Heat transfer by laminar Hartmann flow in thermal entrance region with a step change in wall temperatures: the Graetz problem extended

Abstract Thermally developing laminar Hartmann flow through a parallel-plate channel, including both viscous dissipation, Joule heating and axial heat conduction with a step change in wall temperatures, has been studied analytically. Expressions for the developing temperature and local Nusselt number in the entrance region are obtained in terms of Peclet number Pe, Hartmann number M, Brinkman number Br, under electrically insulating wall conditions, χ=−1 and perfectly conducting wall conditions, χ=0. The associated eigenvalue problem is solved by obtaining explicit forms of eigenfunctions and related expansion coefficients. We show that the nonorthogonal eigenfunctions correspond to Mathieu's functions. We propose a new asymptotic solution for the modified Mathieu's differential equation. The asymptotic eigenfunctions for large eigenvalues are also obtained in terms of Pe and M. Results show that the heat transfer characteristics in the entrance region are strongly influenced by Pe, M, Br and χ.

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