V(7) = matrix of radially dependent velocities V, = effective y-phase velocity when phase interactions are = matrix of effective velocities (Part I, Eq. 21) = matrix of effective velocities in the constant coefficient axial dispersion model ignored Greek Letters E = matrix of hold-up ratios { = normalised axial coordinate 7 = normalised radial coordinate h = Fourier transform variable p = viscosity p =density T = dimensionless time w = matrix of weighting functions for concentration averaging