Particle-laden exchange flows in inclined pipes

Lock-exchange suspension gravity current in a tilted narrow pipe is studied experimentally at very low Reynolds numbers. We investigate the interpenetration of a heavy particle-laden fluid on top of a light pure fluid by considering the effects of the initial volume fraction of particles ${\varphi }_0$ and inclination angle of the pipe from vertical $\beta$. Density mismatch between heavy and light phases is maintained low (Boussinesq limit) to allow a stable quasiparallel viscous flow. Pertaining to whether particles stay mixed in the suspension or sediment, three distinct mixing, transitionary, and sedimentary regimes are identified. Results of the classification over a wide range of particle concentration ${\varphi }_0\in \left[0.05,0.50\right]$ and inclination angle $\beta \in \left[0^{\circ },{88}^{\circ }\right]$ are proposed in dimensionless phase diagrams suitable for industrial calculations. Mixing behavior emerges at sharp inclinations as flow continues steadily along the pipe. Sedimentary behavior is attributed to the decelerating flows that involve dilute or packed suspensions over nearly horizontal angles. To quantify the interpenetration rate, a scaling analysis is performed to approximate the velocity of the particle-laden front for different values of ${\varphi }_0$ and $\beta$. Furthermore, the impacts of the fluid's viscosity and particle size are investigated in detail. Reducing the particle size generally weakens the sedimentary behavior of the flow. Increasing the liquid's viscosity has almost no effect on regime transition, yet it results in slower advancing speed across all tilt angles.

Figure 1

Schematic of the experimental apparatus. The entire system can be tilted at an angle, $\beta$, measured from the vertical direction. The interface shape and particle distribution in the bottom right image are illustrative only.

Figure 2

(a) Sequence of experimental images for pure exchange flow obtained for $\beta ={60}^{\circ }$ at times $\widehat{t}=[0,5,11,\cdots 32,37]$ s with ${\widehat{\rho }}_H=1018.4\mathrm{kg}{\mathrm{m}}^{-3}, {\widehat{\rho }}_L=998.1\mathrm{kg}{\mathrm{m}}^{-3}$, and ${\widehat{\mu }}_H={\widehat{\mu }}_L=1\times {10}^{-3}\mathrm{Pa}\mathrm{s}$ ($\text{At}=0.01$ [Atwood number, $\mathrm{At}=({\widehat{\rho }}_H-{\widehat{\rho }}_L)/({\widehat{\rho }}_H+{\widehat{\rho }}_L)$], ${\varphi }_0=0, {\text{Re}}_t=291, \xi =0$, and $r_p=0$). (b) Array of images over the full range of inclination angle, $\beta$, at $\widehat{t}\approx 35$ s. The field of view is $2000\times 9.53{\mathrm{mm}}^2$ in all the images. The color bar at the top left of the figure shows concentration value C, with 0 and 1 referring to the pure heavy and light fluids, respectively.

Figure 3

(a) Sequence of images at times $\widehat{t}=[0,136,272,\cdots 816,952]$ s—the field of view is $2000\times 9.53{\mathrm{mm}}^2$, (b) spatiotemporal diagram of the depth-averaged concentration, (c) leading front's location with time, and (d) frontal velocity dependency on time for a particle-laden exchange experiment from set A: ${\varphi }_0=0.30, \beta ={88}^{\circ }, \widehat{d}=70\mu \mathrm{m}, {\widehat{\mu }}_f=0.367\mathrm{Pa}\mathrm{s}$, and ${\widehat{u}}_0=0.009{\mathrm{mm}\mathrm{s}}^{-1}$ ($\psi =0.767, {\text{Re}}_t=1.21, \xi =2.01$, and $r_p=0.007$). Stoppage time and distance in part (c) are denoted by ${\widehat{t}}_s$ and ${\widehat{X}}_s$, respectively. Dashed lines in parts (a) and (b) mark the proximate locations of leading, trailing and sedimentary fronts over time; arrows show relative directions. Labels (I), (II), and (III) in the inset of part (c) correspond to inertia, viscous, and sedimentary phases, respectively.

Figure 4

Variation of the stoppage distance ${\widehat{X}}_s$, and time ${\widehat{t}}_s$ (inset), with initial volume fraction of particles: ${\varphi }_0=\left\{0.05,0.15,0.30,0.40,0.50\right\}, \beta ={88}^{\circ }$, using set A, $\widehat{d}=70\mu \mathrm{m}, {\widehat{\mu }}_f=0.367\mathrm{Pa}\mathrm{s}$, and ${\widehat{u}}_0=0.009{\mathrm{mm}\mathrm{s}}^{-1}$ ($\psi =\left\{0.952,0.868,0.767,0.712,0.664\right\}, {\text{Re}}_t=\left\{1.37,1.72,1.21,0.67,0.22\right\}, \xi =2.01$, and $r_p=0.007$). Maximum ${\widehat{X}}_s$ and time ${\widehat{t}}_s$ correspond to the experiment with ${\varphi }_0=0.30$, which is in accordance with findings of Ref. [28].

Figure 5

Experimental images at $\widehat{t}\approx 150\mathrm{s}$ (top) and corresponding spatiotemporal diagrams of the depth-averaged concentration (bottom) for three flow regimes (a) sedimentary, (b) transitionary, and (c) mixing: ${\varphi }_0=0.30$ at $\beta ={88}^{\circ },{75}^{\circ },\text{and}{30}^{\circ }, \widehat{d}=70\mu \mathrm{m}, {\widehat{\mu }}_f=0.367\mathrm{Pa}\mathrm{s}$, and ${\widehat{u}}_0=0.009{\mathrm{mm}\mathrm{s}}^{-1}$ ($\psi =0.767, {\text{Re}}_t=1.21, \xi =2.01$, and $r_p=0.007$). The field of view in snapshots is $2000\times 9.53{\mathrm{mm}}^2$.

Figure 6

Suspension front position, ${\widehat{X}}_f^2\left(\widehat{t}\right)$, over time, $\widehat{t}$, for various angles of inclination corresponding to the experiments in set A. Different colors correspond to sedimentary (green), transitionary (yellow), and mixing (blue) regimes: $\widehat{d}=70\mu \mathrm{m}, {\widehat{\mu }}_f=0.367\mathrm{Pa}\mathrm{s}, {\widehat{u}}_0=0.009{\mathrm{mm}\mathrm{s}}^{-1}$ for (a) ${\varphi }_0=0.05$, (b) ${\varphi }_0=0.15$, (c) ${\varphi }_0=0.30$, (d) ${\varphi }_0=0.40$, and (e) ${\varphi }_0=0.50$ ($\psi =\left\{0.952,0.868,0.767,0.712,0.664\right\}, {\text{Re}}_t=\left\{1.37,1.72,1.21,0.67,0.22\right\}, \xi =2.01$, and $r_p=0.007$). Results in the dimensionless format are shown in the inset.

Figure 7

The ratio of advection timescale ${\widehat{t}}_A$, and sedimentation one ${\widehat{t}}_S$, calculated theoretically for inclination angles $\beta \in \left[2^{\circ },{88}^{\circ }\right]$ in set A: ${\varphi }_0=\left\{0.05,0.15,0.30,0.40,0.50\right\}, \widehat{d}=70\mu \mathrm{m}, {\widehat{\mu }}_f=0.367\mathrm{Pa}\mathrm{s}$, and ${\widehat{u}}_0=0.009{\mathrm{mm}\mathrm{s}}^{-1}$ ($\psi =\left\{0.952,0.868,0.767,0.712,0.664\right\}, {\text{Re}}_t=\left\{1.37,1.72,1.21,0.67,0.22\right\}, \xi =2.01$, and $r_p=0.007$). The red circles are transitionary data points obtained for each line from its corresponding experiments.

Figure 8

Array of experimental images at various angles for same experiments as in Fig. 6 recorded at: (a) $\widehat{t}=484\pm 32\mathrm{s}$, (b) $\widehat{t}=248\pm 12\mathrm{s}$, (c) $\widehat{t}=141\pm 13\mathrm{s}$, (d) $\widehat{t}=101\pm 9\mathrm{s}$, and (e) $\widehat{t}=153\pm 35\mathrm{s}$. The field of view is $2000\times 9.53{\mathrm{mm}}^2$.

Figure 9

Concentration profiles on the right-hand side of the pipe at times $\widehat{t}=[0,23,46\cdots 138,161]$ s, showing the formation of a ridge close to the front region. ${\varphi }_0=0.30, \beta ={15}^{\circ }, {\widehat{\mu }}_f=0.367\mathrm{Pa}\mathrm{s}$, and ${\widehat{u}}_0=0.009{\mathrm{mm}\mathrm{s}}^{-1}$ ($\psi =0.767, {\text{Re}}_t=1.21, \xi =2.01$, and $r_p=0.007$).

Figure 10

(a) Phase diagram, and (b) variation of the mean frontal velocity ${\widehat{V}}_{f,\text{av}}$, versus angle of inclination $\beta$, for experiments in set A: ${\varphi }_0=\left\{0.05,0.15,0.30,0.40,0.50\right\}, \widehat{d}=70\mu \mathrm{m}, {\widehat{\mu }}_f=0.367\mathrm{Pa}\mathrm{s}$, and ${\widehat{u}}_0=0.009{\mathrm{mm}\mathrm{s}}^{-1}$ ($\psi =\left\{0.952,0.868,0.767,0.712,0.664\right\}, {\text{Re}}_t=\left\{1.37,1.72,1.21,0.67,0.22\right\}, \xi =2.01$, and $r_p=0.007$). Right inset (subfigure b): ${\widehat{V}}_{f,\text{av}}$ plotted against the velocity scale ${\widehat{U}}_f$ for experiments in set A; the fitted line is given by Eq. (6) with the slope $C_F=0.435$. Left inset: dimensionless mean frontal velocity $V_{f,\text{av}}$ with $\beta$.

Figure 11

Array of experimental images at various angles corresponding to the experiments of set B: $\widehat{d}=40\mu \mathrm{m}, {\widehat{\mu }}_f=0.367\mathrm{Pa}\mathrm{s}, {\widehat{u}}_0=0.003{\mathrm{mm}\mathrm{s}}^{-1}$ for (a) ${\varphi }_0=0.05$ at $\widehat{t}=778\pm 51\mathrm{s}$, (b) ${\varphi }_0=0.30$ at $\widehat{t}=200\pm 26\mathrm{s}$, and (c) ${\varphi }_0=0.50$ at $\widehat{t}=165\pm 59\mathrm{s}$ ($\psi =\left\{0.952,0.767,0.664\right\}, {\text{Re}}_t=\left\{1.37,1.21,0.22\right\}, \xi =2.01$, and $r_p=0.004$). The field of view is $2000\times 9.53{\mathrm{mm}}^2$.

Figure 12

(a) Phase diagram and (b) variation of the mean frontal velocity ${\widehat{V}}_{f,\text{av}}$, versus angle of inclination $\beta$, for experiments in set B: ${\varphi }_0=\left\{0.05,0.30,0.50\right\}, \widehat{d}=40\mu \mathrm{m}, {\widehat{\mu }}_f=0.367\mathrm{Pa}\mathrm{s}$, and ${\widehat{u}}_0=0.003{\mathrm{mm}\mathrm{s}}^{-1}$ ($\psi =\left\{0.952,0.767,0.664\right\}, {\text{Re}}_t=\left\{1.37,1.21,0.22\right\}, \xi =2.01$, and $r_p=0.004$). The fitted line in the right inset is given by Eq. (6) with the slope $C_F=0.244$.

Figure 13

Array of experimental images at various angles corresponding to the experiments of set C: $\widehat{d}=70\mu \mathrm{m}, {\widehat{\mu }}_f=0.765\mathrm{Pa}\mathrm{s}, {\widehat{u}}_0=0.004{\mathrm{mm}\mathrm{s}}^{-1}$ for (a) ${\varphi }_0=0.05$ at $\widehat{t}=882\pm 52\mathrm{s}$, (b) ${\varphi }_0=0.30$ at $\widehat{t}=323\pm 42\mathrm{s}$, and (c) ${\varphi }_0=0.50$ at $\widehat{t}=251\pm 18\mathrm{s}$ ($\psi =\left\{0.953,0.770,0.668\right\}, {\text{Re}}_t=\left\{0.66,0.58,0.10\right\}, \xi =1.99$, and $r_p=0.007$). The field of view is $2000\times 9.53{\mathrm{mm}}^2$.

Figure 14

(a) Phase diagram and (b) variation of the mean frontal velocity ${\widehat{V}}_{f,\text{av}}$, versus angle of inclination $\beta$, for experiments in set C: ${\varphi }_0=\left\{0.05,0.30,0.50\right\}, \widehat{d}=70\mu \mathrm{m}, {\widehat{\mu }}_f=0.765\mathrm{Pa}\mathrm{s}$, and ${\widehat{u}}_0=0.004{\mathrm{mm}\mathrm{s}}^{-1}$ ($\psi =\left\{0.953,0.770,0.668\right\}, {\text{Re}}_t=\left\{0.66,0.58,0.10\right\}, \xi =1.99$, and $r_p=0.007$). The fitted line in the right inset is given by Eq. (6) with the slope $C_F=0.385$.

Figure 15

Benchmarking result of pure exchange limit against Ref. [6]: (a) dependency of ${\widehat{V}}_f/{\widehat{V}}_t$ on ${\text{Re}}_{t}cos\beta$ for viscous ($•$), transitionary ($\blacksquare$), and diffusive ($▴$) regimes showing complete agreement with the findings of Ref. [6]. Different colors correspond to $\mathrm{At}=0.0035$ (red), $\mathrm{At}=0.01$ (yellow), and $\mathrm{At}=0.04$ (blue); the oblique dashed line corresponds to ${\widehat{V}}_f/{\widehat{V}}_t=\left[(1/16-1/\left(2{\pi }^2\right)\right]{\text{Re}}_{t}cos\beta$ and the horizontal dashed line corresponds to ${\widehat{V}}_f/{\widehat{V}}_t=0.7$ [6]. (b) Variation of the dimensionless macroscopic diffusion coefficient versus inclination angle $\beta$, for various Atwood numbers At. The fitted curve shown is also suggested by Ref. [23] which is given by Eq. (A2).