Free-surface flow past a semi-infinite or a finite-length corrugation in an otherwise flat and ho... more Free-surface flow past a semi-infinite or a finite-length corrugation in an otherwise flat and horizontal open channel is considered. Numerical solutions for the steady flow problem are computed using both a weakly nonlinear and fully nonlinear model. The new solutions are classified in terms of a depth-based Froude number and the four classical flow types (supercritical, subcritical, generalized hydraulic rise, and hydraulic rise) for flow over a small bump. While there is no hydraulic fall solution for semi-infinite topography, we provide strong numerical evidence that such a solution does exist in the case of a finite-length corrugation. Numerical solutions are also found for the other flow types for either semi-infinite or finite-length corrugation. For subcritical flow over a semi-infinite corrugation, the free-surface profile is found to be quasiperiodic downstream.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012
The flow of a viscous liquid layer over an inclined uneven wall heated from below is considered. ... more The flow of a viscous liquid layer over an inclined uneven wall heated from below is considered. The flow is assumed to occur at zero Reynolds number and the thermal Péclet number is taken to be sufficiently small that the temperature field inside the layer is governed by Laplace’s equation. With a prescribed wall temperature distribution and Newton’s Law of cooling imposed at the layer surface, the emphasis is placed on describing the surface profile of the liquid layer and, in particular, on studying how this is affected by wall heating. A linearized theory, valid when the amplitude of the wall topography is small, is derived and this is complemented by some nonlinear results computed using the boundary element method. It is shown that for flow over a sinusoidally shaped wall the liquid layer can be completely flattened by differential wall heating. For flow over a flat wall with a downwards step, it is demonstrated how the capillary ridge that has been identified by previous work...
The low-Reynolds-number flow of a liquid film down an inclined plane wall over a particle attache... more The low-Reynolds-number flow of a liquid film down an inclined plane wall over a particle attached to the wall is considered. The effect of the particle is described on the assumption that the free surface suffers only a minor disturbance from its basic flat state, and the disturbance velocity over the free surface is small, even though the particle size may not be small compared to the unperturbed film thickness. The problem is formulated using the boundary integral method for Stokes flow to describe the surface velocity, the deformation of the film surface, and the distribution of the traction over the particle surface. Results are presented for small particles following the earlier asymptotic analysis of Pozrikidis and Thoroddsen, and for moderate-sized particles with semispherical, spherical, and semispheroidal shapes. The simulations reveal that, in all cases, the free surface causes an upstream hump and a horseshoe type of deformation downstream whose intensity depends on the Bond number and is largely insensitive to the specific particle shape.
The two-dimensional nonlinear evolution of the interface between two superposed layers of viscous... more The two-dimensional nonlinear evolution of the interface between two superposed layers of viscous fluid moving in a channel in the presence of an insoluble surfactant is examined. A pair of coupled weakly nonlinear equations is derived describing the interfacial and surfactant dynamics when one of the two fluid layers is very thin in comparison to the other. In contrast to previous work, the dynamics in the thin film are coupled to the dynamics in the thicker layer through a nonlocal integral term. For asymptotically small Reynolds number, the flow in the thicker layer is governed by the Stokes equations. A linearized analysis confirms the linear instability identified by previous workers and it is proven that the film flow is linearly unstable if the undisturbed surfactant concentration exceeds a threshold value. Numerical simulations of the weakly nonlinear equations reveal the existence of finite amplitude traveling-wave solutions. For order one Reynolds number, the flow in the thicker layer is governed by the linearized Navier-Stokes equations. In this case the weakly nonlinear film dynamics are more complex and include the possibility of periodic traveling-waves and chaotic flow.
The flow of an electrified liquid layer moving over a prescribed topography is studied with the a... more The flow of an electrified liquid layer moving over a prescribed topography is studied with the aim of determining the shape of the free surface. The steady flow is assumed to be inviscid, incompressible, and irrotational. The liquid is assumed to act as a perfect conductor and the air above the layer is assumed to act as a perfect dielectric. The electric field is produced by placing one or more charged electrodes at a distance above the free surface. A weakly nonlinear one-dimensional analysis is used to classify the possible solutions and nonlinear solutions are obtained numerically by boundary integral equation methods. It is found that the shape of the liquid layer's surface can be manipulated (using charged electrodes) to become wave-free.
The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite... more The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite splitter plate is considered. Matched asymptotic expansions are used to analyse the developing flow, which is decomposed into four regions: a boundary layer of Blasius type growing along the plate, an inviscid core, a viscous layer close to the curved wall and a nonlinear corner region. The core solution is found numerically, initially in the long-distance down-pipe limit and thereafter the full problem is solved using down-pipe Fourier transforms. The accuracy in the corners of the semicircular cross-section is improved by subtracting out the singularity in the velocity perturbation. The linear viscous wall layer is solved analytically in terms of a displacement function determined from the core. A plausible structure for the corner region and equations governing the motion therein are presented although no solution is attempted. The presence of the plate has little effect ahead of the...
The effect of an insoluble surfactant on the stability of the gravity-driven flow of a liquid fil... more The effect of an insoluble surfactant on the stability of the gravity-driven flow of a liquid film down an inclined plane is investigated by a normal-mode analysis. Numerical solutions of the Orr-Sommerfeld equation reveal the occurrence of a stable Marangoni mode and a possibly unstable Yih mode, and demonstrate that the primary role of the surfactant is effectively to raise the critical Reynolds number at which instability is first encountered.
The capillary instability of a liquid thread containing a regular array of spherical particles al... more The capillary instability of a liquid thread containing a regular array of spherical particles along the centreline is considered with reference to microencapsulation. The thread interface may be clean or occupied by an insoluble surfactant. The main goal of the analysis is to illustrate the effect of the particle spacing on the growth rate of axisymmetric perturbations and identify the structure of the most unstable modes. A normal-mode linear stability analysis based on Fourier expansions for Stokes flow reveals that, at small particle separations, the interfacial profiles are nearly pure sinusoidal waves whose growth rate is nearly equal to that of a pure thread devoid of particles. Higher harmonics suddenly enter the normal modes for moderate and large particle separations, elevating the growth rates and yielding a stability diagram that consists of a sequence of superposed pure-thread lobes. A complementary numerical stability analysis based on the boundary integral formulation...
Fully developed flows are often used to describe fluid motion in complex geometrical systems, inc... more Fully developed flows are often used to describe fluid motion in complex geometrical systems, including the human macrocirculation. In fact they may frequently be quite inappropriate even for geometrically simple pipes, owing to the unfeasibly large viscous entry lengths required. Inviscid adjustment to changes in geometry, however, occurs on the lengthscale of the pipe diameter. Inviscid idealizations are therefore more likely to apply in relatively short arterial sections. We aim to quantify the distances involved by calculating the rates of spatial decay for a general disturbance superimposed on an idealized base flow. Both irrotational and rotational base flows are examined, although in the latter case there can exist non-decaying inertial waves, so that an arbitrary inflow need not attain an inviscid state independent of the downstream coordinate. In the rotational case, we therefore restrict attention to those flows which settle down to perturbations of such a state, whereas t...
The stability of a core–annular fluid arrangement consisting of two concentric fluid layers surro... more The stability of a core–annular fluid arrangement consisting of two concentric fluid layers surrounding a solid cylindrical rod on the axis of a circular pipe is examined when the interface between the two fluid layers is covered with an insoluble surfactant. The motion is driven either by an imposed axial pressure gradient or by the movement of the rod at a prescribed constant velocity. In the basic state the fluid motion is unidirectional and the interface between the two fluids is cylindrical. A linear stability analysis is performed for arbitrary layer thicknesses and arbitrary Reynolds number. The results show that the flow can be fully stabilized, even at zero Reynolds number, if the base flow shear rate at the interface is set appropriately. This result is confirmed by an asymptotic analysis valid when either of the two fluid layers is thin in comparison to the gap between the pipe wall and the rod. It is found that for a thin inner layer the flow can be stabilized if the inn...
The effect of an insoluble surfactant on the stability of the core-annular flow of two immiscible... more The effect of an insoluble surfactant on the stability of the core-annular flow of two immiscible fluids is investigated by a normal-mode linear analysis and by numerical simulations based on the immersed-interface method for axisymmetric perturbations. The results reveal that, although the Marangoni stress due to surfactant concentration variations is unable to initiate a new type of instability as in the case of twodimensional two-layer channel flow, it does destabilize the interface by broadening the range of growing wavenumbers and by raising the growth rate of unstable perturbations. Numerical simulations for large-amplitude disturbances reveal that the surfactant plays an important role in determining the morphology of the interfacial structures developing in the nonlinear stages of the motion.
We consider the unsteady motion of a viscous incompressible fluid inside a cylindrical tube whose... more We consider the unsteady motion of a viscous incompressible fluid inside a cylindrical tube whose radius is changing in a prescribed manner. We construct a class of exact solutions of the Navier-Stokes equations in the case when the vessel radius is a function of time alone so that the cross-section is circular and uniform along the pipe axis. The Navier-Stokes equations admit solutions which are governed by nonlinear partial differential equations depending on the radial coordinate and time alone, and forced by the wall motion. These solutions correspond to a wide class of bounded radial stagnation-point flows and are of practical importance. In dimensionless terms, the flow is characterized by two parameters: ∆, the amplitude of the oscillation, and R, the Reynolds number for the flow. We study flows driven by a time-periodic wall motion, and find that at small R the flow is synchronous with the forcing and as R increases a Hopf bifurcation takes place. Subsequent dynamics, as R increases, depend on the value of ∆. For small ∆ the Hopf bifurcation leads to quasi-periodic solutions in time, with no further bifurcations occurring-this is supported by an asymptotic high-Reynolds-number boundary layer theory. At intermediate ∆, the Hopf bifurcation is either quasi-periodic (for the smaller ∆) or subharmonic (for larger ∆), and the solutions tend to a chaotic attractor at sufficiently large R; the route to chaos is found not to follow a Feigenbaum scenario. At larger values of ∆, we find that the solution remains time periodic as R increases, with solution branches supporting periods of successive integer multiples of the driving period emerging. On a given branch the flow exhibits a self-similarity in both time and space and these features are elucidated by careful numerics and an asymptotic analysis. In contrast to the two-dimensional case (see Hall & Papageorgiou 1999) chaos is not found at either small or comparatively large ∆.
The effect of an insoluble surfactant on the stability of two-layer viscous flow in an inclined c... more The effect of an insoluble surfactant on the stability of two-layer viscous flow in an inclined channel confined by two parallel walls is considered. A lubrication-flow model applicable to long waves and low-Reynolds-number-flow is developed, and pertinent nonlinear evolution equations for the interface position and surfactant concentration are derived. Linear stability analysis based on the lubrication-flow model and the inclusive equations of Stokes flow confirm the recent findings of Frenkel & Halpern (2002) and Halpern & Frenkel (2003) that the presence of an insoluble surfactant induces a Marangoni instability in certain regions of parameter space defined by the layer thickness and viscosity ratios. Numerical simulations based on both approaches show that the interfacial waves may grow and saturate into steep profiles. The lubrication-flow model is adequate in capturing the essential features of the instability for small and moderate wavenumbers.
The linear stability of core–annular fluid arrangements are considered in which two concentric vi... more The linear stability of core–annular fluid arrangements are considered in which two concentric viscous fluid layers occupy the annular region within a straight pipe with a solid rod mounted on its axis when the interface between the fluids is coated with an insoluble surfactant. The linear stability of this arrangement is studied in two scenarios: one for core–annular flow in the absence of the rod and the second for rod–annular flow when the rod moves parallel to itself along the pipe axis at a prescribed velocity. In the latter case the effect of convective motion on a quiescent fluid configuration is also considered. For both flows the emphasis is placed on non-axisymmetric modes; in particular their impact on the recent stabilization to axisymmetric modes at zero Reynolds number discovered by Bassom, Blyth and Papageorgiou (J. Fluid. Mech., vol. 704, 2012, pp. 333–359) is assessed. It is found that in general non-axisymmetric disturbances do not undermine this stabilization, but...
The stability of a viscous liquid film flowing under gravity down an inclined wall with periodic ... more The stability of a viscous liquid film flowing under gravity down an inclined wall with periodic corrugations is investigated. A long-wave model equation valid at near-critical Reynolds numbers is used to study the film dynamics, and calculations are performed for either sinusoidal or rectangular wall corrugations assuming either a fixed flow rate in the film or a fixed volume of fluid within each wall period. Under the two different flow assumptions, steady solution branches are delineated including subharmonic branches, for which the period of the free surface is an integer multiple of the wall period, and the existence of quasi-periodic branches is demonstrated. Floquet–Bloch theory is used to determine the linear stability of steady, periodic solutions and the nature of any instability is analysed using the method of exponentially weighted spaces. Under certain conditions, and depending on the wall period, the flow may be convectively unstable for small wall amplitudes but under...
The effect of inertia on the Yih-Marangoni instability of the interface between two liquid layers... more The effect of inertia on the Yih-Marangoni instability of the interface between two liquid layers in the presence of an insoluble surfactant is assessed for shear-driven channel flow by a normal-mode linear stability analysis. The Orr-Sommerfeld equation describing the growth of small perturbations is solved numerically subject to interfacial conditions that allow for the Marangoni traction. For general Reynolds numbers and arbitrary wave numbers, the surfactant is found to either provoke instability or significantly lower the rate of decay of infinitesimal perturbations, while inertial effects act to widen the range of unstable wave numbers. The nonlinear evolution of growing interfacial waves consisting of a special pair of normal modes yielding an initially flat interface is analysed numerically by a finite-difference method. The results of the simulations are consistent with the predictions of the linear theory and reveal that the interfacial waves steepen and eventually overturn under the influence of the shear flow.
International Journal of Heat and Mass Transfer, 2003
The effect of irregularities on the rate of heat conduction from a two-dimensional isothermal sur... more The effect of irregularities on the rate of heat conduction from a two-dimensional isothermal surface into a semiinfinite medium is considered. The effect of protrusions, depressions, and surface roughness is quantified in terms of the displacement of the linear temperature profile prevailing far from the surface. This shift, coined the displacement length, is designated as an appropriate global measure of the effect of the surface indentations incorporating the particular details of the possibly intricate geometry. To compute the displacement length, LaplaceÕs equation describing the temperature distribution in the semi-infinite space above the surface is solved numerically by a modified Schwarz-Christoffel transformation whose computation requires solving a system of highly non-linear algebraic equations by iterative methods, and an integral equation method originating from the single-layer integral representation of a harmonic function involving the periodic GreenÕs function. The conformal mapping method is superior in that it is capable of handling with high accuracy a large number of vertices and intricate wall geometries. On the other hand, the boundary integral method yields the displacement length as part of the solution. Families of polygonal wall shapes composed of segments in regular, irregular, and random arrangement are considered, and pre-fractal geometries consisting of large numbers of vertices are analyzed. The results illustrate the effect of wall geometry on the flux distribution and on the overall enhancement in the rate of transport for regular and complex wall shapes.
The buckling and collapse of empty and liquid-filled thin-wall cylindrical tubes resting on a hor... more The buckling and collapse of empty and liquid-filled thin-wall cylindrical tubes resting on a horizontal or inclined plane is considered. The deflection is due to the action of gravity causing the tube to deform under the influence of its own weight, or due to a negative transmural pressure pushing the tube inward on the outside. Classical thin-membrane theory is used to formulate a boundary-value problem describing the shell deformation, and the results illustrate families of deformed shapes of inextensible shells with point or segment contact occurring between the shell and the supporting surface or between two collapsed sections of the shell. The computed two-dimensional deformed shapes are used to reconstruct the three-dimensional shape of a slowly collapsing fluid-conveying vessel in the absence of significant hydrostatic pressure variations over the cross section.
Stagnation-point flow of a semi-infinite viscous fluid against a liquid film resting on a plane w... more Stagnation-point flow of a semi-infinite viscous fluid against a liquid film resting on a plane wall is considered under conditions of Stokes flow and for arbitrary Reynolds numbers. Assuming that the interface remains flat and parallel to the wall at all times, the lateral spatial coordinate is scaled out to yield a simplified system of governing equations in the transverse coordinate normal to the wall, describing the evolution of the flow and displacement of the interface. In this inherently unsteady flow the film keeps thinning in time, while the rate of thinning decreases as the interface approaches the wall. Orthogonal two-dimensional, axisymmetric, three-dimensional, and oblique two-dimensional flow are individually considered. In each case, exact solutions of a similarity form are constructed, and an evolution equation describing the film thickness is formulated and solved by numerical methods. Quasi-steady solutions compare favourably with full calculations of the unsteady flow, suggesting that the unsteady terms have only a minor effect on the rate of film thinning. Dual solutions are uncovered in the case of three-dimensional flow. ''exact'' boundary layer distinguished by the absence of the usual boundary-layer approximation. The more general oblique two-dimensional stagnation-point flow was analyzed by Stuart [20], Tamada [21], and more recently by Dorrepaal [7]. In the case of oblique flow, the problem is formulated in terms of a primary and a suspended ordinary differential equation. Homann [11] considered orthogonal axisymmetric flow, and Howarth [12] and Davey [5] considered the inclusive three-dimensional orthogonal stagnation-point flow. The stability of steady and oscillatory stagnation-point flow has also been considered by several authors in the theoretical and applied literature, as discussed by Brattkus and Davis [3], Lasseigne and Jackson [13] and Riley [17]. Other authors have discussed the effect of in-plane wall oscillations (e.g., Glauert [8]) and far-field oscillations (e.g., Riley and Vasantha [18], Blyth and Hall [2]), and the significance of the wall slip velocity occurring in rarefied flow or under high pressure (Wang [22]).
Free-surface flow past a semi-infinite or a finite-length corrugation in an otherwise flat and ho... more Free-surface flow past a semi-infinite or a finite-length corrugation in an otherwise flat and horizontal open channel is considered. Numerical solutions for the steady flow problem are computed using both a weakly nonlinear and fully nonlinear model. The new solutions are classified in terms of a depth-based Froude number and the four classical flow types (supercritical, subcritical, generalized hydraulic rise, and hydraulic rise) for flow over a small bump. While there is no hydraulic fall solution for semi-infinite topography, we provide strong numerical evidence that such a solution does exist in the case of a finite-length corrugation. Numerical solutions are also found for the other flow types for either semi-infinite or finite-length corrugation. For subcritical flow over a semi-infinite corrugation, the free-surface profile is found to be quasiperiodic downstream.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012
The flow of a viscous liquid layer over an inclined uneven wall heated from below is considered. ... more The flow of a viscous liquid layer over an inclined uneven wall heated from below is considered. The flow is assumed to occur at zero Reynolds number and the thermal Péclet number is taken to be sufficiently small that the temperature field inside the layer is governed by Laplace’s equation. With a prescribed wall temperature distribution and Newton’s Law of cooling imposed at the layer surface, the emphasis is placed on describing the surface profile of the liquid layer and, in particular, on studying how this is affected by wall heating. A linearized theory, valid when the amplitude of the wall topography is small, is derived and this is complemented by some nonlinear results computed using the boundary element method. It is shown that for flow over a sinusoidally shaped wall the liquid layer can be completely flattened by differential wall heating. For flow over a flat wall with a downwards step, it is demonstrated how the capillary ridge that has been identified by previous work...
The low-Reynolds-number flow of a liquid film down an inclined plane wall over a particle attache... more The low-Reynolds-number flow of a liquid film down an inclined plane wall over a particle attached to the wall is considered. The effect of the particle is described on the assumption that the free surface suffers only a minor disturbance from its basic flat state, and the disturbance velocity over the free surface is small, even though the particle size may not be small compared to the unperturbed film thickness. The problem is formulated using the boundary integral method for Stokes flow to describe the surface velocity, the deformation of the film surface, and the distribution of the traction over the particle surface. Results are presented for small particles following the earlier asymptotic analysis of Pozrikidis and Thoroddsen, and for moderate-sized particles with semispherical, spherical, and semispheroidal shapes. The simulations reveal that, in all cases, the free surface causes an upstream hump and a horseshoe type of deformation downstream whose intensity depends on the Bond number and is largely insensitive to the specific particle shape.
The two-dimensional nonlinear evolution of the interface between two superposed layers of viscous... more The two-dimensional nonlinear evolution of the interface between two superposed layers of viscous fluid moving in a channel in the presence of an insoluble surfactant is examined. A pair of coupled weakly nonlinear equations is derived describing the interfacial and surfactant dynamics when one of the two fluid layers is very thin in comparison to the other. In contrast to previous work, the dynamics in the thin film are coupled to the dynamics in the thicker layer through a nonlocal integral term. For asymptotically small Reynolds number, the flow in the thicker layer is governed by the Stokes equations. A linearized analysis confirms the linear instability identified by previous workers and it is proven that the film flow is linearly unstable if the undisturbed surfactant concentration exceeds a threshold value. Numerical simulations of the weakly nonlinear equations reveal the existence of finite amplitude traveling-wave solutions. For order one Reynolds number, the flow in the thicker layer is governed by the linearized Navier-Stokes equations. In this case the weakly nonlinear film dynamics are more complex and include the possibility of periodic traveling-waves and chaotic flow.
The flow of an electrified liquid layer moving over a prescribed topography is studied with the a... more The flow of an electrified liquid layer moving over a prescribed topography is studied with the aim of determining the shape of the free surface. The steady flow is assumed to be inviscid, incompressible, and irrotational. The liquid is assumed to act as a perfect conductor and the air above the layer is assumed to act as a perfect dielectric. The electric field is produced by placing one or more charged electrodes at a distance above the free surface. A weakly nonlinear one-dimensional analysis is used to classify the possible solutions and nonlinear solutions are obtained numerically by boundary integral equation methods. It is found that the shape of the liquid layer's surface can be manipulated (using charged electrodes) to become wave-free.
The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite... more The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite splitter plate is considered. Matched asymptotic expansions are used to analyse the developing flow, which is decomposed into four regions: a boundary layer of Blasius type growing along the plate, an inviscid core, a viscous layer close to the curved wall and a nonlinear corner region. The core solution is found numerically, initially in the long-distance down-pipe limit and thereafter the full problem is solved using down-pipe Fourier transforms. The accuracy in the corners of the semicircular cross-section is improved by subtracting out the singularity in the velocity perturbation. The linear viscous wall layer is solved analytically in terms of a displacement function determined from the core. A plausible structure for the corner region and equations governing the motion therein are presented although no solution is attempted. The presence of the plate has little effect ahead of the...
The effect of an insoluble surfactant on the stability of the gravity-driven flow of a liquid fil... more The effect of an insoluble surfactant on the stability of the gravity-driven flow of a liquid film down an inclined plane is investigated by a normal-mode analysis. Numerical solutions of the Orr-Sommerfeld equation reveal the occurrence of a stable Marangoni mode and a possibly unstable Yih mode, and demonstrate that the primary role of the surfactant is effectively to raise the critical Reynolds number at which instability is first encountered.
The capillary instability of a liquid thread containing a regular array of spherical particles al... more The capillary instability of a liquid thread containing a regular array of spherical particles along the centreline is considered with reference to microencapsulation. The thread interface may be clean or occupied by an insoluble surfactant. The main goal of the analysis is to illustrate the effect of the particle spacing on the growth rate of axisymmetric perturbations and identify the structure of the most unstable modes. A normal-mode linear stability analysis based on Fourier expansions for Stokes flow reveals that, at small particle separations, the interfacial profiles are nearly pure sinusoidal waves whose growth rate is nearly equal to that of a pure thread devoid of particles. Higher harmonics suddenly enter the normal modes for moderate and large particle separations, elevating the growth rates and yielding a stability diagram that consists of a sequence of superposed pure-thread lobes. A complementary numerical stability analysis based on the boundary integral formulation...
Fully developed flows are often used to describe fluid motion in complex geometrical systems, inc... more Fully developed flows are often used to describe fluid motion in complex geometrical systems, including the human macrocirculation. In fact they may frequently be quite inappropriate even for geometrically simple pipes, owing to the unfeasibly large viscous entry lengths required. Inviscid adjustment to changes in geometry, however, occurs on the lengthscale of the pipe diameter. Inviscid idealizations are therefore more likely to apply in relatively short arterial sections. We aim to quantify the distances involved by calculating the rates of spatial decay for a general disturbance superimposed on an idealized base flow. Both irrotational and rotational base flows are examined, although in the latter case there can exist non-decaying inertial waves, so that an arbitrary inflow need not attain an inviscid state independent of the downstream coordinate. In the rotational case, we therefore restrict attention to those flows which settle down to perturbations of such a state, whereas t...
The stability of a core–annular fluid arrangement consisting of two concentric fluid layers surro... more The stability of a core–annular fluid arrangement consisting of two concentric fluid layers surrounding a solid cylindrical rod on the axis of a circular pipe is examined when the interface between the two fluid layers is covered with an insoluble surfactant. The motion is driven either by an imposed axial pressure gradient or by the movement of the rod at a prescribed constant velocity. In the basic state the fluid motion is unidirectional and the interface between the two fluids is cylindrical. A linear stability analysis is performed for arbitrary layer thicknesses and arbitrary Reynolds number. The results show that the flow can be fully stabilized, even at zero Reynolds number, if the base flow shear rate at the interface is set appropriately. This result is confirmed by an asymptotic analysis valid when either of the two fluid layers is thin in comparison to the gap between the pipe wall and the rod. It is found that for a thin inner layer the flow can be stabilized if the inn...
The effect of an insoluble surfactant on the stability of the core-annular flow of two immiscible... more The effect of an insoluble surfactant on the stability of the core-annular flow of two immiscible fluids is investigated by a normal-mode linear analysis and by numerical simulations based on the immersed-interface method for axisymmetric perturbations. The results reveal that, although the Marangoni stress due to surfactant concentration variations is unable to initiate a new type of instability as in the case of twodimensional two-layer channel flow, it does destabilize the interface by broadening the range of growing wavenumbers and by raising the growth rate of unstable perturbations. Numerical simulations for large-amplitude disturbances reveal that the surfactant plays an important role in determining the morphology of the interfacial structures developing in the nonlinear stages of the motion.
We consider the unsteady motion of a viscous incompressible fluid inside a cylindrical tube whose... more We consider the unsteady motion of a viscous incompressible fluid inside a cylindrical tube whose radius is changing in a prescribed manner. We construct a class of exact solutions of the Navier-Stokes equations in the case when the vessel radius is a function of time alone so that the cross-section is circular and uniform along the pipe axis. The Navier-Stokes equations admit solutions which are governed by nonlinear partial differential equations depending on the radial coordinate and time alone, and forced by the wall motion. These solutions correspond to a wide class of bounded radial stagnation-point flows and are of practical importance. In dimensionless terms, the flow is characterized by two parameters: ∆, the amplitude of the oscillation, and R, the Reynolds number for the flow. We study flows driven by a time-periodic wall motion, and find that at small R the flow is synchronous with the forcing and as R increases a Hopf bifurcation takes place. Subsequent dynamics, as R increases, depend on the value of ∆. For small ∆ the Hopf bifurcation leads to quasi-periodic solutions in time, with no further bifurcations occurring-this is supported by an asymptotic high-Reynolds-number boundary layer theory. At intermediate ∆, the Hopf bifurcation is either quasi-periodic (for the smaller ∆) or subharmonic (for larger ∆), and the solutions tend to a chaotic attractor at sufficiently large R; the route to chaos is found not to follow a Feigenbaum scenario. At larger values of ∆, we find that the solution remains time periodic as R increases, with solution branches supporting periods of successive integer multiples of the driving period emerging. On a given branch the flow exhibits a self-similarity in both time and space and these features are elucidated by careful numerics and an asymptotic analysis. In contrast to the two-dimensional case (see Hall & Papageorgiou 1999) chaos is not found at either small or comparatively large ∆.
The effect of an insoluble surfactant on the stability of two-layer viscous flow in an inclined c... more The effect of an insoluble surfactant on the stability of two-layer viscous flow in an inclined channel confined by two parallel walls is considered. A lubrication-flow model applicable to long waves and low-Reynolds-number-flow is developed, and pertinent nonlinear evolution equations for the interface position and surfactant concentration are derived. Linear stability analysis based on the lubrication-flow model and the inclusive equations of Stokes flow confirm the recent findings of Frenkel & Halpern (2002) and Halpern & Frenkel (2003) that the presence of an insoluble surfactant induces a Marangoni instability in certain regions of parameter space defined by the layer thickness and viscosity ratios. Numerical simulations based on both approaches show that the interfacial waves may grow and saturate into steep profiles. The lubrication-flow model is adequate in capturing the essential features of the instability for small and moderate wavenumbers.
The linear stability of core–annular fluid arrangements are considered in which two concentric vi... more The linear stability of core–annular fluid arrangements are considered in which two concentric viscous fluid layers occupy the annular region within a straight pipe with a solid rod mounted on its axis when the interface between the fluids is coated with an insoluble surfactant. The linear stability of this arrangement is studied in two scenarios: one for core–annular flow in the absence of the rod and the second for rod–annular flow when the rod moves parallel to itself along the pipe axis at a prescribed velocity. In the latter case the effect of convective motion on a quiescent fluid configuration is also considered. For both flows the emphasis is placed on non-axisymmetric modes; in particular their impact on the recent stabilization to axisymmetric modes at zero Reynolds number discovered by Bassom, Blyth and Papageorgiou (J. Fluid. Mech., vol. 704, 2012, pp. 333–359) is assessed. It is found that in general non-axisymmetric disturbances do not undermine this stabilization, but...
The stability of a viscous liquid film flowing under gravity down an inclined wall with periodic ... more The stability of a viscous liquid film flowing under gravity down an inclined wall with periodic corrugations is investigated. A long-wave model equation valid at near-critical Reynolds numbers is used to study the film dynamics, and calculations are performed for either sinusoidal or rectangular wall corrugations assuming either a fixed flow rate in the film or a fixed volume of fluid within each wall period. Under the two different flow assumptions, steady solution branches are delineated including subharmonic branches, for which the period of the free surface is an integer multiple of the wall period, and the existence of quasi-periodic branches is demonstrated. Floquet–Bloch theory is used to determine the linear stability of steady, periodic solutions and the nature of any instability is analysed using the method of exponentially weighted spaces. Under certain conditions, and depending on the wall period, the flow may be convectively unstable for small wall amplitudes but under...
The effect of inertia on the Yih-Marangoni instability of the interface between two liquid layers... more The effect of inertia on the Yih-Marangoni instability of the interface between two liquid layers in the presence of an insoluble surfactant is assessed for shear-driven channel flow by a normal-mode linear stability analysis. The Orr-Sommerfeld equation describing the growth of small perturbations is solved numerically subject to interfacial conditions that allow for the Marangoni traction. For general Reynolds numbers and arbitrary wave numbers, the surfactant is found to either provoke instability or significantly lower the rate of decay of infinitesimal perturbations, while inertial effects act to widen the range of unstable wave numbers. The nonlinear evolution of growing interfacial waves consisting of a special pair of normal modes yielding an initially flat interface is analysed numerically by a finite-difference method. The results of the simulations are consistent with the predictions of the linear theory and reveal that the interfacial waves steepen and eventually overturn under the influence of the shear flow.
International Journal of Heat and Mass Transfer, 2003
The effect of irregularities on the rate of heat conduction from a two-dimensional isothermal sur... more The effect of irregularities on the rate of heat conduction from a two-dimensional isothermal surface into a semiinfinite medium is considered. The effect of protrusions, depressions, and surface roughness is quantified in terms of the displacement of the linear temperature profile prevailing far from the surface. This shift, coined the displacement length, is designated as an appropriate global measure of the effect of the surface indentations incorporating the particular details of the possibly intricate geometry. To compute the displacement length, LaplaceÕs equation describing the temperature distribution in the semi-infinite space above the surface is solved numerically by a modified Schwarz-Christoffel transformation whose computation requires solving a system of highly non-linear algebraic equations by iterative methods, and an integral equation method originating from the single-layer integral representation of a harmonic function involving the periodic GreenÕs function. The conformal mapping method is superior in that it is capable of handling with high accuracy a large number of vertices and intricate wall geometries. On the other hand, the boundary integral method yields the displacement length as part of the solution. Families of polygonal wall shapes composed of segments in regular, irregular, and random arrangement are considered, and pre-fractal geometries consisting of large numbers of vertices are analyzed. The results illustrate the effect of wall geometry on the flux distribution and on the overall enhancement in the rate of transport for regular and complex wall shapes.
The buckling and collapse of empty and liquid-filled thin-wall cylindrical tubes resting on a hor... more The buckling and collapse of empty and liquid-filled thin-wall cylindrical tubes resting on a horizontal or inclined plane is considered. The deflection is due to the action of gravity causing the tube to deform under the influence of its own weight, or due to a negative transmural pressure pushing the tube inward on the outside. Classical thin-membrane theory is used to formulate a boundary-value problem describing the shell deformation, and the results illustrate families of deformed shapes of inextensible shells with point or segment contact occurring between the shell and the supporting surface or between two collapsed sections of the shell. The computed two-dimensional deformed shapes are used to reconstruct the three-dimensional shape of a slowly collapsing fluid-conveying vessel in the absence of significant hydrostatic pressure variations over the cross section.
Stagnation-point flow of a semi-infinite viscous fluid against a liquid film resting on a plane w... more Stagnation-point flow of a semi-infinite viscous fluid against a liquid film resting on a plane wall is considered under conditions of Stokes flow and for arbitrary Reynolds numbers. Assuming that the interface remains flat and parallel to the wall at all times, the lateral spatial coordinate is scaled out to yield a simplified system of governing equations in the transverse coordinate normal to the wall, describing the evolution of the flow and displacement of the interface. In this inherently unsteady flow the film keeps thinning in time, while the rate of thinning decreases as the interface approaches the wall. Orthogonal two-dimensional, axisymmetric, three-dimensional, and oblique two-dimensional flow are individually considered. In each case, exact solutions of a similarity form are constructed, and an evolution equation describing the film thickness is formulated and solved by numerical methods. Quasi-steady solutions compare favourably with full calculations of the unsteady flow, suggesting that the unsteady terms have only a minor effect on the rate of film thinning. Dual solutions are uncovered in the case of three-dimensional flow. ''exact'' boundary layer distinguished by the absence of the usual boundary-layer approximation. The more general oblique two-dimensional stagnation-point flow was analyzed by Stuart [20], Tamada [21], and more recently by Dorrepaal [7]. In the case of oblique flow, the problem is formulated in terms of a primary and a suspended ordinary differential equation. Homann [11] considered orthogonal axisymmetric flow, and Howarth [12] and Davey [5] considered the inclusive three-dimensional orthogonal stagnation-point flow. The stability of steady and oscillatory stagnation-point flow has also been considered by several authors in the theoretical and applied literature, as discussed by Brattkus and Davis [3], Lasseigne and Jackson [13] and Riley [17]. Other authors have discussed the effect of in-plane wall oscillations (e.g., Glauert [8]) and far-field oscillations (e.g., Riley and Vasantha [18], Blyth and Hall [2]), and the significance of the wall slip velocity occurring in rarefied flow or under high pressure (Wang [22]).
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Papers by Mark Blyth