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We solve the free boundary problem for the dynamics of a cylindrical, axisymmetric vis-coelastic filament stretching in a gravity-driven extensional flow for the Upper Convected Maxwell and Oldroyd-B constitutive models. Assuming the axial stress in the filament has a spatial dependence provides the simplest coupling of viscoelastic effects to the...
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... will also consider two models that are simplifications of (2.7): (i) a Newtonian fluid (λ = 0 and λ r = 0), and (ii) an Upper Convected Maxwell fluid (λ r = 0). The boundary of the liquid filament is a free surface, which we denote by r = h(z, t), as shown in Figure 3. The fluid surrounding the filament is assumed to be at rest and have negligible viscosity. ...
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Citations
... Mathematical modeling-both numerical and analytical/semi-analytical approaches to the behavior modelling of living cells are difficult tasks, mainly due to the fact that material parameters of living biological tissues are submitted to set and laboratory measurements that are difficult to obtain. Based on the in vitro behavior and appearance of telopodes, an appropriate semi-analytical model for their characterization has been proposed, using viscoelastic elongation [43,46]. For the numerical model, we used the Ansys software (ANSYS, Inc., Pittsburgh, PA, USA) for multiphasic viscoelastic model with the same conditions as in the semi-analytical model in order to compare the two solutions. ...
Background:
Telocytes (TCs) are unique interstitial or stromal cells of mesodermal origin, defined by long cellular extensions called telopodes (Tps) which form a network, connecting them to surrounding cells. TCs were previously found around stem and progenitor cells, and were thought to be most likely involved in local tissue metabolic equilibrium and regeneration. The roles of telocytes are still under scientific scrutiny, with existing studies suggesting they possess various functions depending on their location.
Methods:
Human myometrium biopsies were collected from pregnant and non-pregnant women, telocytes were then investigated in myometrial interstitial cell cultures based on morphological criteria and later prepared for time-lapse microscopy. Semi-analytical and numerical solutions were developed to highlight the geometric characteristics and the behavior of telocytes.
Results:
Results were gathered in a database which would further allow efficient telocyte tracking and indexing in a content-based image retrieval (CBIR) of digital medical images. Mathematical analysis revealed pivotal information regarding the homogeneity, hardness and resistance of telocytes' structure. Cellular activity models were monitored in vitro, therefore supporting the creation of databases of telocyte images.
Conclusions:
The obtained images were analyzed, using segmentation techniques and mathematical models in conjunction with computer simulation, in order to depict TCs behavior in relation to surrounding cells. This paper brings an important contribution to the development of bioinformatics systems by creating software-based telocyte models that could be used both for diagnostic and educational purposes.
... The free surface moves, so the problem domain varies in time. The governing equations together with the conditions corresponding to the free surface determine a coupled ordinary differential equations [6,7,8]. is presented in figures 3 and 4, respectively. ...
Modeling living cells behavior is a delicate mission as it involves alike biological, physical, chemical, electrical aspects. This article presents a semi-analytical modeling for telopodes elongation, based on their appearance and behavior captured from in vitro approaches. The semi-analytical solution is determined solving a system of five coupled differential equations and the results are compared both with the numerical solution and with the results of the laboratory tests. A good agreement is obtained.
... As far as we know, studies for viscoelastic material coiling are only limited to laboratory experiments and scaling analysis, and there is rarely analytically modeling work in this field. Related modeling work includes that some researchers developed 1-D axisymmetric slender jet models with straight axis by using the upper-convected Maxwell and Oldroyd-B model [15,16]. For the curved axis, Reneker et al. [17] developed the discrete viscoelastic dumbbell thread model which described the stretching, but omitted the torsion and bending. ...
Currently, coiling problems for an elastic rope and viscous jet have been extensively studied, but there are few theoretical studies for modeling of viscoelastic jet coiling. In this paper, we have established a continuous one-dimensional (1-D) model for a weakly viscoelastic jet of Maxwell-type, considering jet radius variation and surface tension. The 1-D model is solved numerically by the continuation method to investigate the effects of relaxation time, the falling height, and the flow rate on the coiling frequency and falling time. The numerical results mainly show that for a weakly viscoelastic jet more frequency wiggles and lower frequencies appear than for the Newtonian flow.
... , where w(t) represents the velocity due to application of force F , the conditions corresponding to the free surface R=R(z,t) conditions are[10]: kinematic condition; if a fluid particle is adjacent to a boundary then we must impose a condition which links the velocity of the boundary to that of the particle: ...
Telocytes are a novel interstitial cell type characterized by a small cell body and a number of one to five of extremely long and thin prolongations, named telopodes. This article proposes an analytical and numerical modeling for telopodes elongation, based on their appearance and behavior captured from in vitro approaches. Both the analytical and numerical solutions are developed for a viscoelastic model and they are compared and a good agreement is obtained.
... The relatively low mass of a pendant drop (as compared to the weights used by Matta and Tytus [28] and in the current paper) also results in low imposed deformation rates; for example, the maximum accelerations achieved by Jones et al. were only approximately $0.5 g. A recent detailed analysis of this problem for the Maxwell/Oldroyd-B model shows that as the tensile stresses grow and retard further acceleration of the pendant droplet and the elongating filament, the dynamics can in fact relax back towards a dominant Newtonian balance [43] which again limits the use of this geometry as an extensional rheometer. ...
... The recent analysis of Smolka et al. [43] highlights a key strength of the CFP technique suggested by Matta and Tytus: Applying a high enough initial force allows one to impose filament deformation rates dln L/dt that are sufficiently fast to overcome the relaxation time k and the relaxation processes in the elongating fluid filament. With Weissenberg numbers Wi = kdln L/dt ) 1 the constant force pull ultimately imposes an almost affine Fig. 1. ...
We revisit the rapid stretching of a liquid filament under the action of a constant imposed tensile force, a problem which was first considered by Matta and Tytus [J. Non-Newton. Fluid Mech. 35 (1990) 215-229]. A liquid bridge formed from a viscous Newtonian fluid or from a dilute polymer solution is first established between two cylindrical disks. The upper disk is held fixed and may be connected to a force transducer while the lower cylinder falls due to gravity. By varying the mass of the falling cylinder and measuring its resulting acceleration, the viscoelastic nature of the elongating fluid filament can be probed. In particular, we show that with this constant force pull (CFP) technique it is possible to readily impose very large material strains and strain rates so that the maximum extensibility of the polymer molecules may be quantified. This unique characteristic of the experiment is analyzed numerically using the FENE-P model and two alternative kinematic descriptions; employing either an axially-uniform filament approximation or a quasi two-dimensional Lagrangian description of the elongating thread. In addition, a second order pertubation theory for the trajectory of the falling mass is developed for simple viscous filaments. Based on these theoretical considerations we develop an expression that enables estimation of the finite extensibility parameter characterizing the polymer solution in terms of quantities that can be extracted directly from simple measurement of the time-dependent filament diameter. (C) 2011 Elsevier B.V. All rights reserved.
... A filament can be stretched over many layer thicknesses after the vertical deflection W exceeds the initial layer thickness, h (e.g., Smolka and Belmonte, 2003;Smolka et al., 2004). Unlike that in incompressible Newtonian fluids, downward stretching of a viscoelastic filament results in vertical dilatation and horizontal contraction and creates a dynamic restoring force that must be relaxed before the drop can descend. ...
... Appendix B. Drop theory of a viscoelastic fluid Smolka and Belmonte (2003;Smolka et al., 2004) described and analyzed growth of a drop and connecting filament, which lengthens during growth, using a first-order approximation in terms of springforce equation (y is downward positive): ...
To understand how elasticity affects convective instability of viscoelastic fluids near the viscous limit, we carried out numerical experiments of Rayleigh–Taylor instability of compressible viscoelastic fluids overlying inviscid inelastic substrata (hence with infinite viscosity and elasticity ratios). Unlike incompressible viscous fluids, for which growth becomes super-exponential when perturbations to the thickness of the unstable layer grow to several tens of percent of the thickness, for compressible viscoelastic fluids, super-exponential growth does not appear to develop at relatively low Deborah numbers, De=Δρgh/η∼10−6–10−4 (where Δρ is the density difference between unstable layer and substratum, g is gravity, h is the thickness of the unstable layer, and η is its viscosity) and for very high (>106) ratios of viscosity between layer and substratum, which characterize large-scale geodynamic systems. This behavior differs from that of viscoelastic two-layer systems with higher Deborah numbers (>10−4) and with smaller viscosity ratios (
... 3 (a) and (b)). A comparison of the interfacial motion of this filament for the 780 ppm xanthan gum solution with 0% ≤ [KCl] ≤ 0.047% to an analytic model shows strong quantitative agreement [34]. The model, which also provides the stretch rate in the filament, predicts De > 1 over a transient period while the filament initially stretches [34] . ...
... A comparison of the interfacial motion of this filament for the 780 ppm xanthan gum solution with 0% ≤ [KCl] ≤ 0.047% to an analytic model shows strong quantitative agreement [34]. The model, which also provides the stretch rate in the filament, predicts De > 1 over a transient period while the filament initially stretches [34] . After this transient period, the stretch rate monotonically decreases so that eventually De < 1 [34]. ...
... The model, which also provides the stretch rate in the filament, predicts De > 1 over a transient period while the filament initially stretches [34] . After this transient period, the stretch rate monotonically decreases so that eventually De < 1 [34]. Thus we expect viscoelastic effects to be relevant to the filament motion during some initial time period in the experiment. ...
We experimentally investigate the filament dynamics of non-Newtonian drops falling under gravity through air, and report the effects of semi-dilute concentrations of the polyelectrolyte polymer xanthan gum mixed in 80:20 glycerol/water with varying concentrations of potassium chloride (KCl). We find that the addition of 780 ppm xanthan gum to the glycerol/water solvent dramatically increases the drop length just before pinch-off, by two orders of magnitude. This length is decreased by as much as an order of magnitude with the addition of KCl. The qualitative dynamics of bead formation on the filament is also very sensitive to added salt. These experimental observations are related to well-known changes in the molecular structure of xanthan gum due to charge screening effects.
... Two of the speakers presented research on filaments and jets: Linda Smolka presented her work on liquid filaments in extensional flows [16]. She considered exact, cylindrical solutions in which the radius is time-dependent and used these in a study of various viscoelastic constitutive models. ...
This five-day multidisciplinary workshop focussed on the mathematics of free-surface fluid flow. Building on theoretical and experimental developments of the last few years, the workshop brought together mathematicians and physicists at the forefront of research in experiments, analysis, computation, and modelling. The workshop yielded a rich exchange of ideas in the areas of thin liquid films, dynamic contact lines, slender jets, Hele-Shaw flow, and fluid interfaces.
The human tear film is a multilayer structure in which the dynamics are often strongly affected by a floating lipid layer. That layer has liquid crystalline characteristics and plays important roles in the health of the tear film. Previous models have treated the lipid layer as a Newtonian fluid in extensional flow. Motivated to develop a more realistic treatment, we present a model for the extensional flow of thin sheets of nematic liquid crystal. The rod-like molecules of these substances impart an elastic contribution to the rheology. We rescale a weakly elastic model due to Cummings et al. [“Extensional flow of nematic liquid crystal with an applied electric field,” Eur. J. Appl. Math. 25, 397–423 (2014).] to describe a lipid layer of moderate elasticity. The resulting system of two nonlinear partial differential equations for sheet thickness and axial velocity is fourth order in space, but still represents a significant reduction of the full system. We analyze solutions arising from several different boundary conditions, motivated by the underlying application, with particular focus on dynamics and underlying mechanisms under stretching. We solve the system numerically, via collocation with either finite difference or Chebyshev spectral discretization in space, together with implicit time stepping. At early times, depending on the initial film shape, pressure either aids or opposes extensional flow, which changes the free surface dynamics of the sheet and can lead to patterns reminiscent of those observed in tear films. We contrast this finding with the cases of weak elasticity and Newtonian flow, where the sheet retains the same qualitative shape throughout time.
Motivated by problems arising in tear film dynamics, we present a model for the extensional flow of thin sheets of nematic liquid crystal. The rod-like molecules of these substances impart an elastic contribution to its response. We rescale a weakly elastic model due to Cummings et al. [European Journal of Applied Mathematics 25 (2014): 397-423] to describe a case of moderate elasticity. The resulting system of two nonlinear partial differential equations for sheet thickness and axial velocity is nonlinear and fourth order in space, but still represents a significant reduction of the full system. We analyze solutions arising from several different boundary conditions, motivated by the underlying application, with particular focus on dynamics and underlying mechanisms under stretching. We solve the system numerically, via collocation with either finite difference or Chebyshev spectral discretization in space, together with implicit time stepping. At early times, depending on the initial film shape, pressure either aids or opposes extensional flow, which changes the shape of the sheet and may result in the loss of a minimum or maximum at the moving end. We contrast this finding with the cases of weak elasticity and Newtonian flow, where the sheet retains all extrema from the initial condition throughout time.
