Solution selection of axisymmetric Taylor bubbles

Doak, Alex; Vanden‐Broeck, J.‐M.

doi:10.1017/jfm.2018.156

Cited by 8 publications

(7 citation statements)

References 26 publications

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“…It follows that one could approximate the case of a surrounding fluid of infinite radius (D ! 1) by taking a suitably large value of D. This is further confirmed by considering the dispersion relation (14) for various values of D, as shown in figure 22. It can be seen that the dispersion relations for D = 8 and D !…”

Section: B > Bsupporting

confidence: 60%

“…When there is a minimum in the dispersion curve, we see periodic solutions exhibiting higher mode resonance, as described by equation (16). These solutions exist for integer values of n > 1 when c(k) = c(nk), where c is given by (14). Fixing a value of k and n, we can find a value of B such that this equality is satisfied.…”

Section: B > Bmentioning

confidence: 99%

“…Note that we recover the stability condition found by [27] by taking B = 0 (that all solutions with k < 1 are unstable). The right hand side of equation (14) tends to infinity as k ! 1.…”

Section: Linear Theorymentioning

confidence: 99%

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Travelling wave solutions on an axisymmetric ferrofluid jet

Doak

Vanden‐Broeck

2019

J. Fluid Mech.

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We consider a potential flow model of axisymmetric waves travelling on a ferrofluid jet. The ferrofluid coats a copper wire, through which an electric current is run. The induced azimuthal magnetic field magnetises the ferrofluid, which in turn stabilises the well known Plateau–Rayleigh instability seen in axisymmetric capillary jets. This model is of interest because the stabilising mechanism allows for axisymmetric magnetohydrodynamical solitary waves. A numerical scheme capable of computing steady periodic, solitary and generalised solitary wave solutions is presented. It is found that the solution space for the model is very similar to that of the classical problem of two-dimensional gravity–capillary waves.

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Section: B > Bsupporting

confidence: 60%

Section: B > Bmentioning

confidence: 99%

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Travelling wave solutions on an axisymmetric ferrofluid jet

Doak

Vanden‐Broeck

2019

J. Fluid Mech.

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“…8 Relationship between μ and F for α = 5 (solid curve) and α = 20 (dashed curve) for β = π . The dotted lines are μ = π/2 and μ = π Table 1 A table for the value of μ when α −1 = 0 On the higher order branches of smooth bubbles, the profiles develop oscillations [39]. It is of interest to note that the solution space of axisymmetric Taylor bubbles exhibit much of the same behaviour [39,40].…”

Section: Results For β = πmentioning

confidence: 98%

Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

Doak

Vanden‐Broeck

2021

J Eng Math

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This paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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“…Bubble in a tube. The second physical model is a classical problem of a bubble rising in a semi-finite long cylindrical tube (see references [10] and [14]). More precisely, it describes that an ideal rotational incompressible fluid acted on by gravity falls into a cylindrical nozzle and the free surface starts from some point D on the symmetric axis and goes into the far field, which is shown in Figure 5.…”

Section: Mathematical Setting Of the Physical Problemmentioning

confidence: 99%

The free boundary of steady axisymmetric inviscid flow with vorticity I: near the degenerate point

Li¹,

Huang²,

Pu³

2022

Preprint

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In this paper, we investigate the singularity near the degenerate points of the steady axisymmetric flow with general vorticity of an inviscid incompressible fluid acted on by gravity and with a free surface. We called the points on the free boundary at which the gradient of the stream function vanishes as the degenerate points. The main results in this paper give the different classifications of the singularity near the degenerate points on the free surface. More precisely, we obtained that at the stagnation points, the possible profiles must be a Stokes corner, or a horizontal cusp, or a horizontal flatness. At the degenerate points on the symmetric axis except the origin, the wave profile must be a cusp. At the origin, the possible wave profiles must be a Garabedian pointed bubble, or a horizontal cusp, or a horizontal flatness.

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Solution selection of axisymmetric Taylor bubbles

Cited by 8 publications

References 26 publications

Travelling wave solutions on an axisymmetric ferrofluid jet

Travelling wave solutions on an axisymmetric ferrofluid jet

Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

The free boundary of steady axisymmetric inviscid flow with vorticity I: near the degenerate point

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