Spatiotemporal boundary dissipation measurement in Taylor–Couette flow using diffusing-wave spectroscopy

Abstract

Diffusing-wave spectroscopy (DWS) allows for the direct measurement of the squared strain-rate tensor. When combined with commonly available high-speed cameras, we show that DWS gives direct access to the spatiotemporal variations of the viscous dissipation rate of a Newtonian fluid flow. The method is demonstrated using a Taylor–Couette (TC) cell filled with a lipid emulsion or a TiO₂ suspension. We image the boundary dissipation rate in a quantitative and time-resolved fashion by shining coherent light at the experimental cell and measuring the local correlation time of the speckle pattern. The results are validated by comparison with the theoretical prediction for an ideal TC flow and with global measurements using a photomultiplier tube and a photon correlator. We illustrate the method by characterizing the spatial organization of the boundary dissipation rate past the Taylor–Couette instability threshold, and its spatiotemporal dynamics in the wavy vortex flow that arises beyond a secondary instability threshold. This study paves the way for direct imaging of the dissipation rate in a large variety of flows, including turbulent ones.

Appendix: Boundary conditions and finite-size effects

To exactly compute the correlation function $g_1$, one needs to solve the diffusion equation to determine the probability density of paths length P (Weitz and Pine 1993; Sheng 2006). To do so, we have to choose the initial and boundary conditions (BC) describing the diffusive transport of the light in the cell. We consider a slice of thickness L in the x direction ($0\le x\le L$) and of infinite extent in the y and z directions. For the initial condition, in the case of uniform illumination on the incident face, the initial “diffusive light” (in the sense of being described by the diffusion equation) is often described in the DWS theory by a Dirac (with infinite extent in y and z) at a distance $x_0$ from the incident face. Indeed, the transport of light can be described as diffusive only once the incident light has been scattered. We expect that the first scattering event happens at a distance of order $l^*$ from the incident face, so $x_0\approx l^*$. For the boundary conditions, we can decide to set the flux of diffusive light into the cell to zero at the boundaries, since no scattered light enters the sample from outside. It is even more relevant to set the flux of diffusive light into the cell to a fraction R of the flux of diffusive light leaving the cell, to take into account reflections at the boundaries. This is the partial-current BC. An equivalent BC is the extrapolated BC: the density of diffusive light is set to 0 at an extrapolation length $C=\frac{2}{3}l^*\frac{1+R}{1-R}$ outside the cell (Zhu et al. 1991; Haskell et al. 1994). We found that this solution is in even better agreement with our experimental data than the partial-current BC. Other boundary conditions are possible, such as the absorbing BC, but they usually provide solutions less in agreement with experiments (Weitz and Pine 1993; Pine et al. 1990). The probability density of path lengths P(s) and its Laplace transform (the correlation function $g_1(\tau )$) can be obtained from chapter 14.3 in Carslaw and Jaeger (1959). For the extrapolated BC, in backscattering (i.e., looking at the diffusive light at x = 0), we obtain:

$$\begin{aligned} P(s)= & {} \frac{\pi ^2 l^*\sum \nolimits _{n=1}^{\infty }\sin \left( \frac{n\pi C}{L+2C}\right) \sin \left( \frac{n\pi x_0}{L+2C}\right) \exp \left( -\frac{n^2\pi ^2 l^* s}{3(L+2C)^2}\right) }{3(L+2C)^2\sum \nolimits _{n=1}^{\infty }\frac{1}{n^2}\sin \left( \frac{n\pi C}{L+2C}\right) \sin \left( \frac{n\pi x_0}{L+2C}\right) } \end{aligned}$$

(A.1)

$$\begin{aligned} g_1(\tau )= & {} \frac{\sinh \left( \frac{C}{l^*}\sqrt{6T}\right) \sinh \left( \frac{L+C-x_0}{l^*}\sqrt{6T}\right) }{\left( 1-\frac{x_0+C}{L+2C}\right) \frac{C}{l^*}\sqrt{6T}\sinh \left( \frac{L+2C}{l^*}\sqrt{6T}\right) } \end{aligned}$$

(A.2)

where $T=\tau /\tau _0+\tau ^2/\tau _v^2$. In the limit of a semi-infinite medium ($l^*\ll L$), the correlation function reduces to:

$$\begin{aligned} g_1(\tau )= \frac{l^*}{C\sqrt{6T}}\exp \left(-\frac{x_0+C}{l^*}\sqrt{6T}\right)\sinh \left(\frac{C}{l^*}\sqrt{6T}\right) \end{aligned}$$

(A.3)

In the limit of short times ($T\ll 1$), the decay is almost exponential:

$$\begin{aligned} g_1(\tau )\approx \exp \left(-\gamma \sqrt{6T}\right) \end{aligned}$$

(A.4)

where $\gamma =\frac{x_0+C}{l^*}=\frac{x_0}{l^*}+\frac{2}{3}\frac{1+R}{1-R}$. The dimensionless parameter $\gamma$ is therefore linked to $x_0$ but also to the geometry of the cell and the refractive indices of the fluid and the cell through R. It is also known to depend on the presence of a polarizer or analyzer, since these can foster shorter or longer paths (Pine et al. 1990; Weitz and Pine 1993).

Fig. 6

Normalized correlation function $(g_2(\tau )-1)/\beta =|g_1(\tau )|^2$ of the intensity for $L/l^*=25$ (in blue) corresponding to the TiO₂ suspension setup and $L/l^*=81$ (in green) corresponding to the lipid emulsion setup, from Eq. (A.2). The semi-infinite medium case (in red) from Eq. (A.3) and the exponential approximation (in black) from Eq. (A.4) are plot for comparison. The inset zooms in the early stage of the curve where the finite-size effects are significant

Figure 6 highlights the finite-size effects on the normalized correlation function of the intensity $(g_2(\tau )-1)/\beta =|g_1(\tau )|^2$, for $x_0=l^*$, $C=2l^*/3$ ($R=0$, no reflection) and the corresponding $\gamma =5/3$. When the ratio $L/l^*$ decreases, deviation from the exponential behavior is observed at very short times. It corresponds to a reduction of the contribution of very long paths, since they can be transmitted and therefore lost for backscattering. To avoid these effects, we remove the very first points in our correlation functions and extrapolate the initial value $\beta =g_2(0)-1$ (see Sect. 2.4). Note that at slightly longer times, the slopes are the same and are very close to the exponential approximation. We can also focus on these differences at very short time to measure $l^*$ by fitting Eq. (A.2) to the experimental data, as long as $x_0$ and C are known from a “semi-infinite” measurement fitted with Eq. (A.3).