Abstract
Chaotic advection can effectively enhance the heat transfer rate between a boundary and fluids with high Prandtl number. These fluids are usually highly viscous and thus turbulent agitation is not a viable solution since the energy required to mix the fluid would be prohibitive. Here, we analyze previously obtained results on chaotic advection and heat transfer in two similar 2-D periodic flows and on their corresponding 3-D periodic flows when an axial velocity component is superposed. The two flows studied are the flow between eccentric rotating cylinders and the flow between confocal ellipses. For both of these flows the analysis is simplified because the Stokes equations can be solved analytically to obtain a closed form solution. For both 2-D periodic flows, we show that chaotic heat transfer is enhanced by the displacement of the saddle point location during one period. Furthermore, the enhancement by chaotic advection in the elliptical geometry is approximately double that obtained in the cylindrical geometry because there are two saddle points instead of one. We also explain why, for high eccentricity ratios, there is no heat transfer enhancement in the cylindrical geometry. When an axial velocity component is added to both of these flows so that they become 3-D, previous work has shown that there is an optimum modulation frequency for which chaotic advection and heat transfer enhancement is a maximum. Here we show that the optimum modulation frequency can be derived from results without an axial flow. We also explain by physical arguments other previously unanswered questions in the published data.
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Abbreviations
- \(A\) :
-
Cross-sectional area for flow passage
- \(Pe\) :
-
Peclet number
- \(R_1, R_2\) :
-
Radii of inner and outer cylinders
- \(T\) :
-
Absolute temperature
- \(V_z\), \(\overline{V}_z\) :
-
Axial velocity, average axial velocity
- \({[}\alpha , \beta , z{]}\) :
-
Bipolar coordinate system
- \(\alpha _T\) :
-
Thermal diffusivity of the fluid
- \(\epsilon\) :
-
Eccentricity ratio
- \(\tau\) :
-
Residence time of fluid in mixer
- \(\psi\) :
-
Stream function
- \(\omega\) :
-
Modulation frequency
- \(\Omega\) :
-
Angular velocity
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Vinsard, G., Dufour, S., Saatdjian, E. et al. Chaotic advection and heat transfer in two similar 2-D periodic flows and in their corresponding 3-D periodic flows. Heat Mass Transfer 52, 521–530 (2016). https://doi.org/10.1007/s00231-015-1574-7
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DOI: https://doi.org/10.1007/s00231-015-1574-7