Visco-elastic jets

Last updated August 19, 2024

Draining phenomenon in bead string structure Draining.jpg — Draining phenomenon in bead string structure

Merging phenomenon in bead string structure Merging.jpg — Merging phenomenon in bead string structure

Collision phenomenon in bead string structure Collision1.jpg — Collision phenomenon in bead string structure

Oscillation phenomenon in bead string structure Oscillation.jpg — Oscillation phenomenon in bead string structure

Oscillation phenomenon in bead string structure (Continuation of image on left) Oscillation1.jpg — Oscillation phenomenon in bead string structure (Continuation of image on left)

Visco-elastic jets are the jets of viscoelastic fluids, i.e. fluids that disobey Newton's law of Viscocity. A Viscoelastic fluid that returns to its original shape after the applied stress is released.

Background

Everybody has witnessed a situation where a liquid is poured out of an orifice at a given height and speed, and it hits a solid surface. For example, – dropping of honey onto a bread slice, or pouring shower gel onto one's hand. Honey is a purely viscous, Newtonian fluid: the jet thins continuously and coils regularly.

Jets of non-Newtonian Viscoelastic fluids show a novel behaviour. A viscoelastic jet breaks up much more slowly than a Newtonian jet. Typically, it evolves into the so-called beads-on-string structure, where large drops are connected by thin threads. The jet widens at its base (reverse swell phenomenon) and folds back and forth on itself. The slow breakup process provides the viscoelastic jet sufficient time to exhibit some new phenomena, including drop migration, drop oscillation, drop merging and drop draining.

These properties are a result of the interplay of non-Newtonian properties (viscoelasticity, shear-thinning) with gravitational, viscous, and inertial effects in the jets. Free surface continuous jets of viscoelastic fluids are relevant in many engineering applications involving blood, paints, adhesives, or foodstuff and industrial processes like fiber spinning, bottle-filling, oil drilling, etc. In many of these processes, an understanding of the instabilities a jet undergoes due to changes in fluid parameters like Reynolds number or Deborah number is essential from a process engineering point of view. With the advent of microfluidics, an understanding of the jetting properties of non-Newtonian fluids becomes essential from micro- to macro length scales, and from low to high Reynolds numbers7–9. Like other fluids, When considering viscoelastic flows, the velocity, pressure, and stress must satisfy the mass and momentum equation, supplemented with a constitutive equation involving the velocity and stress.

The temporal evolution of a viscoelastic fluid thread 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)

Oh={\frac {\eta _{0}}{\sqrt[{}]{\rho \gamma R_{0}}}}

, which is the inverse of the Reynolds number based on a characteristic capillary velocity ${\frac {\gamma }{\eta _{0}}}$ and, secondly, the intrinsic Deborah number De,

De=\lambda {\sqrt[{}]{\gamma /(\rho R_{0}^{3})}}

, defined as the ratio of the time scale for elastic stress relaxation, λ, to the “Rayleigh time scale” for inertio-capillary breakup of an inviscid jet, $t_{r}={\sqrt[{}]{\rho R_{0}^{3}/\gamma }}$ . In these expressions, $\rho$ is the fluid density, $\eta _{0}$ is the fluid zero shear viscosity, $\gamma$ is the surface tension, $R_{0}$ is the initial radius of the jet, and $\lambda$ is the relaxation time associated with the polymer solution.

Mathematical Equations governing bead formation, filament thinning & breakup in weakly viscoelastic jets

{\frac {\partial \ R}{\partial t}}+{\frac {\partial \ vR^{2}}{\partial z}}=0.

(1)

\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)

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

\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}}

(4)

\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}}

(5)

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

Beads on string structure

Drop Draining

In drop draining a small bead between two beads gets smaller in size and the fluid particle moves towards the adjacent beads. The smaller bead drains out as shown in the figure.

Drop Merging

In drop merging, a smaller bead and a larger bead move close to each other and merge to form a single bead.

Drop Collision

In drop collision, two adjacent beads collide to form a single bead.

Drop Oscillation

In drop oscillation, two adjacent beads start oscillating and eventually, the distance between them decreases. After some time they merge to form a single bead.

Related Research Articles

A viscometer is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used. Thus, a rheometer can be considered as a special type of viscometer. Viscometers can measure only constant viscosity, that is, viscosity that does not change with flow conditions.

Foams are materials formed by trapping pockets of gas in a liquid or solid.

Hemorheology, also spelled haemorheology, or blood rheology, is the study of flow properties of blood and its elements of plasma and cells. Proper tissue perfusion can occur only when blood's rheological properties are within certain levels. Alterations of these properties play significant roles in disease processes. Blood viscosity is determined by plasma viscosity, hematocrit and mechanical properties of red blood cells. Red blood cells have unique mechanical behavior, which can be discussed under the terms erythrocyte deformability and erythrocyte aggregation. Because of that, blood behaves as a non-Newtonian fluid. As such, the viscosity of blood varies with shear rate. Blood becomes less viscous at high shear rates like those experienced with increased flow such as during exercise or in peak-systole. Therefore, blood is a shear-thinning fluid. Contrarily, blood viscosity increases when shear rate goes down with increased vessel diameters or with low flow, such as downstream from an obstruction or in diastole. Blood viscosity also increases with increases in red cell aggregability.

In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist both shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

In physics, Washburn's equation describes capillary flow in a bundle of parallel cylindrical tubes; it is extended with some issues also to imbibition into porous materials. The equation is named after Edward Wight Washburn; also known as Lucas–Washburn equation, considering that Richard Lucas wrote a similar paper three years earlier, or the Bell-Cameron-Lucas-Washburn equation, considering J.M. Bell and F.K. Cameron's discovery of the form of the equation in 1906.

<span class="mw-page-title-main">Rayleigh–Taylor instability</span> Unstable behavior of two contacting fluids of different densities

The Rayleigh–Taylor instability, or RT instability, is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

In mathematical physics, the gamma matrices, $also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin particles. Gamma matrices were introduced by Paul Dirac in 1928.$

The upper-convected Maxwell (UCM) model is a generalisation of the Maxwell material for the case of large deformations using the upper-convected time derivative. The model was proposed by James G. Oldroyd. The concept is named after James Clerk Maxwell. It is the simplest observer independent constitutive equation for viscoelasticity and further is able to reproduce first normal stresses. Thus, it constitutes one of the most fundamental models for rheology.

In physics and fluid mechanics, a Blasius boundary layer describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Falkner and Skan later generalized Blasius' solution to wedge flow, i.e. flows in which the plate is not parallel to the flow.

In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

In physics, the Young–Laplace equation is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although use of the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It is a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface : $where is the Laplace pressure, the pressure difference across the fluid interface, is the surface tension, is the unit normal pointing out of the surface, is the mean curvature, and and are the principal radii of curvature. Note that only normal stress is considered, because a static interface is possible only in the absence of tangential stress.$

The capillary length or capillary constant is a length scaling factor that relates gravity and surface tension. It is a fundamental physical property that governs the behavior of menisci, and is found when body forces (gravity) and surface forces are in equilibrium.

In fluid mechanics and mathematics, a capillary surface is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid interfaces.

In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime to curved spacetime, a general Lorentzian manifold.

Von Kármán swirling flow is a flow created by a uniformly rotating infinitely long plane disk, named after Theodore von Kármán who solved the problem in 1921. The rotating disk acts as a fluid pump and is used as a model for centrifugal fans or compressors. This flow is classified under the category of steady flows in which vorticity generated at a solid surface is prevented from diffusing far away by an opposing convection, the other examples being the Blasius boundary layer with suction, stagnation point flow etc.

In fluid dynamics, a stagnation point flow refers to a fluid flow in the neighbourhood of a stagnation point or a stagnation line with which the stagnation point/line refers to a point/line where the velocity is zero in the inviscid approximation. The flow specifically considers a class of stagnation points known as saddle points wherein incoming streamlines gets deflected and directed outwards in a different direction; the streamline deflections are guided by separatrices. The flow in the neighborhood of the stagnation point or line can generally be described using potential flow theory, although viscous effects cannot be neglected if the stagnation point lies on a solid surface.

The shear viscosity of a fluid is a material property that describes the friction between internal neighboring fluid surfaces flowing with different fluid velocities. This friction is the effect of (linear) momentum exchange caused by molecules with sufficient energy to move between these fluid sheets due to fluctuations in their motion. The viscosity is not a material constant, but a material property that depends on temperature, pressure, fluid mixture composition, local velocity variations. This functional relationship is described by a mathematical viscosity model called a constitutive equation which is usually far more complex than the defining equation of shear viscosity. One such complicating feature is the relation between the viscosity model for a pure fluid and the model for a fluid mixture which is called mixing rules. When scientists and engineers use new arguments or theories to develop a new viscosity model, instead of improving the reigning model, it may lead to the first model in a new class of models. This article will display one or two representative models for different classes of viscosity models, and these classes are:

Capillary breakup rheometry is an experimental technique used to assess the extensional rheological response of low viscous fluids. Unlike most shear and extensional rheometers, this technique does not involve active stretch or measurement of stress or strain but exploits only surface tension to create a uniaxial extensional flow. Hence, although it is common practice to use the name rheometer, capillary breakup techniques should be better addressed to as indexers.

Schlichting jet is a steady, laminar, round jet, emerging into a stationary fluid of the same kind with very high Reynolds number. The problem was formulated and solved by Hermann Schlichting in 1933, who also formulated the corresponding planar Bickley jet problem in the same paper. The Landau-Squire jet from a point source is an exact solution of Navier-Stokes equations, which is valid for all Reynolds number, reduces to Schlichting jet solution at high Reynolds number, for distances far away from the jet origin.

Becker–Morduchow–Libby solution is an exact solution of the compressible Navier–Stokes equations, that describes the structure of one-dimensional shock waves. The solution was discovered in a restrictive form by Richard Becker in 1922, which was generalized by Morris Morduchow and Paul A. Libby in 1949. The solution was also discovered independently by M. Roy and L. H. Thomas in 1944The solution showed that there is a non-monotonic variation of the entropy across the shock wave. Before these works, Lord Rayleigh obtained solutions in 1910 for fluids with viscosity but without heat conductivity and for fluids with heat conductivity but without viscosity. Following this, in the same year G. I. Taylor solved the whole problem for weak shock waves by taking both viscosity and heat conductivity into account.

References

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.