducting spherical bubbles of negligible viscosity in a viscous conducting liquid under conditions of vanishingly small Reynolds and Marangoni numbers. Recently, Acrivos, Jeffrey & Saville (1990) showed that when all the bubbles are of identical size, the ensemble-averaged migration velocity Ū₁ of a test bubble of radius a1 within the suspension equals U⁽⁰⁾₁[1-3/2c₁+O(c²₁)] , where c1 is the volume fraction of the bubbles and U1(0) is the thermocapillary velocity of single bubble given by Young, Goldstein & Block (1959). Here we extend this result to a bi-disperse suspension containing bubbles of radii a1 and a2 ≡ λa1 in which case Ū₁ = U⁽⁰⁾₁[1-3/2c₁ - S(λ)c₂+…] , where c1 and c2 are the corresponding volume fractions of the two sets of bubbles. Values for S(Λ) are presented for some typical size ratios Λ, and asymptotic expressions for S(Λ) are derived for Λ → 0 and for Λ → ∞.