The formation of drops through viscous instability

We investigate whether barred galaxies are statistically more likely to harbour radial molecular gas flows and what effect those flows have on their global properties. Using 46 galaxies from the ALMA-MaNGA QUEnching and STar formation (ALMaQUEST) survey, we identify galaxies hosting optical bars using a combination of the morphological classifications in Galaxy Zoo 2 and HyperLEDA. In order to detect radial molecular gas flows, we employ full 3D kinematic modelling of the ALMaQUEST 12CO(1–0) data cubes. By combining our bar classifications with our radial bar-driven flow detections, we find that galaxies classed as barred are statistically more likely to host large-scale radial gas motions compared to their un-barred and edge-on galaxy counterparts. Moreover, the majority of barred galaxies require multicomponent surface brightness profiles in their best-fitting models, indicative of the presence of resonance systems. We find that galaxies classed as barred with radial bar-driven flows (‘barred + radial flow’ subset) have significantly suppressed global star-formation efficiencies compared to barred galaxies without radial bar-driven flows and galaxies in the other morphological sub-samples. Our ‘barred + radial flow’ subset galaxies also possess consistently centrally concentrated molecular gas distributions, with no indication of depleted gas mass fractions, suggesting that gas exhaustion is not the cause of their suppressed star formation. Furthermore, these objects have higher median gas mass surface densities in their central 1 kpc, implying that central gas enhancements do not fuel central starbursts in these objects. We propose that dynamical effects, such as shear caused by large-scale inflows of gas, act to gravitationally stabilize the inner gas reservoirs.

I In part I of this thesis similarity solutions to the equations of three phase flow through porous media are examined. The three phases are water, steam, and a noncondensing phase, most likely oil. The main purpose of analyzing such flows is to gain understanding of the steam flooding of oil fields. Provided steam is being injected at a higher pressure than the initial field pressure, it is shown that there will always be at least two saturation shocks. As one increases the pressure of the injected steam several regimes are encountered; first the flow develops a region where all the steam is completely condensed, then the position of two of the shocks are interchanged, and finally one of the shocks grows weaker and is eventually replaced by an expansion fan. In sections 12 and 13 the stability of steadily moving condensation fronts in porous media is analyzed. For one special problem it is found that the sign of the jump in pressure gradient at the interface determines whether the interfaces are stable or unstable. This result is applied with some caution to the similarity solutions found in the earlier sections. II Recently McLean analyzed the shapes of fingers in a Hele-Shaw cell, including the effects of surface tension. His work resolved the question of the uniqueness of the shapes first brought up by Saffman and Taylor in their analysis that did not include surface tension. It is however felt that there are still unresolved problems. In determining the pressure jump across an interface there are two principal radii of curvature. McLean only took into account the effect of the larger of these, assuming that the other was constant along the outline of the finger. Unless the smaller radius is very nearly constant, it should in fact give a larger contribution to the jump in pressure. In this thesis the effect of this smaller radius of curvature is modelled by assuming that it is a function of the normal velocity of the mean two dimensional surface of the finger. It is found that if one only takes into account the smaller radius of curvature, the problem is not uniquely determined, as in the case with no surface tension at all. When both radii of curvature are taken into account, the effect of the smaller radius of curvature is to modify the finger shapes in a way that is qualitatively in agreement with experimental data. Also, McLean's results are checked by an independent numerical scheme, and the results are found to be in excellent agreement. Using both methods of solution a second solution branch other than that described by McLean was also found.

If a less viscous fluid displaces a more viscous one in a porous medium or Hele-Shaw cell, the unfavorable mobility gradient is known to cause the classical ’viscous fingering instability’. In the past, this instability has been investigated mostly on the basis of Darcy’s law, which involves certain averaging procedures that are usually not valid for the case of a Hele-Shaw displacement. In this diploma thesis, miscible fingering in a horizontal Hele-Shaw cell is studied by means of Stokes simulations and linear stability analysis. The difference from former studies is that a nonmonotonic concentration viscosity profile is used which can exist in nature for specific fluid pairs. Here, particular emphasis is placed on identifying parameter regimes in terms of the Peclet-number Pe, the viscosity contrast R, the value and position of the viscosity maximum μm and cm. Λ is an additional parameter depending on the end-point derivatives. The existing nonlinear Stokes code for monotonic exponential viscosity profiles developed by N. Goyal was modified to incorporate any general viscosity concentration functional relationship. These twodimensional simulations lead to a moving finger with a quasisteady state near the tip of the displacement front for sufficiently large Peclet number and high viscosity ratios. The front thickness d0 and the tip velocity vtip of the quasisteady state is primarily affected by the Peclet number Pe and the viscosity contrast R. The scale of d0 with Pe−1/2 which is found for the monotonic profile is deviating for the nonmonotonic profile. A higher value of the maximum of the nonmonotonic viscosity profile leads to a slower tip and thicker front, while a shift of the maximum from the less viscous fluid to the more viscous fluid leads to a increase of the tip velocity and a decrease of the front thickness. In a second step the results of the direct numerical Stokes flow simulation are used to examine the stability of the quasisteady front to spanwise perturbations. In the linear stability analysis the viscosity contrast R is the dominant factor that affects the stability of the flow. Manickam&Homsy [10] investigated that for the Darcy flow a negative Λ leads to a more stable configuration than a positive Λ, while we see a reversal of this behavior. Only higher values of Λ lead to strong changes for the growth rate σmax of the most dangerous mode and the corresponding wavenumber βmax. Because the shape of the profile is also changing a lot with high values of Λ it is really hard to figure out what leads to this change. The exact reason could not be identified and would need further investigation.

Viscous fingering instabilities, common in confined environments such as porous media or Hele-Shaw cells, surprisingly also occur in unconfined, non-porous settings as revealed by recent experiments. These novel instabilities involve free-surface flows of dissimilar viscosity. We demonstrate that such a free-surface flow, involving a thin film of viscous fluid spreading over a substrate that is prewetted with a fluid of higher viscosity, is susceptible to a similar type of novel viscous fingering instability. Such flows are relevant to a range of geophysical, industrial and physiological applications from the small scales of thin-film coating applications and nasal drug delivery to the large scales of lava flows. In developing a theoretical framework, we assume that the intruding layer and the liquid film over which it flows are both long and thin, the effects of inertia and surface tension are negligible, and both layers are driven by gravity and resisted by viscous shear stress so that the principles of lubrication theory hold. We investigate the stability of axisymmetric similarity solutions, describing the base flow, by examining the growth of small-amplitude non-axisymmetric perturbations. We characterise regions of instability across parameter space and find that these instabilities emerge above a critical viscosity ratio. That is, a fluid of low viscosity intruding into another fluid of sufficiently high viscosity is susceptible to instability, akin to traditional viscous fingering in a porous medium. We identify the mechanism of instability, compare with other frontal instabilities and demonstrate that high enough density differences suppress the instability completely.

The simplified mathematical model for two-phase displacement in a Hele–Shaw cell, i.e., slow flow between closely spaced parallel plates, has been extended to include effects arising from a constant rate of rotation of the system. A linear stability analysis for an initially concentric circular drop reveals that it is unstable both to translation and, depending on the rotation rate, to a number of fingering modes. The nonlinear evolution with time of two of these modes is traced using a boundary-integral numerical scheme. Ultimately, the liquid mass will break up into a number of drops. Apart from small distortion due to the Coriolis force, the patterns formed are essentially independent of both the liquid viscosity and the cell plate spacing or porous-medium permeability. Potential application areas include certain aspects of coating flows, mold filling, and a class of liquid-filled spinning projectiles with permeable packing.

The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell

Heat transfer induced by buoyancy from a pipe buried in a semi-infinite porous medium with a superimposed fluid layer has been numerically examined in this study. Due to the complexity involved, finite difference method along with body-fitted coordinate systems has been employed. The Brinkman-extended Darcy equations are used to model flow in the porous medium while Navier-Stokes equations are used for the fluid layer. The conditions applied at the interface between the fluid and porous layers are the continuity of temperature, heat flux, normal and tangential velocity, shear stress and pressure. A parametric study has been performed to investigate the effects of Rayleigh number, Prandtl number, Darcy number, and fluid layer thickness on the flow patterns and heat transfer rates. The results show that heat transfer increases with the Rayleigh number, but the convective strength decreases with the Darcy number. The heat transfer rate is smaller when the superimposed fluid is air instead of water. For a porous layer with Da ≤ 0.0005 and an overlaying fluid layer thickness of L/ri ≥ 1, convection is initiated in the fluid layer and it may develop into multiple recirculating cells at a moderate Rayleigh number (i.e., Ra ≤ 104), and may further develop into a single cell at a higher Rayleigh number of 105.

A thorough understanding of the microscopic flow process in porous and fractured media is significant for oil and gas development, geothermal energy extraction and subsurface CO 2 storage etc. In CO 2 geological sequestration, the CO 2 is often injected at the supercritical state (scCO 2 ), which will displace the connate fluids in the pore spaces during the drainage process. However, when CO 2 injection stops, the connate brine or water flows back to displace the scCO 2 . Therefore, the configuration of migration paths in a specific reservoir plays a significant role in affecting the connectivity and storage efficiency of scCO 2 . In this paper, the two-phase (scCO 2 and water) boundary has been defined using the phase field method, and the COMSOL Multiphysics simulator is applied to study the migration of scCO 2 in porous/fractured media at the pore scale. The geological conditions of Shiqianfeng formation in the CO 2 capture and storage pilot site of the Ordos Basin in China is selected as the engineering background. Before using the actual microscopic geometry based on thin-section of Shiqianfeng sandstone, we get the general understanding on scCO 2 migration in fractured porous media that has the highly simplified configuration with circular particles, considering the impacts of wettability, geometry of formation mineral grains, interfacial tension, injection rates, and fracture geometry. Results show that the CO 2 preferential flow occurs at locations with high CO 2 flow rates and high CO 2 pore pressure. The preferential flow of scCO 2 occurs adjacent to the wall of grains while minimal or little flow takes place through the interior between the grains, considering the grains with irregular shapes. The interfacial tension of porous media plays a significant role in controlling the spatial distribution of the scCO 2 . A much lower interfacial tension results in a much thinner scCO 2 flow band with a much higher saturation. The geometry of fractures in porous media increases the complexity of the scCO 2 flow paths at the pore scale. Cited as : Liu, H., Zhu, Z., Patrick, W., Liu, J., Lei, H., Zhang, L. Pore-scale numerical simulation of supercritical CO 2 migration in porous and fractured media saturated with water. Advances in Geo-Energy Research, 2020, 4(4): 419-434, doi: 10.46690/ager.2020.04.07

The Saffman--Taylor finger problem is to predict the shape and, in particular, width of a finger of fluid travelling in a Hele--Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem sufficiently difficult that asymptotic or numerical methods must be used. We adapt numerical methods used to solve the Saffman--Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele--Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as for the surface tension problem. References P. G. Saffman and G. Taylor. The penetration of a fluid into a porous medium or Hele--Shaw cell containing a more viscous liquid. Proc. R. Soc. London, Ser. A , 245:312--329, 1958. doi:10.1098/rspa.1958.0085 J. W. McLean and P. G. Saffman. The effect of surface tension on the shape of fingers in a Hele--Shaw cell. J. Fluid. Mech. , 102:455--469, 1981. doi:10.1017/S0022112081002735 J. M. Vanden-Broeck. Fingers in a Hele--Shaw cell with surface tension. Phys. Fluids , 26(8):2033--2034, 1983. doi:10.1063/1.864406 S. J. Chapman. On the role of Stokes lines in the selection of Saffman--Taylor fingers with small surface tension. Eur. J. Appl. Math. , 10(06):513--534, 1999. doi:10.1017/S0956792599003848 R. Combescot, T. Dombre, V. Hakim, Y. Pomeau, and A. Pumir. Shape selection of Saffman--Taylor fingers. Phys. Rev. Lett. , 56(19):2036--2039, May 1986. doi:10.1103/PhysRevLett.56.2036 R. Combescot, V. Hakim, T. Dombre, Y. Pomeau, and A. Pumir. Analytic theory of the Saffman--Taylor fingers. Phys. Rev. A , 37(4):1270--1283, February 1988. doi:10.1103/PhysRevA.37.1270 M. Reissig, S. V. Rogosin, and F. Hubner. Analytical and numerical treatment of a complex model for Hele--Shaw moving boundary value problems with kinetic undercooling regularization. Eur. J. Appl. Math. , 10(6):561--579, 1999. doi:10.1017/S0956792599003939 S. J. Chapman and J. R. King. The selection of Saffman--Taylor fingers by kinetic undercooling. J. Eng. Math. , 46(1):1--32, 2003. doi:10.1023/A:1022860705459 D. Bensimon. Stability of viscous fingering. Phys. Rev. A , 33(2):1302--1309, 1986. doi:10.1103/PhysRevA.33.1302 D. A. Kessler and H. Levine. Theory of the Saffman--Taylor ``finger'' pattern. {I}. Phys. Rev. A , 33(4):2621--2633, 1986. doi:10.1103/PhysRevA.33.2621 D. A. Kessler and H. Levine. Theory of the Saffman--Taylor ``finger'' pattern. {II}. Phys. Rev. A , 33(4):2634--2639, 1986. doi:10.1103/PhysRevA.33.2634 A. J. Degregoria and L. W. Schwartz. A boundary integral method for two-phase displacement in Hele--Shaw cells. J. Fluid Mech. , 164:383--400, 1986. doi:10.1017/S0022112086002604 T. Y. Hou, Z. Li, S. Osher, and H. Zhao. A hybrid method for moving interface problems with application to the Hele--Shaw flow. J. Comput. Phys. , 134(2):236--252, 1997. doi:10.1006/jcph.1997.5689 G. Tryggvason and H. Aref. Numerical experiments on Hele--Shaw flow with a sharp interface. J. Fluid Mech. , 136:1--30, 1983. doi:10.1017/S0022112083002037 N. Whitaker. Some numerical methods for the Hele--Shaw equations. J. Comput. Phys. , 111(1):81--88, 1994. doi:10.1006/jcph.1994.1046

We conducted an experimental investigation to examine the immiscible radial displacement flows of air invading three-dimensional foam in a Hele-Shaw cell. Our study successfully identified three distinct flow regimes. In the initial regime, characterized by relatively low fingertip velocities, the foam underwent a slow displacement through plug flow. During this process, the three-phase contact lines slipped at the cell walls. Notably, we discovered that the air injection pressure exhibited a proportional relationship with the power of the fingertip velocity. This relationship demonstrated excellent agreement with a power law, where the exponent was determined to be 2/3. Transitioning to the second regime, we observed relatively high velocities, resulting in the displacement of the foam as a plug within single layers of foam bubbles. The movement of these bubbles near the cell walls was notably slower. Similar to the first regime, the behavior in this regime also adhered to a power law. In the third regime, which manifested at higher air injection pressures, the development of air fingers occurred through narrow channels. These channels had the potential to isolate the air fingers as they underwent a process of "healing." Furthermore, our results unveiled a significant finding that the width of the air fingers exhibited a continuous scaling with the air injection pressure, irrespective of the flow regimes being observed.

Non-linear effects on some unsteady non-Darcian flows through porous media

Convection in a closed domain driven by a dense buoyancy source along the upper boundary soon starts to wane owing to the increase of the average interior density. In this paper, theoretical and numerical models are developed of the subsequent long period of shutdown of convection in a two-dimensional porous medium at high Rayleigh number $\mathit{Ra}$. The aims of this paper are twofold. Firstly, the relationship between this slowly evolving ‘one-sided’ shutdown system and the statistically steady ‘two-sided’ Rayleigh–Bénard (RB) cell is investigated. Numerical measurements of the Nusselt number $\mathit{Nu}$ from an RB cell (Hewitt et al., Phys. Rev. Lett., vol. 108, 2012, 224503) are very well described by the simple parametrization $\mathit{Nu}= 2. 75+ 0. 0069\mathit{Ra}$. This parametrization is used in theoretical box models of the one-sided shutdown system and found to give excellent agreement with high-resolution numerical simulations of this system. The dynamical structure of shutdown can also be accurately predicted by measurements from an RB cell. Results are presented for a general power-law equation of state. Secondly, these ideas are extended to model more complex physical systems, which comprise two fluid layers with an equation of state such that the solution that forms at the (moving) interface is more dense than either layer. The two fluids are either immiscible or miscible. Theoretical box models compare well with numerical simulations in the case of a flat interface between the fluids. Experimental results from a Hele-Shaw cell and numerical simulations both show that interfacial deformation can dramatically enhance the convective flux. The applicability of these results to the convective dissolution of geologically sequestered ${\mathrm{CO} }_{2} $ in a saline aquifer is discussed.

Swirl and buoyancy effects on mixing performance of baffle-plate-type miniature confined multijet

We consider the one-phase problem on radial viscous fingering structures in a Hele–Shaw cell with surface tension. This problem is a nonlinear free-boundary problem for elliptic equations. Unlike the Stefan problem for the heat equation, we deal with a problem of hydrodynamic type. We establish the classical solvability of the one-phase Hele–Shaw problem with radial geometry by using the same method as that used for the Stefan problem and justifying the vanishing coefficient of the time-derivative in the parabolic equation.

We consider the flow of two immiscible fluids lying between concentric cylinders when the outer cylinder is fixed and the inner one rotates. The interface is assumed to be concentric with the cylinders, and gravitational effects are neglected. We present a numerical study of the effect of different viscosities, different densities and surface tension on the linear stability of the Couette flow. Our results indicate that, with surface tension, a thin layer of the less-viscous fluid next to either cylinder is linearly stable and that it is possible to have stability with the less dense fluid lying outside. The stable configuration with the less-viscous fluid next to the inner cylinder is more stable than the one with the less-viscous fluid next to the outer cylinder. The onset of Taylor instability for one-fluid flow may be delayed by the addition of a thin layer of less-viscous fluid on the inner wall and promoted by a thin layer of more-viscous fluid on the inner wall.