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Description

A jet of a Newtonian fluid, such as honey poured from a bottle, thins continuously and coils regularly.¹ In contrast, a viscoelastic jet breaks up much more slowly. Typically, it evolves into a "beads-on-a-string" structure, where large drops are connected by thin threads. The slow breakup process provides the viscoelastic jet sufficient time to exhibit other phenomena, including:

• drop draining – a small bead between two beads shrinks as its fluid particles move towards the adjacent beads ("drains away");
• drop merging – a smaller bead and a larger bead move close to each other and merge to form a single bead;
• drop collision – a moving bead collides and combines with an adjacent bead;
• drop oscillation – two adjacent beads start oscillating, their separation gradually decreases, and they eventually merge to form a single bead.

The behaviors of non-Newtonian fluids result from the interplay of non-Newtonian properties (e.g. viscoelasticity, shear-thinning) with gravitational, viscous, and inertial effects.²[needs update]

The evolution of a viscoelastic fluid thread over time depends on the relative magnitude of the viscous, inertial, and elastic stresses and the capillary pressure. To study the inertio-elasto-capillary balance for a jet, two dimensionless parameters are defined: the Ohnesorge number (Oh)

O h = η 0 ρ γ R 0 {\displaystyle Oh={\frac {\eta _{0}}{\sqrt[{}]{\rho \gamma R_{0}}}}}

which is the inverse of the Reynolds number based on a characteristic capillary velocity γ η 0 {\displaystyle {\frac {\gamma }{\eta _{0}}}} ; and the intrinsic Deborah number (De), defined as

D e = λ ρ R 0 3 / γ {\displaystyle De={\frac {\lambda }{\sqrt[{}]{\rho R_{0}^{3}/\gamma }}}}

where t r = ρ R 0 3 / γ {\displaystyle t_{r}={\sqrt[{}]{\rho R_{0}^{3}/\gamma }}} is the "Rayleigh time scale" for inertio-capillary breakup of an inviscid jet. In these expressions, ρ {\displaystyle \rho } is the fluid density, η 0 {\displaystyle \eta _{0}} is the fluid zero shear viscosity, γ {\displaystyle \gamma } is the surface tension, R 0 {\displaystyle R_{0}} is the initial radius of the jet, and λ {\displaystyle \lambda } is the relaxation time associated with the polymer solution.

Equations

Like other fluids, when considering viscoelastic flows, the velocity, pressure, and stress must satisfy equations of mass and momentum, supplemented with a constitutive equation involving the velocity and stress.

The behaviors of weakly viscoelastic jets can be described by the following set of mathematical equations:

∂   R ∂ t + ∂   v R 2 ∂ z = 0 {\displaystyle {\frac {\partial \ R}{\partial t}}+{\frac {\partial \ vR^{2}}{\partial z}}=0}

1

ρ ( ∂   v ∂ t + v ∂ ∂ z ) = − γ ∂ κ ∂ t + 3 η s R 2 ∗ ∂ ( R 2 ∂ v ∂ z ) ∂ z + 1 R 2 ∂ ( R 2 ( σ z z − σ r r ) ) ∂ z {\displaystyle \rho ({\frac {\partial \ v}{\partial t}}+{\frac {v\partial }{\partial z}})=-\gamma {\frac {\partial \kappa }{\partial t}}+{\frac {3\eta _{s}}{R^{2}}}*{\frac {\partial (R^{2}{\frac {\partial v}{\partial z}})}{\partial z}}+{\frac {{\frac {1}{R^{2}}}\partial (R^{2}(\sigma _{zz}-\sigma _{rr}))}{\partial z}}}

2

κ = 1 R ( 1 + R z 2 ) 1 2 − R z z ( 1 + R z z 2 ) 3 2 {\displaystyle \kappa ={\frac {1}{R(1+R_{z}^{2})^{\frac {1}{2}}}}-{\frac {R_{zz}}{(1+R_{zz}^{2})^{\frac {3}{2}}}}}

3

where ( z , t ) {\displaystyle (z,t)} is the axial velocity; η s {\displaystyle \eta _{s}} and η p {\displaystyle \eta _{p}} are the solvent and polymer contribution to the total viscosity, respectively (total viscosity η 0 = η s + η p {\displaystyle \eta _{0}=\eta _{s}+\eta _{p}} ); R z {\displaystyle R_{z}} indicates the partial derivative ∂ R ∂ z {\displaystyle {\frac {\partial R}{\partial z}}} ; and σ z z {\displaystyle \sigma _{zz}} and σ r r {\displaystyle \sigma _{rr}} are the diagonal terms of the extra-stress tensor. Equation (1) represents mass conservation, and Equation (2) represents the momentum equation in one dimension. The extra-stress tensors σ z z {\displaystyle \sigma _{zz}} and σ r r {\displaystyle \sigma _{rr}} can be calculated as follows:

σ z z + λ ( ∂ σ z z ∂ t + v ∂ σ z z ∂ z − 2 ∂ v ∂ z σ z z ) + α λ η p σ z z 2 = 2 η p ∂ v ∂ z {\displaystyle \sigma _{zz}+\lambda ({\frac {\partial \sigma _{zz}}{\partial t}}+v{\frac {\partial \sigma _{zz}}{\partial z}}-2{\frac {\partial v}{\partial z}}\sigma _{zz})+{\frac {\alpha \lambda }{\eta _{p}}}\sigma _{zz}^{2}=2\eta _{p}{\frac {\partial v}{\partial z}}} σ r r + λ ( ∂ σ r r ∂ t + v ∂ σ r r ∂ z + ∂ v ∂ z σ r r ) + α λ η p σ r r 2 = − η p ∂ v ∂ z {\displaystyle \sigma _{rr}+\lambda ({\frac {\partial \sigma _{rr}}{\partial t}}+v{\frac {\partial \sigma _{rr}}{\partial z}}+{\frac {\partial v}{\partial z}}\sigma _{rr})+{\frac {\alpha \lambda }{\eta _{p}}}\sigma _{rr}^{2}=-\eta _{p}{\frac {\partial v}{\partial z}}}

where λ {\displaystyle \lambda } is the relaxation time of the liquid, and α {\displaystyle \alpha } is the mobility factor, a positive dimensionless parameter corresponding to the anisotropy of the hydrodynamic drag on the polymer molecules.

References

1. McKinley, Gareth (Nov 18, 2013). "Viscoelastic Jet". Gareth McKinley's Non-Newtonian Fluid Dynamics Research Group. MIT. Retrieved 2025-01-24. https://web.mit.edu/nnf/research/phenomena/viscoelastic_jet.html ↩

2. McKinley, Gareth (Nov 18, 2013). "Viscoelastic Jet". Gareth McKinley's Non-Newtonian Fluid Dynamics Research Group. MIT. Retrieved 2025-01-24. https://web.mit.edu/nnf/research/phenomena/viscoelastic_jet.html ↩