Density-stable displacement flow of immiscible fluids in inclined pipes

We study experimentally the isoviscous displacement flow of two immiscible Newtonian fluids in an inclined pipe. The less dense displacing fluid is placed above the denser displaced fluid in a density-stable configuration. The displacing and displaced solutions are oil- and water-based, respectively. The former exhibits nonwetting behavior in the vicinity of the pipe wall, whereas the latter is wetting. The pipe has a small diameter-to-length ratio. The mixing and interpenetration of two fluids have been studied over a wide range of controlling parameters, revealing remarkable results. Compared to the previously studied miscible limit, we observe behavior at the interface between the two fluids where the displaced fluid stays “pinned” to the lower wall of the pipe upon pumping the displacing one. This phenomenon, which is observed over the full range of investigated flow rates, tilt angles, and density contrasts, is associated with the wetting characteristic of the displacing liquid and is also present when light and heavy viscosity mineral oils are used as the displacing fluid. Ultrasonic Doppler velocimetry revealed a segmented velocity profile at the interface of the immiscible fluids. Due to pinning, the efficiency of the removal of displaced fluid in the immiscible limit can be lowered by 14% compared to the miscible case due to the combined effects of the density-stable configuration and the immiscibility of the flow. Within the family of immiscible fluids, the maximum efficiency is achieved at close-to-vertical tilt angles, large density contrasts, and counterintuitively low imposed flow rates, which is of great importance in industrial design.

Figure 1

Schematic view of the experimental setup used. Interface shape is illustrative only.

Figure 2

(a) Immiscible density-stable, (b) immiscible density-unstable, and (c) miscible density-stable displacements. The bottommost image in each panel is a color bar of concentration C, with 0 and 1 referring to the displaced and displacing fluids, respectively. For comparison, panels (a) and (c) are of similar Reynolds number. (a) Snapshots of the displacement flow for $\beta =45{}^{\circ }$, ${\widehat{V}}_0=456.9\mathrm{mm}$ ${\mathrm{s}}^{-1}$, ${\widehat{\rho }}_H=1181\mathrm{kg}$ ${\mathrm{m}}^{-3}$, and ${\widehat{\rho }}_L=918\mathrm{kg}{\mathrm{m}}^{-3}$, at times $\widehat{t}=[$0.42, 0.84, 1.26,..., 2.94, 3.36] s ($\text{At}=-0.125$, $\text{Re}=914$, $\text{Fr}=4.22$, $\text{Ca}=0.65$, $\theta =56{}^{\circ }$). The field of view is 1732 $\times$ $9.53{\mathrm{mm}}^2$. (b) Snapshots of the displacement flow for $\beta =60{}^{\circ }$, ${\widehat{V}}_0=81.4\mathrm{mm}$ ${\mathrm{s}}^{-1}$, ${\widehat{\rho }}_H=1181\mathrm{kg}$ ${\mathrm{m}}^{-3}$, and ${\widehat{\rho }}_L=918\mathrm{kg}$ ${\mathrm{m}}^{-3}$, at times $\widehat{t}$=[0.41, 3.39, 6.38,..., 18.27, 21.25] s ($\text{At}=0.125$, $\text{Re}=163$, $\text{Fr}=0.75$, $\text{Ca}=0.12$, $\theta =56{}^{\circ }$). The field of view is $1950\times 9.53{\mathrm{mm}}^2$. (Figure taken from Hasnain et al. [15].) (c) Snapshots of the displacement flow for $\beta =70{}^{\circ }$, ${\widehat{V}}_0=20.9\mathrm{mm}$ ${\mathrm{s}}^{-1}$, ${\widehat{\rho }}_H=1007\mathrm{kg}$ ${\mathrm{m}}^{-3}$, and ${\widehat{\rho }}_L=999\mathrm{kg}$ ${\mathrm{m}}^{-3}$, at times $\widehat{t}=[$21.5, 24.75, 28.00,..., 34.50] s ($\text{At}=-0.0035$, $\text{Re}=855.2$, $\text{Fr}=0.82$). The field of view is 430 $\times$ 19 ${\mathrm{mm}}^2$, taken 800 mm below the gate valve. (Figure taken from Alba et al. [16].)

Figure 3

Spatiotemporal diagrams of cross-sectional averaged concentration field, ${\displaystyle {\overline{C}}_{\widehat{y}}(\widehat{x},\widehat{t}}$), obtained from the same experiments as shown in Fig. 2. (a) Figure taken from Hasnain et al. [15]. (c) Figure taken from Alba et al. [16],

Figure 4

(a) Experimental profiles of normalized $\widehat{h}(\widehat{x},\widehat{t}$), at times $\widehat{t}=[$1.68, 1.96, 2.24,..., 3.08, 3.36] s and streamwise location, $\widehat{x}$, for immiscible density-stable fluids, measured from the gate valve for the same experiment as shown in Fig. 2, where $\beta ={45}^{\circ }$ and ${\widehat{V}}_0=456.9\mathrm{mm}$ ${\mathrm{s}}^{-1}$. (b) Experimental profiles of normalized $\widehat{h}(\widehat{x},\widehat{t}$) at times $\widehat{t}$ = [68, 74.5,..., 126.5, 133] s and streamwise location, $\widehat{x}$, for miscible density-stable fluids (taken from Alba et al. [16]). (c) Normalized $\widehat{h}(\widehat{x},\widehat{t}$)/$\widehat{D}$ profiles for the same experiment as Fig. 4 plotted against $(\widehat{x}-{\widehat{V}}_0\widehat{t})/\widehat{t}$. (Inset) $\widehat{h}(\widehat{x},\widehat{t}$)/$\widehat{D}$ plotted against $(\widehat{x}-{\widehat{V}}_0\widehat{t})$. (d) Collapse of the normalized $\widehat{h}(\widehat{x},\widehat{t}$)/$\widehat{D}$ profiles with $(\widehat{x}-{\widehat{V}}_0\widehat{t})/\widehat{t}$ (taken from Alba et al. [16]). (Inset) $\widehat{h}(\widehat{x},\widehat{t}$)/$\widehat{D}$ plotted against $(\widehat{x}-{\widehat{V}}_0\widehat{t})$ (e) Schematic representation of pinning and nonpinning behavior in miscible and immiscible density-stable flows, respectively. (f) Experimental evolution of the displacing front velocity value ${\widehat{V}}_f$ with time for the same density-stable experiment as in Fig. 4.

Figure 5

Experimental profiles of $\widehat{h}(\widehat{x},\widehat{t}$)/$\widehat{D}$ for varying values of $\beta$ with ${\widehat{\rho }}_H=998\mathrm{kg}$ ${\mathrm{m}}^{-3}$ and ${\widehat{\rho }}_L=918\mathrm{kg}$ ${\mathrm{m}}^{-3}$ ($\text{At}=-0.042$). (a) ${\widehat{V}}_0\approx$ 94 mm ${\mathrm{s}}^{-1}$ ($\text{Re}\approx 172$). (b) ${\widehat{V}}_0\approx$ 459 mm ${\mathrm{s}}^{-1}$ ($\text{Re}\approx 838$). (c) ${\widehat{V}}_0\approx 909\mathrm{mm}$ ${\mathrm{s}}^{-1}$ ($\text{Re}\approx 1659$). (Inset, left) $\widehat{h}(\widehat{x},\widehat{t}$)/$\widehat{D}$ plotted against similarity parameter ($\widehat{x}-{\widehat{V}}_0\widehat{t}$). (Inset, right) Snapshots of the corresponding displacement flows.

Figure 6

Snapshots of change in isoviscous displacement flow with $\beta$ for immiscible fluids in a density-stable configuration, obtained for ${\widehat{V}}_0\approx$ 475.8 mm ${\mathrm{s}}^{-1}$, ${\widehat{\rho }}_H=998\mathrm{kg}$ ${\mathrm{m}}^{-3}$, and ${\widehat{\rho }}_L=918\mathrm{kg}$ ${\mathrm{m}}^{-3}$, at time $\widehat{t}\approx$ 2.5 s. (At = $-0.042$, $\text{Re}\approx$ 868, $\text{Fr}\approx$ 7.61, $\text{Ca}\approx$ 0.68, $\theta =56{}^{\circ }$) The field of view is $1659\times 9.53$ ${\mathrm{mm}}^2$. The color bar at the top left of the figure shows the corresponding concentration value, C, with 0 referring to the pure displaced fluid and 1 to the pure displacing fluid.

Figure 7

Experimental profiles of $\widehat{h}(\widehat{x},\widehat{t}$)/$\widehat{D}$, for varying values of (a) At and (b) ${\widehat{V}}_0$ for $\beta ={60}^{\circ }$. Arrows indicate direction of increasing $\left|\text{At}\right|$ and ${\widehat{V}}_0$, respectively. (a) Similar ${\widehat{V}}_0$ (${\widehat{V}}_0\approx$ 450.7 mm ${\mathrm{s}}^{-1}$) and different $\text{At}<0$ ($\text{Re}\approx 859$, $\text{Fr}=[7.79,3.86]$, $\text{Ca}\approx$ 0.64, $\theta =56{}^{\circ }$). (b) $\text{At}=-0.125$ with different ${\displaystyle {\widehat{V}}_0}$ = [90.4, 417.6, 862.4] mm ${\mathrm{s}}^{-1}(\text{Re}=[181,835,1724]$, $\text{Fr}=[0.84,7.79,7.97]$, $\text{Ca}=[0.13,0.69,1.23]$, $\theta =56{}^{\circ }$). (Inset, left) $\widehat{h}(\widehat{x},\widehat{t}$)/$\widehat{D}$ plotted against similarity parameter ($\widehat{x}-{\widehat{V}}_0\widehat{t}$). (Inset, right) Snapshots of the corresponding displacement flows.

Figure 8

Front velocity values, ${\widehat{V}}_f$, plotted against the imposed flow velocity, ${\widehat{V}}_0$, for the full range of experiments ($\text{At}=[-0.042\left(•\right),-0.075\left(♦\right),-0.125(\square )],\theta =[52,56,59]{}^{\circ })$. The dashed line is the linear fit ${\widehat{V}}_f=1.22{\widehat{V}}_0$. The solid line represents the ideal displacement case with ${\widehat{V}}_f={\widehat{V}}_0$.

Figure 9

(a) Experimental snapshots for $\beta ={60}^{\circ }$, ${\widehat{V}}_0=893\mathrm{mm}$ ${\mathrm{s}}^{-1}$, ${\widehat{\rho }}_H=998\mathrm{kg}$ ${\mathrm{m}}^{-3}$, and ${\widehat{\rho }}_L=918\mathrm{kg}{\mathrm{m}}^{-3}$ at times $\widehat{t}=[0.18,0.36,0.55,...,1.09]\mathrm{s}$ ($\text{At}=-0.042$, $\text{Re}=1629$, $\text{Fr}=14.29$, $\text{Ca}=1.28$, $\theta ={56}^{\circ }$), with a field of view of $1736\times 9.53$ ${\mathrm{mm}}^2$. The bottommost image is a color bar of concentration $C$, with 0 and 1 referring to the displaced and displacing fluids respectively. (b) Full range of experiments ($\mathrm{At}=[-0.042$, $-0.125]$, $\theta ={56}^{\circ }$), with the presence of a ridged feature marked with ($\blacksquare$) and the absence of a ridged feature marked with ($•$). The dashed line marks $\text{Re}cos\beta /\text{Fr}$ = $0.15\text{Re}-67$, experimentally shown to divide between the presence and absence of a ridge feature.

Figure 10

Snapshots of change in isoviscous displacement flow with $\beta$ for immiscible fluids in a density-stable configuration, obtained for At = 0.125. (a) Silicone oil displacing saltwater with ${\widehat{\rho }}_H=1181\mathrm{kg}$ ${\mathrm{m}}^{-3}$, ${\widehat{\rho }}_L=918\mathrm{kg}$ ${\mathrm{m}}^{-3}$, $\widehat{\mu }=0.005\mathrm{Pa}\mathrm{s}$, $\widehat{\sigma }=3.5\mathrm{mN}$ ${\mathrm{m}}^{-1}$, and $\theta ={56}^{\circ }$ ($\text{Re}\in [835,914]$, $\text{Fr}\in [3.86,4.22]$, $\text{Ca}\in [0.60,0.65]$). From left to right, ${\widehat{V}}_0=417$, 457, 418 mm ${\mathrm{s}}^{-1}$ and $\widehat{t}=1.3$, 1.4, 1.4 s. (b) Light viscosity mineral oil displacing saltwater with ${\widehat{\rho }}_H=1097\mathrm{kg}$ ${\mathrm{m}}^{-3}$, ${\widehat{\rho }}_L=853\mathrm{kg}$ ${\mathrm{m}}^{-3}$, $\widehat{\mu }=0.018\mathrm{Pa}\mathrm{s}$, $\widehat{\sigma }=10.1\mathrm{mN}$ ${\mathrm{m}}^{-1}$, and $\theta ={52}^{\circ }$ ($\text{Re}\in [220,258],\text{Fr}\in [3.94,4.63],\text{Ca}\in [0.76,0.89]$). From left to right, ${\widehat{V}}_0=426$, 487, 500 mm ${\mathrm{s}}^{-1}$ and $\widehat{t}=1.4$, 1.2, 1.1 s. (c) Heavy viscosity mineral oil displacing saltwater with ${\widehat{\rho }}_H=1110\mathrm{kg}$ ${\mathrm{m}}^{-3}$, ${\widehat{\rho }}_L=864\mathrm{kg}$ ${\mathrm{m}}^{-3}$, $\widehat{\mu }=0.129\mathrm{Pa}\mathrm{s}$, $\widehat{\sigma }=33.4\mathrm{mN}$ ${\mathrm{m}}^{-1}$, and $\theta ={59}^{\circ }$ ($\text{Re}\in [12,18]$, $\text{Fr}\in [1.51,2.30]$, $\text{Ca}\in [0.63,0.96]$). From left to right, ${\widehat{V}}_0=249$, 247, 162 mm ${\mathrm{s}}^{-1}$ and $\widehat{t}=1.1$, 1.6, 1.7 s. The field of view is $1903\times 9.53$ ${\mathrm{mm}}^2$.

Figure 11

(a) Experimental snapshots for $\beta ={75}^{\circ }$, ${\widehat{V}}_0=426\mathrm{mm}$ ${\mathrm{s}}^{-1}$, ${\widehat{\rho }}_H=1181\mathrm{kg}$ ${\mathrm{m}}^{-3}$, and ${\widehat{\rho }}_L=918\mathrm{kg}$ ${\mathrm{m}}^{-3}$ at different times $\widehat{t}=[1.4,1.9,2.4,2.9]\mathrm{s}$ ($\text{At}=-0.125$, $\text{Re}=853$, $\text{Fr}=3.94$, $\text{Ca}=0.61$, $\theta ={56}^{\circ }$). The field of view is $1692\times 9.53{\mathrm{mm}}^2$. The bottommost image is a color bar of concentration $C$, with 0 and 1 referring to the displaced and displaced fluids, respectively. Each snapshot is assigned a letter B to E. The representative UDV profiles to these flows are plotted in panels (b) to (e). (b) Velocity profile averaged over $\widehat{t}$ = [1.4, 1.9] s, when only the displaced densified water flows under the UDV probe. (c) Velocity profile at $\widehat{t}\approx$ 1.82 s, the moment at which the front arrives at the probe. (d) Velocity profile averaged over $\widehat{t}$ = [1.9, 2.4] s, as the interface leaves the probe area. ($e$) Velocity profile averaged over $\widehat{t}$ = [2.4, 2.9] s, when displacing oil has begun to dominate the flow profile. The dotted lines in panels (b) to (e) for $\widehat{y}\lesssim 1$ are guides to the eye correcting the UDV refraction errors close to the lower wall.