6 results
The absolute instability of an inviscid compound jet
- ANUJ CHAUHAN, CHARLES MALDARELLI, DEMETRIOS T. PAPAGEORGIOU, DAVID S. RUMSCHITZKI
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- Journal of Fluid Mechanics / Volume 549 / 25 February 2006
- Published online by Cambridge University Press:
- 08 February 2006, pp. 81-98
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This paper examines the emergence of the absolute instability from convectively unstable states of an inviscid compound jet. A compound jet is composed of a cylindrical jet of one fluid surrounded by a concentric annulus of a second, immiscible fluid. For all jet velocities $v$, there are two convectively unstable modes. As in the single-fluid jet, the compound jet becomes absolutely unstable below a critical dimensionless velocity or Weber number $V ({:=}\,\sqrt{v^2\,{\rho_1 R_1}/\sigma_1}$ where $\rho_{1}$, $R_{1}$ and $\sigma _{1}$ are the core density, radius and core–annular interfacial tension), which is a function of the annular/core ratios of densities $\beta$, surface tensions $\gamma$ and radii $a$. At $V\,{=}\,0$, the absolutely unstable modes and growth recover the fastest growing temporal waves. We focus specifically on the effect of $\gamma$ at $a\,{=}\,2$ and $\beta\,{=}\,1$ and find that when the outer tension is significantly less than the inner $(0.1\,{<}\,\gamma\,{<}\,0.3)$, the critical Weber number $V_{\hbox{\scriptsize{\it crit}}}$ decreases with <$\gamma$, whereas for higher ratios $(0.3\,{<}\,\gamma\,{<}\,3)$ it increases. The values (1.2–2.3) of $V_{\hbox{\scriptsize{\it crit}}}$ for the compound jet include the parameter-independent critical value of 1.77 for the single jet. Therefore, increasing the outer tension can access the absolute instability at higher dimensional velocities than for a single jet with the same radius and density as the core and a surface tension equal to the compound jet's liquid–liquid tension. We argue that this potentially facilitates distinguishing experimentally between absolute and convective instabilities because higher velocities and surface tension ratios higher than 1 extend the breakup length of the convective instability. In addition, for $0.3\,{<}\,\gamma\,{<}\,1.16$, the wavelength for the absolute instability is roughly half that of the fastest growing convectively unstable wave. Thus choosing $\gamma$ in this range exaggerates its distinction from the convective instability and further aids the potential observation of absolute instability.
The effects of insoluble surfactants on the linear stability of a core–annular flow
- HSIEN-HUNG WEI, DAVID S. RUMSCHITZKI
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- Journal:
- Journal of Fluid Mechanics / Volume 541 / 25 October 2005
- Published online by Cambridge University Press:
- 11 October 2005, pp. 115-142
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The effects of an insoluble surfactant on the linear stability of a two-fluid core–annular flow in the thin annulus limit, for axisymmetric disturbances with wavelengths large compared to the annulus thickness, $h_{0}$, are the focus of this investigation. A base shear flow affects the interfacial surfactant distribution, thereby inducing Marangoni forces that, along with capillary forces, affect the fluid–fluid interface stability. The resulting system's stability differs markedly from that of the same system with zero base flow. In the thin-annulus limit (the ratio $\varepsilon$ of the undisturbed annulus thickness to core radius tends to zero), common in applications, a scaling and asymptotic analysis yields a coupled set of equations for the perturbed fluid–fluid interface shape and surfactant concentration. The linear dynamics of the annular film fully determine these equations, i.e. the core dynamics are slaved to the film dynamics. The theory provides a unified view of the mechanism of stability in three different regimes of capillary number ${\it Ca}$ (defined as the product of the core viscosity, $\mu _1$, and the centreline velocity, $W_0 $, divided by the interface tension, $\sigma _0^\ast$, that corresponds to an undisturbed (signified by the subscript 0) uniform surfactant concentration, $\Gamma _0^\ast$). In the absence of a base flow or in the limit of small ${\it Ca}({\ll}\varepsilon ^{2})$, Marangoni forces deriving from non-uniformities in the interface concentration of insoluble surfactants oppose the net capillary forces. These latter forces normally stabilize the longitudinal curvature and destabilize the circumferential curvature of perturbations to the interface. In the limit of large ${\it Ca}({\gg}\varepsilon ^{2})$, Marangoni forces destabilize disturbances with wavelengths that are large compared to the annulus thickness. For moderately small ${\it Ca}({\sim} \varepsilon^2)$, increasing the Marangoni number Ma (defined as the product of $(-\partial\sigma^\ast/\partial \Gamma^\ast )_0$ and $\Gamma_0^\ast $, divided by $\mu _1 W_0 )$ from zero increases the growth rates of all disturbances (with wavelengths ${\gg}h_{0})$ and, consequently, reduces the marginal wavelength below that typical of the capillary instability. However a further increase in Ma eventually reverses these trends. A very large value for Ma stiffens the interface, which opposes any local variation of the tangential velocity along the interface, and this is true whether or not there is a base flow. In the limit of infinite Ma, the growth rate of the instability is 1/4 of that of the clean interface and the marginal wavenumber, non-dimensionalized by the undisturbed core circumference, returns to its clean interface (capillary) value of 1. All trends are explained physically.
The linear stability of a core–annular flow in an asymptotically corrugated tube
- HSIEN-HUNG WEI, DAVID S. RUMSCHITZKI
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- Journal of Fluid Mechanics / Volume 466 / 10 September 2002
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- 12 September 2002, pp. 113-147
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This paper examines the core–annular flow of two immiscible fluids in a straight circular tube with a small corrugation, in the limit where the ratio ε of the mean undisturbed annulus thickness to the mean core radius and the corrugation (characterized by the parameter σ) are both asymptotically small and where the surface tension is small. It is motivated by the problems of liquid–liquid displacement in irregular rock pores such as occur in secondary oil recovery and in the evolution of the liquid film lining the bronchii in the lungs whose diameters vary over different generations of branching. We investigate the asymptotic base flow in this limit and consider the linear stability of its leading order (in the corrugation parameter) solution. For the chosen scalings of the non-dimensional parameters the core's base flow slaves that of the annulus. The equation governing the leading-order interfacial position for a given wall corrugation function shows a competition between shear and capillarity. The former tends to align the interface shape with that of the wall and the latter tends to introduce a phase shift, which can be of either sign depending on whether the circumferential or the longitudinal component of capillarity dominates. The asymptotic linear stability of this leading-order base flow reduces to a single partial differential equation with non-constant coefficients deriving from the non-uniform base flow for the time evolution of an interfacial disturbance. Examination of a single mode k wall function allows the use of Floquet theory to analyse this equation. Direct numerical solutions of the above partial differential equation agree with the predictions of the Floquet analysis. The resulting spectrum is periodic in α- space, α being the disturbance wavenumber space. The presence of a small corrugation not only modifies (at order σ2) the primary eigenvalue of the system. In addition, short-wave order-one disturbances that would be stabilized flowing to capillarity in the absence of corrugation can, in the presence of corrugation and over time scales of order ln(1/σ), excite higher wall harmonics (α±nk) leading to the growth of unstable long waves. Similar results obtain for more complicated wall shape functions. The main result is that a small corrugation makes a core–annular flow unstable to far more disturbances than would destabilize the same uncorrugated flow system. A companion paper examines that competition between this added destabilization due to pore corrugation with the wave steepening and stabilization in the weakly nonlinear regime.
The weakly nonlinear interfacial stability of a core–annular flow in a corrugated tube
- HSIEN-HUNG WEI, DAVID S. RUMSCHITZKI
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- Journal:
- Journal of Fluid Mechanics / Volume 466 / 10 September 2002
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- 12 September 2002, pp. 149-177
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A core–annular flow, the concurrent axial flow of two immiscible fluids in a circular tube or pore with one fluid in the core and the other in the wetting annular region, is frequently used to model technologically important flows, e.g. in liquid–liquid displacements in secondary oil recovery. Most of the existing literature assumes that the pores in which such flows occur are uniform circular cylinders, and examine the interfacial stability of such systems as a function of fluid and interfacial properties. Since real rock pores possess a more complex geometry, the companion paper examined the linear stability of core–annular flows in axisymmetric, corrugated pores in the limit of asymptotically weak corrugation. It found that short-wave disturbances that were stable in straight tubes could couple to the wall's periodicity to excite unstable long waves. In this paper, we follow the evolution of the axisymmetric, linearly unstable waves for fluids of equal densities in a corrugated tube into the weakly nonlinear regime. Here, we ask whether this continual generation of new disturbances by the coupling to the wall's periodicity can overcome the nonlinear saturation mechanism that relies on the nonlinear (kinematic-condition-derived) wave steepening of the Kuramoto–Sivashinsky (KS) equation. If it cannot, and the unstable waves still saturate, then do these additional excited waves make the KS solutions more likely to be chaotic, or does the dispersion introduced into the growth rate correction by capillarity serve to regularize otherwise chaotic motions?
We find that in the usual strong surface tension limit, the saturation mechanism of the KS mechanism remains able to saturate all disturbances. Moreover, an additional capillary-derived nonlinear term seems to favour regular travelling waves over chaos, and corrugation adds a temporal periodicity to the waves associated with their periodical traversing of the wall's crests and troughs. For even larger surface tensions, capillarity dominates over convection and a weakly nonlinear version of Hammond's no-flow equation results; this equation, with or without corrugation, suggests further growth. Finally, for a weaker surface tension, the leading-order base flow interface follows the wall's shape. The corrugation-derived excited waves appear able to push an otherwise regular travelling wave solution to KS to become chaotic, whereas its dispersive properties in this limit seem insufficiently strong to regularize chaotic motions.
Marangoni effects on the motion of an expanding or contracting bubble pinned at a submerged tube tip
- HARRIS WONG, DAVID RUMSCHITZKI, CHARLES MALDARELLI
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- Journal:
- Journal of Fluid Mechanics / Volume 379 / 25 January 1999
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- 25 January 1999, pp. 279-302
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This work studies the motion of an expanding or contracting bubble pinned at a submerged tube tip and covered with an insoluble Volmer surfactant. The motion is driven by constant flow rate Q into or out of the tube tip. The purpose is to examine two central assumptions commonly made in the bubble and drop methods for measuring dynamic surface tension, those of uniform surfactant concentration and of purely radial flow. Asymptotic solutions are obtained in the limit of the capillary number Ca→0 with the Reynolds number Re=o(Ca−1, non-zero Gibbs elasticity (G), and arbitrary Bond number (Bo). (Ca=μQ/a2σc, where μ is the liquid viscosity, a is the tube radius, and σc is the clean surface tension.) This limit is relevant to dynamic-tension experiments, and gives M→∞, where M=G/Ca is the Marangoni number. We find that in this limit the deforming bubble at each instant in time takes the static shape. The surfactant distribution is uniform, but its value varies with time as the bubble area changes. To maintain a uniform distribution at all times, a tangential flow is induced, the magnitude of which is more than twice that in the clean case. This is in contrast to the surface-immobilizing effect of surfactant on an isolated translating bubble. These conclusions are confirmed by a boundary integral solution of Stokes flow valid for arbitrary Ca, G and Bo. The uniformity in surfactant distribution validates the first assumption in the bubble and drop methods, but the enhanced tangential flow contradicts the second.
Theory and experiment on the low-Reynolds-number expansion and contraction of a bubble pinned at a submerged tube tip
- HARRIS WONG, DAVID RUMSCHITZKI, CHARLES MALDARELLI
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- Journal:
- Journal of Fluid Mechanics / Volume 356 / 10 February 1998
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- 10 February 1998, pp. 93-124
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The expansion and contraction of a bubble pinned at a submerged tube tip and driven by constant gas flow rate Q are studied both theoretically and experimentally for Reynolds number Re[Lt ]1. Bubble shape, gas pressure, surface velocities, and extrapolated detached bubble volume are determined by a boundary integral method for various Bond (Bo=ρga2/σ) and capillary (Ca=μQ/σa2) numbers, where a is the capillary radius, ρ and μ are the liquid density and viscosity, σ is the surface tension, and g is the gravitational acceleration.
Bubble expansion from a flat interface to near detachment is simulated for a full range of Ca (0.01–100) and Bo (0.01–0.5). The maximum gas pressure is found to vary almost linearly with Ca for 0.01[les ]Ca[les ]100. This correlation allows the maximum bubble pressure method for measuring dynamic surface tension to be extended to viscous liquids. Simulated detached bubble volumes approach static values for Ca[Lt ]1, and asymptote as Q3/4 for Ca[Gt ]1, in agreement with analytic predictions. In the limit Ca→0, two singular time domains are identified near the beginning and the end of bubble growth during which viscous and capillary forces become comparable.
Expansion and contraction experiments were conducted using a viscous silicone oil. Digitized video images of deforming bubbles compare well with numerical solutions. It is observed that a bubble contracting at high Ca snaps off.