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Faster Diffusion Across an Irregular Boundary View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by HAL-Polytechnique Faster Diffusion across an Irregular Boundary Anna Rozanova-Pierrat, Denis Grebenkov, Bernard Sapoval To cite this version: Anna Rozanova-Pierrat, Denis Grebenkov, Bernard Sapoval. Faster Diffusion across an Irreg- ular Boundary. Physical Review Letters, American Physical Society, 2012, 108, pp.240602. <10.1103/PhysRevLett.108.240602>. <hal-00760296> HAL Id: hal-00760296 https://hal-ecp.archives-ouvertes.fr/hal-00760296 Submitted on 5 Dec 2012 HAL is a multi-disciplinary open access L'archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destin´eeau d´ep^otet `ala diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publi´esou non, lished or not. The documents may come from ´emanant des ´etablissements d'enseignement et de teaching and research institutions in France or recherche fran¸caisou ´etrangers,des laboratoires abroad, or from public or private research centers. publics ou priv´es. Faster Diffusion across an Irregular Boundary A. Rozanova-Pierrat∗ Laboratoire de Physique de la Mati`ere Condens´ee, C.N.R.S. – Ecole Polytechnique, 91128 Palaiseau, France and Laboratoire de Mathematiques Appliquees aux Systemes, Ecole Centrale Paris, 92295 Chatenay-Malabry, France D. S. Grebenkov† and B. Sapoval‡ Laboratoire de Physique de la Mati`ere Condens´ee, C.N.R.S. – Ecole Polytechnique, 91128 Palaiseau, France (Dated: December 5, 2012) We investigate how the shape of a heat source may enhance global heat transfer at short time. An experiment is described that allows to obtain a direct visualization of heat propagation from a prefractal radiator. We show, both experimentally and numerically, that irregularly-shaped passive coolers rapidly dissipate at short times, but their efficiency decreases with time. The de Gennes’ scaling argument is shown to be only a large scale approximation which is not sufficient to de- scribe adequately the temperature distribution close to the irregular frontier. This work shows that radiators with irregular surfaces permits increased cooling of pulsed heat sources. PACS numbers: 05.60.-k, 44.05.+e, 66.10.C- Propagation of heat as well as particle diffusion are of ported for diffusive transfer at fractal boundaries [11–17]. direct interest in every-day life, chemical industry, life We present a direct experimental visualization of heat sciences, and pollution spreading. In electronics or civil propagation from a “fractal” radiator in 2D. engineering, one needs to improve the cooling of micro- Fractals are complex shapes that contain a large va- processors and to build efficient radiators for housing. riety of geometrical feature sizes [18], and it has been Empirically, a large surface is known to dissipate more already shown that deterministic and random shapes ex- so that irregularly-shaped metallic radiators are placed hibiting the same fractal dimensions, behave similarly onto microprocessors in order to remove heat. But this [19]. is the result of both conduction and convection. In this We check, for the first time, the validity of the de letter, we present an experimental, theoretical and nu- Gennes’ scaling argument [20] which is often used to un- merical study of purely diffusive heat transfer that aims derstand heat propagation and diffusion from an irreg- at revealing the role of the shape of the radiator. ular surface [5, 7, 21]. Relying on experimental results and numerical solutions of the underlying Fourier equa- This question has been intensively investigated in a tion, we conclude that this argument can only be a large steady state regime in which the diffusive transfer to- scale approximation which is not sufficient to describe wards a boundary is governed by Laplace equation. In temperature distribution close to the irregular frontier. this case, a general mathematical result known as the The heat propagation in 2D has been studied ex- Makarov’s theorem states that, in 2D, the information perimentally using liquid crystal temperature imaging dimension of the harmonic measure (which describes the [22, 23]. The heat source is constituted by a highly con- distribution of particles arrived onto the boundary) is ductive, 4mm thick, aluminum sheet (Fig 1a). A complex strictly equal to 1 for any simply connected set, whatever perforation has been laser machined through this plate the complexity of its boundary [1]. In physical terms, this to obtain the third generation of the quadratic von Koch means that, contrary to intuition, regular and irregular snowflake of fractal dimension 3/2. The aluminum heat boundaries work similarly in steady state regime [2–4]. ◦ source is heated around Ts 40 C. It is then put on the ≈ ◦ In this letter, we report the first experimental observa- liquid crystal plate (at temperature T0 20 C) which tion that regular and irregular boundaries behave differ- was set on a Styrofoam block with very≈ small heat con- ently in the transitory (or time-dependent) regime. We ductivity. This constitutes, to a good approximation, a show that the role of irregular boundaries is important 2D heat propagation system. A video camera monitors and provide quantitative explanations based on theory the time evolution of the temperature distribution by fol- and numerical simulations. Note that transitory diffu- lowing the displacement of the visible isotherm lines im- sion from an irregular surface has been already stud- aged by their color (Fig. 1(b)-(f)). The entire film can ied in the context of impedance spectroscopy in electro- be seen on www... At each time t, one can extract by a chemistry [5, 6], NMR relaxation in porous media [7, 8], thresholding procedure an isotherm curve rt of a given Brownian motion near fractal surfaces [9] and heat trans- color (at temperature T ). Since a quantitative measure fer [10]. Rigorous short-time asymptotics have been re- of the heat Q(t) transferred from the source from time 0 10 (a) tot (t)/S (a) (b) T S experiment simulation (2/π0.5)L (Kt)0.5 p 2 2 0.25 −1 2L (Kt/L ) 10 −5 −4 −3 −2 −1 10 10 10 10 10 Kt/L2 0 (c) (d) 10 (b) ❘ ∞ ❆❑❆❆❯❆ ■ Q(t)/Q( ) (2/π0.5)L (Kt)0.5 ✛✲ p 2L2(Kt/L2)0.25 heat S (t)/S 0.1 tot S (t)/S (e) (f) 0.5 tot S (t)/S 0.9 tot FIG. 1: (Color online) Experimental visualization of heat Q (t)/Q (∞) −1 0 0 10 propagation in 2D. (a) Schematic representation the experi- −5 −4 −3 −2 −1 mental set-up. A 4mm thick aluminum plate profiled follow- 10 10 10 10 10 2 ing a fractal pattern is put above a liquid crystal film itself Kt/L put above a thermally insulating Styrofoam layer. The heat source large heat capacity and large thermal conductivity as- sure the uniformity of the source temperature; (b)-(f) Motion FIG. 2: (Color online) Time evolution of the heat Q(t) trans- of the isocolor/isothermal line as a function of time (at 5, 10, ferred from the source. (a) Evolution of the area ST (t) of the 20, 40 and 46 seconds, respectively). The differents lenghts layer delimited by the visualized isotherm: experiment (cir- of the white arrows in (e) illustrates the confinement effect cles) and numerical simulations (solid line) of the layer corre- discussed in the text. The white bar presents a length scale sponding to the isotherm level θ = 0.9 (a.u.), best fit for the of 1 cm. experimental curve, see text). Short-time asymptotic (4) and intermediate asymptotic (5) slopes are shown by dashes and dash-dotted lines. (b) Numerical computation of Q(t) (solid 0 to t is not directly accessible, one has to resort to an blue line) and of the three areas S0.1(t), S0.5(t), S0.9(t) (sym- approximation. The first guess is that Q(t) is approxi- bols), all computed. The curves ST (t) are shifted along the horizontal axis by factors 5, 1 (no shift) and 0.11, respectively. mated by the surface area ST (t) between the prefractal source and the isotherm curve r : As explained in the text, these shifts account for the coeffi- t cient β(T ) whose dependence on the chosen isotherm level is unknown for the prefractal boundary. The thermal response Q(t) cδ(Ts T0)ST (t), (1) ≃ − Q0(t)/Q0( ) of the square initiator of the same area is shown by solid black∞ line (see Supplementary Materials for details). where T T is the temperature drop between the heat s − 0 Note that Q(t) and ST (t) are divided by their maximum val- source and the domain, c the film specific heat and δ its ues to be comparable. thickness. The estimation of ST (t) from the experimental images is shown in Fig. 2a on which one observes succes- sive power laws. In order to explain this behavior and over the domain (denoted as Ω) is thus the solution of to check the validity of the approximation (1), we recall the following problem [24, 25]: the theoretical background, the currently admitted scal- ing argument and present direct numerical simulations of ∂ heat propagation. T (r,t) K∆T (r,t)=0 (r Ω), ∂t In the experiment, due to the very large contrast be- − ∈ T (r,t)= T (r Γ), (2) tween the thermal diffusivities of aluminum and the liq- s ∈ r uid crystal film, the temperature over the aluminum T ( ,t =0)= T0, sheet and at its frontier Γ can be considered as con- stant (Ts), imposing a Dirichlet boundary condition on where ∆ is the 2D Laplace operator and K the liquid the heat source. The temperature distribution T (r,t) crystal thermal diffusivity (of order 0.1 cm2s−1).
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