COMSOL 6.4 - Viscous Catenary

The word catenary is derived from the Latin word catena, which means chain. The catenary is the geometrical shape that corresponds to the curve followed by an idealized chain or cable supported at both ends and hanging under its own weight. This shape played an important role in the history of mathematics and physics, highlighting the relationships that occur between geometry and mechanics. It is also important in structural engineering. Robert Hooke first realized that the shape represents the optimal form of an arch of constant cross section and this discovery was likely employed by Christopher Wren in the rebuilding of St Paul’s Cathedral in London. Despite announcing his achievement to the Royal Society in 1671 and publishing it in an encrypted form as a Latin anagram in 1675, the anagram’s meaning was not revealed until after Hooke’s executor published it posthumously in 1705. Although aware that the catenary shape was not a parabola, Hooke could not derive its mathematical form.This was ultimately discovered independently by Gottfried Leibniz, Christiaan Huygens, and Johann Bernoulli in response to a challenge issued by Jakob Bernoulli in 1690.

The viscous catenary problem describes the motion of a cylinder of highly viscous fluid, supported at its ends as it flows under gravity. In the last decade this problem has generated significant theoretical and experimental interest in the academic community. Solutions to the problem in one dimension have been derived (Ref. 1 and Ref. 2). The rich phenomena that occur in this apparently simple problem are industrially important in a range of applications such as filament spinning and glass manufacture.

Model Definition

The model consists of a cylinder of Newtonian fluid, with initial diameter 0.6 mm and length 21.5 mm, falling under its own weight. The fluid density is 1000 kg/m3 and its viscosity is 100 Pa·s. The surface tension of the fluid is 22 mN/m. The capillary number of the flow is high, so the contact angle of the fluid with the supporting surfaces is unimportant — here a nominal value of 90° is used. At the surfaces of the cylinder, the Navier slip boundary condition is used, this means that the supporting edges of the cylinder can move slightly. This displacement is small compared to the overall displacement of the catenary, and can be subtracted from the results if necessary.

Figure 1: Model geometry.

The model geometry is shown in Figure 1. The cylinder is split into two halves to set up points at its center that can be used to compute the height of the lowest point on the catenary.

Results and Discussion

Figure 2 shows the fluid velocity within the catenary after 2 s of falling. The arrows show that the fluid velocity is predominantly vertical. The pressure within the catenary is shown in Figure 3. The pressure variations occur mainly in the small region adjacent to the anchors, where the radius of curvature of the surface is greatest. The deformed mesh is shown in Figure 4. The mesh deformation is large in the region close to the anchor, and the mesh quality is consequently undesirably low. Ideally the model should be stopped at approximately 1 s and a re-meshed before the solver is allowed to run on.

Figure 2: Velocity in the center plane of the catenary after 2s.

Figure 3: Pressure in the center plane of the catenary after 2 s.

Figure 4: Plot showing the deformed mesh after 2 s. The mesh quality in the regions adjacent to the anchors is undesirably low.

Figure 5: Plot showing the position of the centerline of the catenary at 0.02 s intervals during the simulation.

Figure 6: Displacement of the center of the catenary, relative to the anchors, as a function of time. The results are compared to experimental data from Ref. 2, which has an estimated error of approximately 15% (not shown in the plot).

Figure 5 shows the location of the midplane of the catenary as a function of time. The parabolic profile of the catenary for the central part of its length is clearly apparent. This is predicted by the 1-dimensional theory for intermediate time scales (in this model these occur after approximately 0.1 s; see Ref. 1 and Ref. 2).

Figure 6 shows the vertical distance between the midpoint of the catenary and its anchors, as a function of time. This is compared with experimental data from Ref. 2. The agreement between the two datasets is within experimental error, without the scaling factor employed by the authors to obtain agreement with their theory. An additional parametric sweep performed on the surface tension coefficient shows that the surface tension is related to this scaling factor (surface tension was neglected in the theoretical treatments of Ref. 1 and Ref. 2). This example shows how COMSOL can provide fundamental insights into complex problems — having the flexibility to solve the problem without assumptions can give valuable insight into validity of assumptions made in simpler, lower-dimensional models.

References

1. J. Teichman and L. Mahadevan, “The Viscous Catenary,” J. Fluid Mech., vol. 478, pp. 71–80, 2003.

2. J.P. Koulakis, C.D. Mitescu, F. Brouchard-Wyart, P.-G. De Gennes, and E. Guyon, “The Viscous Catenary Revisited: Experiments and Theory,” J. Fluid Mech., vol. 609, pp. 87–110, 2008.

CFD_Module/Multiphase_Flow/viscous_catenary

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > > .

Click .

Click

In the tree, select > .

Click

Set up the model parameters.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
D0	0.6[mm]	6E-4 m	Initial diameter
L0	21.5[mm]	0.0215 m	Initial length
rho0	1000[kg/m^3]	1000 kg/m³	Fluid density
mu0	100[Pa*s]	100 Pa·s	Fluid viscosity
sigma0	22[mN/m]	0.022 N/m	Surface tension coefficient