Visco-elastic jets

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A jet of a Newtonian fluid, such as honey poured from a bottle, thins continuously and coils regularly.{{Cite web |last=McKinley |first=Gareth |date=Nov 18, 2013 |title=Viscoelastic Jet |url=https://web.mit.edu/nnf/research/phenomena/viscoelastic_jet.html |access-date=2025-01-24 |website=Gareth McKinley's Non-Newtonian Fluid Dynamics Research Group |publisher=MIT}} 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{{Snd}}a small bead between two beads shrinks as its fluid particles move towards the adjacent beads ("drains away");
• drop merging{{Snd}}a smaller bead and a larger bead move close to each other and merge to form a single bead;
• drop collision{{Snd}}a moving bead collides and combines with an adjacent bead;
• drop oscillation{{Snd}}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.{{Needs update|date=January 2025|reason=Site claims "This research is in the process of being published" as of 2013. The resultant published paper, if any, would be a more reliable (and informative) source.}}

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 ({{Mvar|Oh}})

:$Oh=\backslash frac\{\backslash eta\_0\}\{\backslash sqrt[]\{\backslash rho\backslash gamma\; R\_0\}\}$

which is the inverse of the Reynolds number based on a characteristic capillary velocity $\backslash frac\{\backslash gamma\}\{\backslash eta\_0\}$; and the intrinsic Deborah number ({{Mvar|De}}), defined as

:$De\; =\; \backslash frac\{\backslash lambda\}\{\backslash sqrt[]\{\backslash rho\; R\_0^3/\; \backslash gamma\}\}$

where $t\_r=\backslash sqrt[]\{\backslash rho\; R\_0^3/\backslash gamma\}$ is the "Rayleigh time scale" for inertio-capillary breakup of an inviscid jet. In these expressions, $\backslash rho$ is the fluid density, $\backslash eta\_0$ is the fluid zero shear viscosity, $\backslash gamma$ is the surface tension, $R\_0$ is the initial radius of the jet, and $\backslash lambda$ is the relaxation time associated with the polymer solution.

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:

{{NumBlk|:|$\backslash frac\{\backslash partial\; \backslash \; R\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash partial\; \backslash \; vR^2\}\{\backslash partial\; z\}=0$|{{EquationRef|1}}}}

{{NumBlk|:|$\backslash rho(\backslash frac\{\backslash partial\; \backslash \; v\}\{\backslash partial\; t\}\; +\; \backslash frac\{v\backslash partial\}\{\backslash partial\; z\})=-\backslash gamma\backslash frac\{\backslash partial\; \backslash kappa\}\{\backslash partial\; t\}\; +\; \backslash frac\{3\backslash eta\_s\}\{R^2\}*\; \backslash frac\{\backslash partial\; (R^2\backslash frac\{\backslash partial\; v\}\{\backslash partial\; z\})\}\{\backslash partial\; z\}\; +\; \backslash frac\{\backslash frac\{1\}\{R^2\}\backslash partial\; (R^2(\backslash sigma\_\{zz\}-\backslash sigma\_\{rr\}))\}\{\backslash partial\; z\}$|{{EquationRef|2}}}}

{{NumBlk|:|$\backslash kappa=\backslash frac\{1\}\{R(1+R\_z^2)^\backslash frac\{1\}\{2\}\}-\backslash frac\{R\_\{zz\}\}\{(1+R\_\{zz\}^2)^\backslash frac\{3\}\{2\}\}$|{{EquationRef|3}}}}

where $(z,t)$ is the axial velocity; $\backslash eta\_s$ and $\backslash eta\_p$ are the solvent and polymer contribution to the total viscosity, respectively (total viscosity $\backslash eta\_0=\backslash eta\_s+\backslash eta\_p$); $R\_z$ indicates the partial derivative $\backslash frac\{\backslash partial\; R\}\{\backslash partial\; z\}$; and $\backslash sigma\_\{zz\}$ and $\backslash sigma\_\{rr\}$ are the diagonal terms of the extra-stress tensor. Equation ({{EquationNote|1}}) represents mass conservation, and Equation ({{EquationNote|2}}) represents the momentum equation in one dimension. The extra-stress tensors $\backslash sigma\_\{zz\}$ and $\backslash sigma\_\{rr\}$ can be calculated as follows:

:$\backslash sigma\_\{zz\}+\backslash lambda(\backslash frac\{\backslash partial\; \backslash sigma\_\{zz\}\}\{\backslash partial\; t\}+v\backslash frac\{\backslash partial\; \backslash sigma\_\{zz\}\}\{\backslash partial\; z\}\; -\; 2\backslash frac\{\backslash partial\; v\}\{\backslash partial\; z\}\backslash sigma\_\{zz\})\; +\; \backslash frac\{\backslash alpha\; \backslash lambda\}\{\backslash eta\_p\}\backslash sigma\_\{zz\}^2\; =\; 2\backslash eta\_p\backslash frac\{\backslash partial\; v\}\{\backslash partial\; z\}$

:$\backslash sigma\_\{rr\}+\backslash lambda(\backslash frac\{\backslash partial\; \backslash sigma\_\{rr\}\}\{\backslash partial\; t\}+v\backslash frac\{\backslash partial\; \backslash sigma\_\{rr\}\}\{\backslash partial\; z\}\; +\; \backslash frac\{\backslash partial\; v\}\{\backslash partial\; z\}\backslash sigma\_\{rr\})\; +\; \backslash frac\{\backslash alpha\; \backslash lambda\}\{\backslash eta\_p\}\backslash sigma\_\{rr\}^2\; =\; -\backslash eta\_p\backslash frac\{\backslash partial\; v\}\{\backslash partial\; z\}$

where $\backslash lambda$ is the relaxation time of the liquid, and $\backslash alpha$ is the mobility factor, a positive dimensionless parameter corresponding to the anisotropy of the hydrodynamic drag on the polymer molecules.

{{Reflist}}

Category:Elasticity (physics)

Category:Non-Newtonian fluids