Analytical solutions of the velocity around bodies oscillating harmonically with small amplitude in a stagnant fluid are derived from the boundary-layer theory. Steady secondary flows around a circular cylinder and a sphere are calculated and shown schematically.
Generalized expressions for the local and overall mass transfer from these bodies in steady, oscillating, and pulsating flows are derived on the basis of the equation of mass balance by using the solutions mentioned above and those reported previously for steady flow. The integrals included in these expressions being evaluated numerically, the local mass-transfer distributions around bodies a d the overall expressions are shown for each flow. These results are compared with the analytical and experimental ones reported previously, and shown to be rather satisfactory. The approximate expressions for pulsating flow are
Sh= [(0.615) ^1.74+ (0.728z^1/3) ^1.74] ^1/1.74 Sc^1/3 Re_p^1/2 (circular cylinder)
and Sh=2+ [(0.654) ^1.85+ (0.648 z^1/3) ^1.85] ^1/1.85 Sc^1/3 Re_p^1/2 (sphere),
where sh=2r₀k_f/D, z= (aω/U∞) ^3/2 (a/r₀) ^1/2, Sc=ν/D, and Re_p=2r₀U∞/ν (a, amplitude; D, di ffusivity; k_f, mass-transfer coefficient; ro, radius; U∞, free stream velocity; ν, kinematic viscosity; ω, angular frequency)
These expressions become ones for steady flow as z→0, and ones for oscillating flow as z→∞.