PHYSICAL REVIEW E VOLUME 49, NUMBER 2 FEBRUARY 1994 Experimental evaluation of the intrinsic noise in the Couette-Taylor system with an axial flow Avraham Tsameret, Galia Goldner, and Victor Steinberg Department of Nuclear Physics, The Weizmann Institute of Science, Rehovot 76100, Israel (Received 3 June 1993; revised manuscript received 7 September 1993) The intrinsic noise in the Couette-Taylor system with axial flow is evaluated experimentally by several methods, which include a comparison of experimental data with numerical simulations of the amplitude equation with a noise term and the application of an external source of stochastic perturbations at the in- let. The intensity of the intrinsic noise is found in our system to be dependent on the through-flow veloc- ity in the following manner: for large enough through-flow velocities (Reynolds number Re) 2) the in- tensity of the noise drastically increases with Re, whereas for small Re the noise amplitude is indepen- dent of Re and reaches a constant value of =0.02 pm/s, which is of the order of magnitude of the theoretically estimated value for the thermal noise. The amplitude of the intrinsic noise at large through-flow velocities (Re=3) is found in our system to be larger than the thermal noise by more than one order of magnitude. Its origin is suggested to be associated with the perturbations of the flow at the inlet boundary. PACS number(s): 47.20. — k, 47.60. +i, 43.50. +y I. INTRODUt. i iON The role of the intrinsic fluctuations in a pattern for- mation near the bifurcation point in systems far from equilibrium has been the subject of numerous theoretical and experimental studies for the past two decades [1 — 3]. Until very recently it was regarded that thermal fluctua- tions cannot play any significant role in pattern formation in hydrodynamical systems and that they are unobserv- ably small due to the extremely small ratio between a mi- croscopic thermal energy kz T and the macroscopic kinetic energy of hydrodynamical flows, e. g. , in a convec- tive roll pdv. The ratio between these values reaches usually 10-11 orders of magnitude, e.g. , for Rayleigh- Benard convection [3]. This conclusion was based on both theoretical estimates [1 — 3, 5] and various experimen- tal results [2, 6 — 8]. The theoretical estimates suggested that in order to observe fluctuations of the velocity or temperature fields, which are caused by thermal fluctua- tions, on a macroscopic level, one should approach the close vicinity of the transition where the fluctuations grow enormously, similarly to the critical behavior in equilibrium systems. However, this suggestion was con- sidered unrealizable experimentally due to various geometrical and thermal inhomogeneities existing in the real system, e. g. , in thermal convection, such as thickness nonuniformity, imperfections due to finite-size effects, de- viations from horizontality, various sources of thermal imperfections in convection, and experimental noise. These effects lead to a rounding of the transition, mask the role of fluctuations, and do not allow one to approach closely the transition point. Therefore a macroscopic Quid motion, caused by the imperfection and which exist- ed before the transition, wipes out the contribution of the hydrodynamical fluctuations [6 — 8]. A new insight on the role of stochastic effects on pat- tern formation was obtained in Ref. [2]. The evolution of patterns in a thermal convection from the basic to the ordered state was studied under boundary conditions which eliminated the effects of sidewall forcing and pro- vided evidence of the stochastic nature of the pattern for- mation process. However, a comparison of the data with solutions of model equations [5] gives the value of the noise intensity, necessary to fit the experiment, more'n four orders of magnitude larger than the thermal noise in the Navier-Stokes equations. Therefore, the thermal noise fluctuations were ruled out as the driving force of the pattern formation in a stationary convection. Nevertheless, it has been reported recently [9] that thermal noise fluctuations were observed and measured in electroconvection in nematic liquid crystals. This system has been shown to be particularly sensitive to noise. The effect of noise fluctuations is relatively larger because the elastic constant of a liquid crystal is small and because a layer of rather small thickness can be used. Electrocon- vection of traveling waves was observed in this system in the form of patches of weak convection rolls with ran- domly varying amplitudes. These patches were shown to be thermal noise fluctuations on the basis of a compar- ison with the stochastic Ginzburg-Landau (GL) equation. Measurement of intensity of the director fluctuations below the onset of electroconvection [10] also identified the origin of these Quctuations with the thermal noise. The above-mentioned experiments were carried out in absolutely unstable and absolutely stable (electroconvec- tion in liquid crystals} systems. A different approach to the measurement of thermal fluctuations is to study them in convectively unstable systems. In a convectively unsta- ble system a small perturbation grows exponentially as it propagates downstream. The noise in the convectively unstable region therefore experiences an amplification process. When the system is sufBciently long, even a mi- croscopic perturbation can be amplified to produce a macroscopic pattern downstream. Therefore convective- ly unstable system are more convenient to study experi- mentally the effect of hydrodynamical fluctuations in gen- 1063-651X/94/49(2)/1309(11)/$06. 00 49 1309 1994 The American Physical Society