**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30663

##### The Influence of Gravity on The Temporal Instability of Viscoelastic Liquid Curved Jets

**Authors:**
Abdullah Madhi Alsharif,
Jamal Uddin

**Abstract:**

A liquid curved jet has many applications in different industrial and engineering processes, such as the prilling process for generating small spherical pellets (fertilizer or magnesium). The liquids used are usually molten and contain small quantities of polymers and therefore can be modelled as non-Newtonian liquids. In this paper, we model the viscoelastic liquid jet by using the Oldroyd- B model. An asymptotic analysis has been used to simplify the governing equations. Furthermore, the trajectory and a linear temporal stability in the presence of gravity and rotation have been determined.

**Keywords:**
Rotation,
Gravity,
prilling,
viscoelastic jets

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1087326

**References:**

[1] Ardekani, A. M., Sharma, V., and Mckinley, G. H., 2010, Dynamics of
bead formation, filament thinning and breakup in weakly viscoelastic
jets. J. Fluid Mech. 665, 46-56

[2] Ashgriz, N. and Mashayek, F., 1995. Temporal analysis of capillary jet
breakup. J. Fluid Mech. 291, 163-190.

[3] Cheong, B. S., Howes, T., 2004, Capillary jet instability under influence
of gravity, Chemmical Engineering Science, 59, 2145-2157

[4] Clasen, C., Eggers, J., Fonlelos, M. A., Li, j., and Mckinley, G. H., 2006,
The beads-on-string structure of viscoelastic threads. J. Fluid Mech.,
556, 283-308

[5] Cooper-White, J. J., Fagan, J. E., Tirtaatmadja, V., Lastert, D. R., and
Boger, D. V., 2002, Drop formation dynamics of constant low-viscosity,
elastic fluids, J. Non-Newtonian fluid Mech., 106,29-59

[6] Davidson, M. R., Harvie, J. E., and Cooper-White, J. J., 2006, Simulation
of pendant drop formation of a viscoelastic liquid, Korea-Australia
Rheology Journal, 18, (2), 41-49

[7] Decent, S. P., King, A. C. and Wallwork, I. M., 2002, Free jets spun
from a prilling tower, Journal of Engineering Mathematics, 42, 265-282

[8] Decent, S. P., King, A. C., Simmons, M. H., P˘ar˘au, E. I., Wong, D. C.
Y., Wallwork, I. M., Gurney, C., and Uddin, J., 2007, The trajectory and
stability of a spiralling liquid jet: Part II. Viscous Theory, Appl. Math.
Modelling, 33, (12), 4283-4302

[9] Eggers, J., 1997. Nonlinear dynamics and breakup of free surface flows.
Rev. Mod. Physis, 69, (3), 865-929.

[10] Goldin, M., Yerushalmi, J., Pfeffer, R., and Shinner, R., 1969, Breakup
of a viscoelastic fluid. J. Fluid Mech. 38, 689-711.

[11] Grant, R. P., and Middleman, S., 1965, Newtonian jet stability. A. I. Ch.
E. Journal, 2, 669.

[12] Larson, R. G., 1992, Instabilities in viscoelastic flows, Rheol., Acta,31

[13] Liu, Z., and Liu, Z., 2008, Instability of a viscoelastic liquid jet
with axisymmetric and asymmetric disturbance, International Journal
of Multiphase Flow, 34, 42-60

[14] Mageda, J. J., and Larson, R. G., 1988, Atransition occurring in ideal
elastic liquids during shear flow, J. Non-Newtonian Fluid Mech., 30,1-19

[15] Middleman, S., 1965, Stability of a viscoelastic jet, Chem. Eng. Sci. 20,
1037-1040.

[16] Morrison, N. F., Harlen, O. G. 2010, Viscoelasticity in inkjet printing,
Rheol., Acta, 49, 619-632

[17] Papageorgiou, D. T., 1995, On the breakup of viscous liquid threads,
Phys. Fluids, 7, 1529 Nonlinear travelling waves on a spiralling liquid
jet, Wave Motion, 43, 599-613.

[18] P˘ar˘au, E. I., Decent, S. P., Simmons, M. J. H., Wong, D. C. Y. and
King, A. C., 2007, Nonlinear viscous liquid jets from a rotating orifice,
J. Of Eng. Maths., 57, 159-179

[19] Rayleigh, W. S., 1878, On the instability of jets, Proc. Lond. Math. Soc
10,4.

[20] Renardy, M., 1995, A numerical study of the asymptotic evolution and
breakup of Newtonian and viscoelastic jets, J. Non-Newtonian Fluid
Mech., 59, 267-282

[21] Renardy, M., 2008, Stability of viscoelastic shear flows in the limit
of high Weissenberg and Reynolds numbers, J. Non-Newtonian Fluid
Mech., 155, 124-129

[22] Uddin, J., 2007. An investigation into methods to control breakup
and droplet formation in single and compound liquid jets. PhD thesis,
University of Birmingham.

[23] Uddin, J., and Decent, S. P., 2010, Instability of non-Newtonian liquid
jets curved by gravity. Mathematics in industry, 15, 597-602.

[24] Wallwork, I. M., 2002a, The trajectory and stability of a spiralling liquid
jet, Ph.D. Thesis, University of Birmingham, Birmingham.

[25] Wallwork, I. M., Decent, S.P., King, A. C. and Schulkes, R. M. S.,
2002b, The trajectory and stability of a spiralling liquid jet. Part 1,
Inviscid Theory, J. Fluid Mech., 459,43-65.
Math. Mech, 11, 136-154.

[26] Weber, C., 1931, Zum Zerfall eines Flussigkeitsstrahles. Z. Angew.