Convected coordinates and elongational flow

Kaye, A.

doi:10.1016/0377-0257(91)87026-t

Cited by 16 publications

(17 citation statements)

References 1 publication

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“…Equations equivalent to (4.6) and (4.7) were given by Kaye (1991). In practice, the Lagrangian system (4.5) and (4.2) is much easier to solve.…”

Section: Slender-drop Approximationmentioning

confidence: 99%

“…Wilson (1988) suggested for a similar problem that this non-physical infinity could be removed by putting inertia back into the problem, and also identified the crisis time with the time at which the drop breaks. Kaye (1991) considered some problems of viscous extensional flow both with and without inertia, and Cram (1984) included inertia in a numerical study of falling drops, but neither discussed the effect of inertia on the crisis time or acceleration. More recent references on drops that are falling and/or in extensional flow include Henderson et al (2000), Wilkes, Phillips & Basaran (1999) and Sarkar & Schowalter (2001).…”

Section: Introductionmentioning

confidence: 99%

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The role of inertia in extensional fall of a viscous drop

Stokes

Tuck

2004

J. Fluid Mech.

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In flows of very viscous fluids, it is often justifiable to neglect inertia and solve the resulting creeping-flow or Stokes equations. For drops hanging beneath a fixed wall and extending under gravity from an initial rest state, an inevitable consequence of neglect of inertia and surface tension is that the drop formally becomes infinite in length at a finite crisis time, at which time the acceleration of the drop, which has been assumed small relative to gravity g, formally also becomes infinite. This is a physical impossibility, and the acceleration must in fact approach the (finite) free-fall value g. However, we verify here, by a full Navier-Stokes computation and also with a slender-drop approximation, that the crisis time is a good estimate of the time at which the bulk of the drop goes into free fall. We also show that the drop shape at the crisis time is a good approximation to the final shape of the freely falling drop, prior to smoothing by surface tension. Additionally, we verify that the drop has an initial acceleration of g, which quickly decreases as viscous forces in the drop become dominant during the early stages of fall.

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“…Equations equivalent to (4.6) and (4.7) were given by Kaye (1991). In practice, the Lagrangian system (4.5) and (4.2) is much easier to solve.…”

Section: Slender-drop Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The role of inertia in extensional fall of a viscous drop

Stokes

Tuck

2004

J. Fluid Mech.

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“…Kaye (1991) included gravity and solved the problem with weak inertial effects, but did not examine the case in which inertia becomes significant. Huang et al (2003Huang et al ( , 2007 discussed the pulling of externally heated glass tubes in the case in which inertia is negligible.…”

Section: Introductionmentioning

confidence: 98%

“…Matovich & Pearson (1969) and DeWynne, Ockendon & Wilmott (1992) formally derived the appropriate long-wavelength equations to model the extensional flow of long thin Newtonian threads which have since been used by many authors. Kaye (1991) and Renardy (1994) derived the equations with a general constitutive law for the study of non-Newtonian fluid threads. Wilson (1988) and Stokes, Tuck & Schwartz (2000) studied the slender initial boundary value problem for a viscous Newtonian drop elongating under gravity when inertia is negligible; the role of inertia in the problem was examined by Stokes & Tuck (2004).…”

Section: Introductionmentioning

confidence: 99%

The evolution of a viscous thread pulled with a prescribed speed

2016

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We examine the extension of an axisymmetric viscous thread that is pulled at both ends with a prescribed speed such that the effects of inertia are initially small. After neglecting surface tension, we derive a particularly convenient form of the long-wavelength equations that describe long and thin threads. Two generic classes of initial thread shape are considered as well as the special case of a circular cylinder. In these cases, we determine explicit asymptotic solutions while the effects of inertia remain small. We further show that inertia will ultimately become important only if the long-time asymptotic form of the pulling speed is faster than a power law with a critical exponent. The critical exponent can take two possible values depending on whether or not the initial minimum of the thread radius is located at the pulled end. In addition, we obtain asymptotic expressions for the solution at large times in the case in which the critical exponent is exceeded and hence inertia becomes important. Despite the apparent simplicity of the problem, the solutions exhibit a surprisingly rich structure. In particular, in the case in which the initial minimum is not at the pulled end, we show that there are two very different types of solution that exhibit very different extension mechanics. Both the small-inertia solutions and the large-time asymptotic expressions compare well with numerical solutions.

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“…It is noted that the equations may also be applied to single-layer preforms. A similar approach to those described above has been used to construct an area-averaged extrusion model (Lin & Jou 1995) Each of the studies discussed above employed an Eulerian co-ordinate system; however, many studies of extensional flow, including the present work, make use of a Lagrangian co-ordinate system (Wilson 1988;Kaye 1991). Lagrangian descriptions have been used to study the gravitational stretching of axisymmetric slender drops neglecting surface tension and inertia (Wilson 1988;Stokes 2000), with inertia (Stokes & Tuck 2004) and with surface tension (Wilson 1988;Stokes et al 2011).…”

Section: Introductionmentioning

confidence: 99%

Gravitational extension of a fluid cylinder with internal structure

et al. 2016

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Motivated by the fabrication of microstructured optical fibres, a model is presented for the extension under gravity of a slender fluid cylinder with internal structure. It is shown that the general problem decouples into a two-dimensional surface-tension-driven Stokes flow that governs the transverse shape and an axial problem that depends upon the transverse flow. The problem and its solution differ from that obtained for fibre drawing, because the problem is unsteady and the fibre tension depends on axial position. Solutions both with and without surface tension are developed and compared, which show that the relative importance of surface tension depends upon both the parameter values and the geometry under consideration. The model is compared to experimental data and is shown to be in good agreement. These results also show that surface-tension effects are essential to accurately describing the cross sectional shape.

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Convected coordinates and elongational flow

Cited by 16 publications

References 1 publication

The role of inertia in extensional fall of a viscous drop

The role of inertia in extensional fall of a viscous drop

The evolution of a viscous thread pulled with a prescribed speed

Gravitational extension of a fluid cylinder with internal structure

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