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Bioprinting technologies rely on the formation of soft gel drops for printing tissue scaffolds and the dynamics of these drops can affect the process. A model is developed to describe the oscillations of a spherical gel drop with finite shear modulus, whose interface is held by surface tension. The governing elastodynamic equations are derived and a solution is constructed using displacement potentials decomposed into a spherical harmonic basis. The resulting nonlinear characteristic equation depends upon two dimensionless numbers, elastocapillary and compressibility, and admits two types of solutions, (i) spheroidal (or shape change) modes and (ii) torsional (rotational) modes. The torsional modes are unaffected by capillarity, whereas the frequency of shape oscillations depend upon both the elastocapillary and compressibility numbers. Two asymptotic dispersion relationships are derived and the limiting cases of the inviscid Rayleigh drop and elastic globe are recovered. For a fixed polar wavenumber, there exists an infinity of radial modes that each transition from an elasticity wave to a capillary wave upon increasing the elastocapillary number. At the transition, there is a qualitative change in the deformation field and a set of recirculation vortices develop at the free surface. Two special modes that concern volume oscillations and translational motion are characterized. A new instability is documented that reflects the balance between surface tension and compressibility effects due to the elasticity of the drop.
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Plateau–Rayleigh instability in a soft viscoelastic material
A soft cylindrical interface endowed with surface tension can be unstable to wavy undulations. This is known as the Plateau–Rayleigh instability (PRI) and for solids the instability is governed by the competition between elasticity and capillarity. A dynamic stability analysis is performed for the cases of a soft (i) cylinder and (ii) cylindrical cavity assuming the material is viscoelastic with power-law rheology. The governing equations are made time-independent through the Laplace transform from which a solution is constructed using displacement potentials. The dispersion relationships are then derived, which depend upon the dimensionless elastocapillary number, solid Deborah number, and compressibility number, and the static stability limit, critical disturbance, and maximum growth rate are computed. This dynamic analysis recovers previous literature results in the appropriate limits. Elasticity stabilizes and compressibility destabilizes the PRI. For an incompressible material, viscoelasticity does not affect stability but does decrease the growth rate and shift the critical wavenumber to lower values. The critical wavenumber shows a more complex dependence upon compressibility for the cylinder but exhibits a predictable trend for the cylindrical cavity.
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- Award ID(s):
- 1750208
- PAR ID:
- 10219184
- Date Published:
- Journal Name:
- Soft Matter
- ISSN:
- 1744-683X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1039/D1SM00019E
