The combined effects of conjugation and magnetic field on the heat transfer enhancement in a laminar liquid metal flow past a thermally conducting and sinusoidally oscillating infinite flat plate are investigated. The wall materials used are compatible with the liquid metals and are assumed to be of finite thickness. Analytical solutions are obtained for the velocity and the temperature distributions. The combined effects of thermal conductivity, the thickness of the plate, and the transverse magnetic field on the net heat flux transported are analyzed in detail and it is found that such effects are same as those on the transverse temperature gradient at any frequency. Due to oscillation, the heat flux is enhanced by O(10 3 ). The optimum value of wall thickness and the corresponding boundary layer thickness for which the maximum heat flux is obtained are reported. The heat flux transported can be increased by choosing a wall of low thermal conductivity. A maximum increase of 52.03% in heat flux can be achieved by optimizing the wall thickness. These information may be useful while designing magnetohydrodynamic liquid metal heat transfer systems. All the results obtained are in good agreement with the results reported earlier. C⃝

The combined effect of conjugation, external magnetic field and oscillation on the enhancement of heat transfer in the laminar flow of liquid metals between parallel plate channels is analyzed. In order to make our results useful to the design engineers, we have considered here only the wall materials that are widely employed in liquid metal heat exchangers. Indeed, all the results obtained through this mathematical investigation are in excellent agreement with the available experimental results. The effective thermal diffusivity κ e is increased by 3 · 10 6 times due to oscillation and that the heat flux as high as 1.5 · 10 10 (W/m 2 ) can be achieved. Based on our investigation, we have recommended the best choice of liquid metal heat carrier, wall material and its optimum thickness along with the optimum value of the frequency to maximize the heat transfer rate. At the optimum frequency, by choosing a wall of high thermal conductivity and optimum thickness, an increase of 19.98% in κ e can be achieved. Our results are directly relevant to the design of a heat transfer device known as electromagnetic dream pipe which is a very recent development.

A mathematical investigation on the combined effect of oscillation and conjugation on the enhancement of heat transfer in a heat pipe called Dream Pipe is carried out, when viscoelastic fluids (CPyCl/NaSal) are employed as the heat carriers. Closed-form solutions for the momentum and heat equations are presented. The physical and thermal properties of the polymer solution used are obtained by experiments. The effects of thermal conductivity and thickness of the wall, fluid thickness, Womersley number (α), Deborah number and Prandtl number on the enhancement of heat transfer are examined. Results obtained in the present analysis are in excellent agreement with those of the existing literature. The effective thermal diffusivity (κe) is maximized at optimum α where the fluid flow exhibits a resonant behavior. Several maxima occur in κe for several resonant frequencies and the dramatic increase in κe due to oscillation for the viscoelastic fluid is 5.63 x 10 9 times higher than that obtained by the molecular motion. This increase is much higher than that (1.84 x 10 4 times) obtained for the Newtonian fluid. κe is increased with increasing wall thermal conductivity and thickness in the viscous regime whereas in the elastic regime the effect of conjugation is saturated. In the viscous regime, a maximum increase of 50.63% in κe is obtained by optimizing the wall thickness. Also κe increases with increasing molar ratio of concentrations of counterion to surfactant. A maximum heat flux of 4.54 x 10 10 W/m 2 is achieved using a viscoelastic fluid with thermally conducting wall and this highest heat flux is 207 times higher than that (2.19 x 10 8 W/m2) obtained with the Newtonian fluid (liquid metal). Hence, viscoelastic fluids are preferable to liquid metals as working fluids in the Dream Pipe. The new insights gained by the present investigation are useful while designing viscoelastic Dream Pipes and micro channel heat exchangers.

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