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Exact Analysis of Unsteady m.h.d. Convective Diffusion

N. Annapurna, A. S. Gupta

1979
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Proceedings of the Royal Society A
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The paper presents an ex act analysis of th e dispersion of a solute in an electrically conducting fluid flowing betw een tw o parallel plates in th e presence of a uniform transverse m agnetic field. Using a generalized dis persion model, which is valid for all tim e after th e injection of th e solute in th e flow we evaluate th e longitudinal dispersion coefficients as functions of tim e. F or small values of th e H artm an n num ber M , th e values of th e dispersion coefficients show rapid
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... fluctuations which decay w ith increase in M , and for m oderate and large values of M , these coefficients monotonically decrease w ith increase in M . In a non-conducting fluid, such rapid fluctuations are absent and th e dispersion coefficients for a pipe flow are less th a n th e corresponding values for a channel flow. I n t r o d u c t i o n The dispersion of a soluble m atter in an incompressible viscous fluid flowing in a circular pipe under laminar conditions was investigated by Taylor (1953) . He represented the unsteady convective diffusion in a steady flow by the dispersion where t is tim e, x is th e axial distance, Cm is th e area average concentration, u m is th e average velocity and k is the dispersion coefficient which depends param eters b u t n o t on t and x. However, T aylor's conceptual model is asy m p to ti cally valid for large t. Neglecting axial molecular diffusion, Lighthill (1966), on th e other hand, obtained an exact solution of th e unsteady convective diffusion equation, which is asym ptotically valid for small t. To be more specific, T aylor's dispersion model is valid if the tim e after th e injection of th e solute exceeds 0 {a is th e tube radius, and Dt h e molecular diffusivity) so th a t th e distanc across th e tu b e during this tim e is com parable w ith th e tu b e diam eter and diffusion acts to average out th e velocity of th e different particles of th e solute. L ighthill's model, on the other hand, takes account of th e initial action of diffusion on fro n t of th e concentration distribution and is valid for tim e less th a n ab o u t 0.1 a 2/D . Gill & Sankarasubram anian (1970) constructed a dispersion model for th e above steady [ 281 ]

doi:10.1098/rspa.1979.0088
fatcat:jyqspnstfrbztjpcrrkjpwb2ha