Abstract

We present a finite element analysis of a diffusion problem involving a coated cylinder enabling the rotation of heat fluxes. The coating consists of a heterogeneous anisotropic conductivity deduced from a geometric transformation in the time dependent heat equation. In contrast to thermal cloak and concentrator, specific heat and density are not affected by the transformation in the rotator. Therein, thermal flux diffuses from region of lower temperature to higher temperature, leading to an apparent negative conductivity analogous to what was observed in transformed thermostatics. When a conducting object lies inside the rotator, it appears as if rotated by certain angle to an external observer, what can be seen as a thermal illusion. A structured rotator is finally proposed inspired by earlier designs of thermostatic and microwave rotators.

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  1. J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  2. D. Van Dantzig, “Electromagnetism, independent of metrical geometry,” Proc. Kon. Ned. Akad. v. Wet.37, 521–531 (1934).
  3. U. Leonhardt, “Optical Conformal Mapping,” Science312, 1777–1780 (2006).
    [CrossRef] [PubMed]
  4. A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E95, 016623 (2005).
    [CrossRef]
  5. G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London Ser. A462, 3027–3059 (2006).
    [CrossRef]
  6. N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express15, 6314–6323 (2007).
    [CrossRef] [PubMed]
  7. S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express20, 8207–8218 (2012).
    [CrossRef] [PubMed]
  8. H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford University Press, 1959).
  9. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90, 241105 (2007).
    [CrossRef]
  10. H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009).
    [CrossRef] [PubMed]
  11. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24, 413–419 (2003).
    [CrossRef] [PubMed]
  12. C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett.92, 251907 (2008).
    [CrossRef]
  13. S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett.108, 214303 (2012).
    [CrossRef] [PubMed]
  14. R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: Molding the flow of heat,” arXiv:1210.2810

2012 (2)

S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett.108, 214303 (2012).
[CrossRef] [PubMed]

S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express20, 8207–8218 (2012).
[CrossRef] [PubMed]

2009 (1)

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009).
[CrossRef] [PubMed]

2008 (1)

C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett.92, 251907 (2008).
[CrossRef]

2007 (2)

2006 (3)

G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London Ser. A462, 3027–3059 (2006).
[CrossRef]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical Conformal Mapping,” Science312, 1777–1780 (2006).
[CrossRef] [PubMed]

2005 (1)

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E95, 016623 (2005).
[CrossRef]

2003 (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24, 413–419 (2003).
[CrossRef] [PubMed]

1934 (1)

D. Van Dantzig, “Electromagnetism, independent of metrical geometry,” Proc. Kon. Ned. Akad. v. Wet.37, 521–531 (1934).

Alu, A.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E95, 016623 (2005).
[CrossRef]

Amra, C.

Ao, X.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009).
[CrossRef] [PubMed]

Botten, L. C.

Carslaw, H. S.

H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford University Press, 1959).

Chan, C. T.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009).
[CrossRef] [PubMed]

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90, 241105 (2007).
[CrossRef]

Chen, H.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009).
[CrossRef] [PubMed]

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90, 241105 (2007).
[CrossRef]

Chen, S.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009).
[CrossRef] [PubMed]

Engheta, N.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E95, 016623 (2005).
[CrossRef]

Fan, C.Z.

C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett.92, 251907 (2008).
[CrossRef]

Gao, Y.

C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett.92, 251907 (2008).
[CrossRef]

Greenleaf, A.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24, 413–419 (2003).
[CrossRef] [PubMed]

Guenneau, S.

S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express20, 8207–8218 (2012).
[CrossRef] [PubMed]

R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: Molding the flow of heat,” arXiv:1210.2810

Hou, B.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009).
[CrossRef] [PubMed]

Huang, J. P.

C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett.92, 251907 (2008).
[CrossRef]

Jaeger, J. C.

H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford University Press, 1959).

Kadic, M.

R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: Molding the flow of heat,” arXiv:1210.2810

Lassas, M.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24, 413–419 (2003).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, “Optical Conformal Mapping,” Science312, 1777–1780 (2006).
[CrossRef] [PubMed]

McPhedran, R. C.

Milton, G.

G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London Ser. A462, 3027–3059 (2006).
[CrossRef]

Milton, G. W.

Narayana, S.

S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett.108, 214303 (2012).
[CrossRef] [PubMed]

Nicorovici, N. A. P.

N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express15, 6314–6323 (2007).
[CrossRef] [PubMed]

G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London Ser. A462, 3027–3059 (2006).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Sato, Y.

S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett.108, 214303 (2012).
[CrossRef] [PubMed]

Schittny, R.

R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: Molding the flow of heat,” arXiv:1210.2810

Shurig, D.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Uhlmann, G.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24, 413–419 (2003).
[CrossRef] [PubMed]

Van Dantzig, D.

D. Van Dantzig, “Electromagnetism, independent of metrical geometry,” Proc. Kon. Ned. Akad. v. Wet.37, 521–531 (1934).

Veynante, D.

Wegener, M.

R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: Molding the flow of heat,” arXiv:1210.2810

Wen, W.

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett.90, 241105 (2007).
[CrossRef]

C.Z. Fan, Y. Gao, and J. P. Huang, “Shaped graded materials with an apparent negative thermal conductivity,” Appl. Phys. Lett.92, 251907 (2008).
[CrossRef]

Opt. Express (2)

Phys. Rev. E (1)

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E95, 016623 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

S. Narayana and Y. Sato, “Heat flux manipulation with engineered thermal materials,” Phys. Rev. Lett.108, 214303 (2012).
[CrossRef] [PubMed]

H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, “Design and Experimental Realization of a Broadband Transformation Media Field Rotator at Microwave Frequencies,” Phys. Rev. Lett.102(18), 183903 (2009).
[CrossRef] [PubMed]

Physiol. Meas. (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Meas.24, 413–419 (2003).
[CrossRef] [PubMed]

Proc. Kon. Ned. Akad. v. Wet. (1)

D. Van Dantzig, “Electromagnetism, independent of metrical geometry,” Proc. Kon. Ned. Akad. v. Wet.37, 521–531 (1934).

Proc. R. Soc. London Ser. A (1)

G. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London Ser. A462, 3027–3059 (2006).
[CrossRef]

Science (2)

U. Leonhardt, “Optical Conformal Mapping,” Science312, 1777–1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Other (2)

H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids (Oxford University Press, 1959).

R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, “Experiments on transformation thermodynamics: Molding the flow of heat,” arXiv:1210.2810

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Figures (4)

Fig. 1
Fig. 1

Normalized temperature field for a source located on top i.e. along the line x0 = (x, 4.10−4m) which diffuses heat in a medium with diffusivity κ/() = 1.10−5m2.s−1 containing a rotator of center (0, 0) and radii R1 = 5.10−5m and R2 = 3.10−4m, and rotation angle θ0 = 3π/4. The temperature is normalized throughout time on the upper side of the cell. Snapshots of temperature distribution at t = 0.005s (a), t = 0.01s (b) and t = 0.1s (c); (d) The mesh formed by streamlines and isothermal values in the long time regime t ≥ 0.1s illustrates the deformation of the transformed thermal space.

Fig. 2
Fig. 2

Normalized temperature field for a source located on top i.e. along the line x0 = (x, 4.10−4m) which diffuses heat in a medium with diffusivity κ/() = 1.10−5m2.s−1 containing a rotator of center (0, 0) and radii R1 = 1.10−4m and R2 = 3.10−4m, and rotation angle θ0 = π/2. The temperature is normalized throughout time on the upper side of the cell. Snapshots of temperature distribution at t = 0.005s (a), t = 0.01s (b) and t = 0.1s (c); (d) The mesh formed by streamlines and isothermal values in the long time regime t ≥ 0.1s illustrates the deformation of the transformed thermal space.

Fig. 3
Fig. 3

Normalized temperature field for a source located on top i.e. along the line x0 = (x, 4.10−4m) which diffuses heat in a medium with diffusivity κ/() = 1.10−5m2.s−1 containing a rotator of center (0, 0) and radii R1 = 2.10−4m and R2 = 3.10−4m, and rotation angle θ0 = π/2. The temperature is normalized throughout time on the upper side of the cell. Snapshots of temperature distribution at t = 0.005s (a), (b) and t = 0.1s (c), (d); The rectangular object has a diffusivity one hundred times smaller than that of the surrounding medium and is rotated by an angle π/2 in (b) and (d) compared to (a) and (c); The temperature field in (a),(b) and (c),(d) is exactly the same oustide the rotator.

Fig. 4
Fig. 4

Normalized temperature field for a source located on the left-hand side i.e. along the line x0 = (−7.10−4m, y) which diffuses heat in a medium with diffusivity κ/() = 1.10−5m2.s−1 containing a structured ring of center (0, 0) and radii R1 = 2.10−4m and R2 = 4.10−4m consisting of four rows with 50 insulating sectors (diffusivity one hundred times smaller than that of the surrounding medium) in each row; Rows are slightly rotated with respect to one another (through an angle θ = π/20), and separated by three thin highly conducting wires (diffusivity 100 times that of surrounding medium) directed along the azimuthal angle. The temperature is normalized throughout time on the left-hand side of the cell. Snapshots of temperature distribution at t = 0.005s (a) and t = 0.02s (b); White curves represent the trajectories of heat fluxes. The achieved rotation of heat flux is π/4.

Equations (7)

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ρ ( x ) c ( x ) u t = ( κ ( x ) u ) + p ( x , t ) ,
( d x d y ) = J x x ( d x d y ) , with J x x = ( x , y ) ( x , y ) .
κ _ _ = J 1 κ J t det ( J ) = κ J 1 J T det ( J ) = κ T 1 ,
ρ ( x ) c ( x ) det ( J ) u t = ( κ T 1 u ) + det ( J ) p ( x , t ) .
r = r , θ = θ + θ 0 , r < R 1 , r = r , θ = θ , r > R 2 , r = r , θ = θ + θ 0 f ( R 2 ) f ( r ) f ( R 2 ) f ( R 1 ) , R 1 < r < R 2 ,
T 1 = ( ( T 1 ) 11 ( T 1 ) 12 ( T 1 ) 21 ( T 1 ) 22 ) ,
( T 1 ) 11 = 1 2 t cos ( θ ) sin ( θ ) + t 2 cos 2 ( θ ) , ( T 1 ) 22 = 1 + 2 t cos ( θ ) sin ( θ ) + t 2 sin 2 ( θ ) , ( T 1 ) 12 = ( T 1 ) 21 = t 2 cos ( θ ) sin ( θ ) t ( cos 2 ( θ ) sin 2 ( θ ) ) ,

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