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Flow visualization of natural convection of magnetic fluid in a rectangular Hele-Shaw cell
Journal of Magnetism and Magnetic Materials 252 (2002) 206–208
Flow visualization of natural convection of magnetic fluid in a rectangular Hele-Shaw cell C.-Y. Wen*, C.-Y. Chen, S.-F. Yang Department of Mechanical Engineering, Da-Yeh University, Chang-Hwa 51505, Taiwan
Abstract The nature convection of a magnetic fluid in a Hele-Shaw cell with aspect ratio of one is studied experimentally. Results obtained from heat transfer measurements and shadowgraphs revealed that the vertically imposed magnetic field has a destabilizing influence. The flow instability mode becomes different from that without the magnetic field. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Magnetic fluids; Natural convection; Flow instability; Hele-Shaw cell; Shadowgraph
1. Introduction and governing equations Magnetic fluids with both magnetic and flow properties have been demonstrated with many promising applications in thermo-fluid engineering [1]. The heat transfer characteristics associated with the hydrodynamic instability of a magnetic fluid has then been a subject of interest to engineers as well as scientists for a great many years [2–4]. In the early experimental and numerical investigations of Yamaguchi et al. [3,4], the magnetic fluid heat transfer characteristics and flow behavior of natural convection in a two-dimensional rectangular cavity with an imposition of an even vertical magnetic field was studied. The measured heat transfer rates showed that the vertically imposed magnetic field has a destabilizing influence, and at the super critical state the flow mode becomes substantially different from that with no magnetic field. A numerical analysis based on the finite difference method was also carried out to verify the experimental heat transfer results and to depict the relevant flow-fields associated with the experimental conditions. The difficulties of visualizing the magnetic flow fields experimentally are due to the
*Corresponding author. Tel.: +886-4-852-8469x2111; fax: +886-4-852-6301. E-mail address: [email protected] (C.-Y. Wen).
opaqueness and selectivity of wavelength of the light source of the magnetic fluids. To visualize the magnetic flow-fields, in the present paper, the natural convection of a magnetic fluid in a bottom-heated square Hele-Shaw cell with two insulated side walls is studied with the shadowgraphy. Heat transfer measurements with and without the externally imposed vertical magnetic field are also presented. The Hele-Shaw cell flow of the natural convection of the magnetic fluid is governed by the following equations [3–5]: r u ¼ 0;
ð1Þ
12Z u þ r0 ½1 þ aðT T0 Þ g h2 þ m0 ðM rÞH; DT qM DH þ m0 T ¼ lr2 T; r0 C Dt qT V ;H Dt rp ¼
r B ¼ 0;
B ¼ m0 ðM þ HÞ;
ð2Þ ð3Þ ð4Þ
where Z is the viscosity, r0 the density, a the volumetric expansion coefficient, m0 the permeability in vacuum, C the specific heat and l the thermal conductivity. Continuity equation (1) for incompressible fluid, energy equation (3), and Maxwell equations (4) for electrically
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 6 7 1 - 6
C.-Y. Wen et al. / Journal of Magnetism and Magnetic Materials 252 (2002) 206–208
nonconductive media are the same as those for the natural convection of a magnetic fluid in a twodimensional rectangular cavity [3,4]. Only momentum equation (2) is replaced by Darcy’s law [5]. Eqs. (1)–(4) can be then nondimensionalized with nondimensional parameters as follows: 1 h 2 r0 gabd 4 Ra ¼ ; kZ 12 d 1 h 2 m0 H0 Ms d 2 Ram ¼ ; 12 d kZ d qT ; ð5Þ Nu ¼ DT qz z¼0 where Ra is the Rayleigh number, Ram the magnetic Rayleigh number and Nu the Nusselt number with DT ¼ TH TC ; b ¼ ðTH TC Þ=d; d is the height of cell, k the thermal diffusivity, H0 the strength of external magnetic field and Ms the saturation magnetization. Ra and Ram in Eq. (5) differ from those defined in 1 Yamaguchi et al. [3,4] by the factor of 12 (h=d)2 which implies the very small thickness in the third dimension, h; in the case of Hele-Shaw cell flow has to come into play. The characteristics of flow instability in Hele-Shaw cell and two-dimensional rectangular cavity [3,4] may be different.
Fig. 1. Experimental setup of heat transfer measurements.
207
2. Experimental apparatus The dimensions of the Hele-Shaw cell are 15
15 0.3 mm3. The magnetic fluid used in the current experiment is the water-based Fe3O4 magnetic fluid with r0 ¼ 1070 kg/m3, Z ¼ 3:75 103 Pa s and Ms ¼ 5:69 emu/g [6]. The experimental apparatus of heat transfer measurements and magnetic field imposition shown in Fig. 1 is similar to that of Yamaguchi et al. [3,4]. The magnetic field was imposed to the cell by the electromagnet vertically. The bottom of the cell was heated through a copper sheet while the upper wall of the cell was cooled through another copper sheet whose upper surface was opened to the ambient air. Nu obtained from the experiment is the local Nu that can be calculated based on the mean heat flux to the cell and the local temperature gradient measured at the position of the temperature-measuring point in the cell by three thermistors. The traditional shadowgraphy [7] was used to visualize the flow fields of natural convection with a He–Ne laser (Melles Griot 05-LHP-991, l ¼ 632:8 nm) as the light source.
3. Results and discussion Figs. 2a and b show the flow fields at the onset of transition and at a higher Ra without the externally imposed vertical magnetic field, respectively. Fig. 2c shows the flow field with Ram ¼ 670 case (which corresponds to H0 ¼ 11:5 G). As seen in Figs. 2a and b, a plume rises at the center of the cell and then evolves into a pair of counter-rotating vortices. By imposing the magnetic field, a pair of symmetric counter-rotating vortices is also clearly observed. The flow features are similar to those corresponding to the second instability mode in the numerical simulations of Yamaguchi et al. [4]. Note that in the two-dimensional square cavity case studied by Yamaguchi et al. [4], one convection vortex is generated at the first mode, instead of a pair of vortices in the present Hele-Shaw cell case. In Fig. 3, the heat transfer characteristics, with and without the imposed magnetic field, are depicted as the
Fig. 2. Flow field obtained from shadowgraphy: (a) Ra ¼ 4:1; Ram ¼ 0; (b) Ra ¼ 31:9; Ram ¼ 0; (c) Ra ¼ 26:3; Ram ¼ 670:
208
C.-Y. Wen et al. / Journal of Magnetism and Magnetic Materials 252 (2002) 206–208 Table 1 Critical Rayleigh numbers at the first flow transition Magnetic Rayleigh number, Ram
Critical Rayleigh number, Rac
0 670
2.01 1.06
4. Conclusion
Fig. 3. Heat transfer characteristics.
Nu versus Ra and are fitted with dotted line and solid line, respectively. It is noted that the experimental results are taken for the steady state condition. As seen, by imposing the magnetic field the heat transfer data shift towards the left indicating the increase of the heat transfer rate at the supercritical state. The heat transfer measurements suggest that the magnetic field has a destabilizing hydrodynamic effect on the flow. These results show close similarity with those of Yamaguchi et al. [3,4]. The critical Rayleigh numbers, Rac ; are estimated by extrapolating the data by the following relation: NuBðRa Rac Þ1=2 and are listed in Table 1. The critical Rayleigh numbers in the present Hele-Shaw cell, with and without the imposed magnetic field, are different from those in the two-dimensional square cavity case [3,4]. As mentioned in Section 1, the very small thickness in the third dimension in the case of Hele-Shaw cell flow affects the momentum equation and has great influence on the characteristics of flow instability.
The macroscopic magnetic flow fields in the HeleShaw cell are visualized with shadowgraphy for the first time. A pair of symmetric counter-rotating vortices is observed for the first instability mode. The heat transfer measurements suggest that the magnetic field has a destabilizing hydrodynamic effect on the flow. The very small thickness in the present case shows great influence on the characteristics of flow instability.
References [1] B.M. Berkovsky, Magnetic Fluids Engineering Applications, Oxford University Press, New York, 1993, p. 214. [2] B.A. Finlayson, J. Fluid Mech. 40 (1970) 753. [3] H. Yamaguchi, I. Kobori, Y. Uehata, K. Shimada, J. Magn. Magn. Mater. 201 (1999) 264. [4] H. Yamaguchi, I. Kobori, Y. Uehata, J. Thermophys. Heat Transf. 13 (1999) 501. [5] C. Tan, G. Homsy, Phys. Fluids 30 (1987) 1239. [6] C.Y. Hong, C.H. Ho, H.E. Horng, C.H. Chen, S.Y. Yang, Y.P. Chiu, H.C. Yang, Magn. Gidro. 35 (1999) 364. [7] W. Merzkirch, Flow Visualization, Academic Press, Orlando, FL, 1987, p. 126.
Flow visualization of natural convection of magnetic fluid in a rectangular Hele-Shaw cell C.-Y. Wen*, C.-Y. Chen, S.-F. Yang Department of Mechanical Engineering, Da-Yeh University, Chang-Hwa 51505, Taiwan
Abstract The nature convection of a magnetic fluid in a Hele-Shaw cell with aspect ratio of one is studied experimentally. Results obtained from heat transfer measurements and shadowgraphs revealed that the vertically imposed magnetic field has a destabilizing influence. The flow instability mode becomes different from that without the magnetic field. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Magnetic fluids; Natural convection; Flow instability; Hele-Shaw cell; Shadowgraph
1. Introduction and governing equations Magnetic fluids with both magnetic and flow properties have been demonstrated with many promising applications in thermo-fluid engineering [1]. The heat transfer characteristics associated with the hydrodynamic instability of a magnetic fluid has then been a subject of interest to engineers as well as scientists for a great many years [2–4]. In the early experimental and numerical investigations of Yamaguchi et al. [3,4], the magnetic fluid heat transfer characteristics and flow behavior of natural convection in a two-dimensional rectangular cavity with an imposition of an even vertical magnetic field was studied. The measured heat transfer rates showed that the vertically imposed magnetic field has a destabilizing influence, and at the super critical state the flow mode becomes substantially different from that with no magnetic field. A numerical analysis based on the finite difference method was also carried out to verify the experimental heat transfer results and to depict the relevant flow-fields associated with the experimental conditions. The difficulties of visualizing the magnetic flow fields experimentally are due to the
*Corresponding author. Tel.: +886-4-852-8469x2111; fax: +886-4-852-6301. E-mail address: [email protected] (C.-Y. Wen).
opaqueness and selectivity of wavelength of the light source of the magnetic fluids. To visualize the magnetic flow-fields, in the present paper, the natural convection of a magnetic fluid in a bottom-heated square Hele-Shaw cell with two insulated side walls is studied with the shadowgraphy. Heat transfer measurements with and without the externally imposed vertical magnetic field are also presented. The Hele-Shaw cell flow of the natural convection of the magnetic fluid is governed by the following equations [3–5]: r u ¼ 0;
ð1Þ
12Z u þ r0 ½1 þ aðT T0 Þ g h2 þ m0 ðM rÞH; DT qM DH þ m0 T ¼ lr2 T; r0 C Dt qT V ;H Dt rp ¼
r B ¼ 0;
B ¼ m0 ðM þ HÞ;
ð2Þ ð3Þ ð4Þ
where Z is the viscosity, r0 the density, a the volumetric expansion coefficient, m0 the permeability in vacuum, C the specific heat and l the thermal conductivity. Continuity equation (1) for incompressible fluid, energy equation (3), and Maxwell equations (4) for electrically
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 6 7 1 - 6
C.-Y. Wen et al. / Journal of Magnetism and Magnetic Materials 252 (2002) 206–208
nonconductive media are the same as those for the natural convection of a magnetic fluid in a twodimensional rectangular cavity [3,4]. Only momentum equation (2) is replaced by Darcy’s law [5]. Eqs. (1)–(4) can be then nondimensionalized with nondimensional parameters as follows: 1 h 2 r0 gabd 4 Ra ¼ ; kZ 12 d 1 h 2 m0 H0 Ms d 2 Ram ¼ ; 12 d kZ d qT ; ð5Þ Nu ¼ DT qz z¼0 where Ra is the Rayleigh number, Ram the magnetic Rayleigh number and Nu the Nusselt number with DT ¼ TH TC ; b ¼ ðTH TC Þ=d; d is the height of cell, k the thermal diffusivity, H0 the strength of external magnetic field and Ms the saturation magnetization. Ra and Ram in Eq. (5) differ from those defined in 1 Yamaguchi et al. [3,4] by the factor of 12 (h=d)2 which implies the very small thickness in the third dimension, h; in the case of Hele-Shaw cell flow has to come into play. The characteristics of flow instability in Hele-Shaw cell and two-dimensional rectangular cavity [3,4] may be different.
Fig. 1. Experimental setup of heat transfer measurements.
207
2. Experimental apparatus The dimensions of the Hele-Shaw cell are 15
15 0.3 mm3. The magnetic fluid used in the current experiment is the water-based Fe3O4 magnetic fluid with r0 ¼ 1070 kg/m3, Z ¼ 3:75 103 Pa s and Ms ¼ 5:69 emu/g [6]. The experimental apparatus of heat transfer measurements and magnetic field imposition shown in Fig. 1 is similar to that of Yamaguchi et al. [3,4]. The magnetic field was imposed to the cell by the electromagnet vertically. The bottom of the cell was heated through a copper sheet while the upper wall of the cell was cooled through another copper sheet whose upper surface was opened to the ambient air. Nu obtained from the experiment is the local Nu that can be calculated based on the mean heat flux to the cell and the local temperature gradient measured at the position of the temperature-measuring point in the cell by three thermistors. The traditional shadowgraphy [7] was used to visualize the flow fields of natural convection with a He–Ne laser (Melles Griot 05-LHP-991, l ¼ 632:8 nm) as the light source.
3. Results and discussion Figs. 2a and b show the flow fields at the onset of transition and at a higher Ra without the externally imposed vertical magnetic field, respectively. Fig. 2c shows the flow field with Ram ¼ 670 case (which corresponds to H0 ¼ 11:5 G). As seen in Figs. 2a and b, a plume rises at the center of the cell and then evolves into a pair of counter-rotating vortices. By imposing the magnetic field, a pair of symmetric counter-rotating vortices is also clearly observed. The flow features are similar to those corresponding to the second instability mode in the numerical simulations of Yamaguchi et al. [4]. Note that in the two-dimensional square cavity case studied by Yamaguchi et al. [4], one convection vortex is generated at the first mode, instead of a pair of vortices in the present Hele-Shaw cell case. In Fig. 3, the heat transfer characteristics, with and without the imposed magnetic field, are depicted as the
Fig. 2. Flow field obtained from shadowgraphy: (a) Ra ¼ 4:1; Ram ¼ 0; (b) Ra ¼ 31:9; Ram ¼ 0; (c) Ra ¼ 26:3; Ram ¼ 670:
208
C.-Y. Wen et al. / Journal of Magnetism and Magnetic Materials 252 (2002) 206–208 Table 1 Critical Rayleigh numbers at the first flow transition Magnetic Rayleigh number, Ram
Critical Rayleigh number, Rac
0 670
2.01 1.06
4. Conclusion
Fig. 3. Heat transfer characteristics.
Nu versus Ra and are fitted with dotted line and solid line, respectively. It is noted that the experimental results are taken for the steady state condition. As seen, by imposing the magnetic field the heat transfer data shift towards the left indicating the increase of the heat transfer rate at the supercritical state. The heat transfer measurements suggest that the magnetic field has a destabilizing hydrodynamic effect on the flow. These results show close similarity with those of Yamaguchi et al. [3,4]. The critical Rayleigh numbers, Rac ; are estimated by extrapolating the data by the following relation: NuBðRa Rac Þ1=2 and are listed in Table 1. The critical Rayleigh numbers in the present Hele-Shaw cell, with and without the imposed magnetic field, are different from those in the two-dimensional square cavity case [3,4]. As mentioned in Section 1, the very small thickness in the third dimension in the case of Hele-Shaw cell flow affects the momentum equation and has great influence on the characteristics of flow instability.
The macroscopic magnetic flow fields in the HeleShaw cell are visualized with shadowgraphy for the first time. A pair of symmetric counter-rotating vortices is observed for the first instability mode. The heat transfer measurements suggest that the magnetic field has a destabilizing hydrodynamic effect on the flow. The very small thickness in the present case shows great influence on the characteristics of flow instability.
References [1] B.M. Berkovsky, Magnetic Fluids Engineering Applications, Oxford University Press, New York, 1993, p. 214. [2] B.A. Finlayson, J. Fluid Mech. 40 (1970) 753. [3] H. Yamaguchi, I. Kobori, Y. Uehata, K. Shimada, J. Magn. Magn. Mater. 201 (1999) 264. [4] H. Yamaguchi, I. Kobori, Y. Uehata, J. Thermophys. Heat Transf. 13 (1999) 501. [5] C. Tan, G. Homsy, Phys. Fluids 30 (1987) 1239. [6] C.Y. Hong, C.H. Ho, H.E. Horng, C.H. Chen, S.Y. Yang, Y.P. Chiu, H.C. Yang, Magn. Gidro. 35 (1999) 364. [7] W. Merzkirch, Flow Visualization, Academic Press, Orlando, FL, 1987, p. 126.