^{1}, H. Ding

^{1}, P. Valluri

^{1}and O. K. Matar

^{1,a)}

### Abstract

We study the effect of buoyancy on pressure-driven flow of two miscible fluids in inclined channels via direct numerical simulations. The flowdynamics are governed by the continuity and Navier–Stokes equations, without the Boussinesq approximation, coupled to a convective-diffusion equation for the concentration of the more viscous fluid through a concentration-dependent viscosity and density. The effect of varying the density ratio, Froude number, and channel inclination on the flowdynamics is examined, for moderate Reynolds numbers. We present results showing the spatiotemporal evolution of the flow together with an integral measure of mixing.

We acknowledge support from the Engineering and Physical Sciences Research Council (through Grant No. EP/D503051/1), and the Department of Trade and Industry of the U.K.

I. INTRODUCTION

II. FORMULATION

A. Governing equations

B. Numerical procedure

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Viscosity
- 44.0
- Flow instabilities
- 31.0
- Core annular flows
- 21.0
- Solubility
- 15.0
- Channel flows
- 14.0

## Figures

Schematic showing the initial flow configuration: fluid 1 occupies the entire channel, inclined at an angle to the horizontal, and is about to be displaced by fluid 2.

Schematic showing the initial flow configuration: fluid 1 occupies the entire channel, inclined at an angle to the horizontal, and is about to be displaced by fluid 2.

Mass fraction of the displaced fluid (a) and temporal evolution of the position of the leading front separating the two fluids, , (b) obtained using different mesh densities for , , , , , and : The dotted lines in panels (a) and (b) represent the limiting case given by and , respectively. This corresponds to the case wherein fluid 1 is displaced by fluid 2 in plug flow and prior to the sharp, vertical interface separating the two fluids exiting the channel.

Mass fraction of the displaced fluid (a) and temporal evolution of the position of the leading front separating the two fluids, , (b) obtained using different mesh densities for , , , , , and : The dotted lines in panels (a) and (b) represent the limiting case given by and , respectively. This corresponds to the case wherein fluid 1 is displaced by fluid 2 in plug flow and prior to the sharp, vertical interface separating the two fluids exiting the channel.

Spatiotemporal evolution of the concentration contours for the same parameters as those used to generate Fig. 2.

Spatiotemporal evolution of the concentration contours for the same parameters as those used to generate Fig. 2.

Evolution of the vorticity (top panel) and velocity vectors (bottom panel) for the same parameter values as those used to generate Fig. 3. The inset at the bottom, at , represents an enlarged view of the velocity vectors for .

Evolution of the vorticity (top panel) and velocity vectors (bottom panel) for the same parameter values as those used to generate Fig. 3. The inset at the bottom, at , represents an enlarged view of the velocity vectors for .

Effect of channel inclination, , on the mass fraction of the displaced fluid 1 (a), the temporal evolution of the position of the leading front separating the two fluids (b) for two different viscosity contrasts (solid lines: , dashed lines: ); variation of the velocity of the leading front in a moving reference frame with , (c), for : The rest of the parameter values are , , , and . The dotted lines in panels (a) and (b) are the analogs of those shown in Figs. 2(a) and 2(b), respectively. The dashed line in panel (c) is a best fit: .

Effect of channel inclination, , on the mass fraction of the displaced fluid 1 (a), the temporal evolution of the position of the leading front separating the two fluids (b) for two different viscosity contrasts (solid lines: , dashed lines: ); variation of the velocity of the leading front in a moving reference frame with , (c), for : The rest of the parameter values are , , , and . The dotted lines in panels (a) and (b) are the analogs of those shown in Figs. 2(a) and 2(b), respectively. The dashed line in panel (c) is a best fit: .

The effect of inclination angle, , on the concentration contours at , for : The rest of the parameter values remain unchanged from those used to generate Fig. 5.

The effect of inclination angle, , on the concentration contours at , for : The rest of the parameter values remain unchanged from those used to generate Fig. 5.

Streamwise variation of the depth-averaged concentration, , for the same parameters as those used to generate Fig. 6. Here, the -coordinate has been rescaled to .

Streamwise variation of the depth-averaged concentration, , for the same parameters as those used to generate Fig. 6. Here, the -coordinate has been rescaled to .

Transverse variation of the axial-averaged concentration, , for the same parameters as those used to generate Fig. 6.

Transverse variation of the axial-averaged concentration, , for the same parameters as those used to generate Fig. 6.

Effect of density ratio, , and Froude number, Fr, on the mass fraction of the displaced fluid 1 [(a) and (c)], the temporal evolution of the position of the leading front separating the two fluids [(b) and (d)]: The rest of the parameter values remain unchanged from those used to generate Fig. 2. The dotted lines in (a) and (c) are analogs of those shown in Figs. 2(a) and 5(a), and the dotted lines in (b) and (d) are analogs of those shown in Figs. 2(b) and 5(b). The lines associated with the “no gravity” legend correspond to the case wherein the gravitational terms in the governing equations have been neglected.

Effect of density ratio, , and Froude number, Fr, on the mass fraction of the displaced fluid 1 [(a) and (c)], the temporal evolution of the position of the leading front separating the two fluids [(b) and (d)]: The rest of the parameter values remain unchanged from those used to generate Fig. 2. The dotted lines in (a) and (c) are analogs of those shown in Figs. 2(a) and 5(a), and the dotted lines in (b) and (d) are analogs of those shown in Figs. 2(b) and 5(b). The lines associated with the “no gravity” legend correspond to the case wherein the gravitational terms in the governing equations have been neglected.

Effect of channel inclination, , on the mass fraction of the displaced fluid 1 (a), the temporal evolution of the position of the leading front separating the two fluids (b) for two different density contrasts (solid lines: , dashed lines: ), variation of the velocity of the leading front, in a moving reference frame with (c) for : The rest of the parameter values remain unchanged from those used to generate Fig. 6. The dotted line in (a) is the analog of those shown in Figs. 2(a), 5(a), 9(a), and 9(c); and the dotted line in (b) is the analog of those shown in Figs. 2(b), 5(b), 9(b), and 9(d). The dashed line in panel (c) is a best fit: .

Effect of channel inclination, , on the mass fraction of the displaced fluid 1 (a), the temporal evolution of the position of the leading front separating the two fluids (b) for two different density contrasts (solid lines: , dashed lines: ), variation of the velocity of the leading front, in a moving reference frame with (c) for : The rest of the parameter values remain unchanged from those used to generate Fig. 6. The dotted line in (a) is the analog of those shown in Figs. 2(a), 5(a), 9(a), and 9(c); and the dotted line in (b) is the analog of those shown in Figs. 2(b), 5(b), 9(b), and 9(d). The dashed line in panel (c) is a best fit: .

The effect of inclination angle, , on the concentration contours at , for : The rest of the parameter values remain unchanged from those used to generate Fig. 10.

The effect of inclination angle, , on the concentration contours at , for : The rest of the parameter values remain unchanged from those used to generate Fig. 10.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content