Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids

We study the shape differentiability of a cost function for the flow of an incompressible viscous fluid of power-law type. The fluid is confined to a bounded planar domain surrounding an obstacle. For smooth perturbations of the shape of the obstacle we express the shape gradient of the cost function which can be subsequently used to improve the initial design

Simplex space-time meshes in thermally coupled two-phase flow simulations of mold filling

The quality of plastic parts produced through injection molding depends on many factors. Especially during the filling stage, defects such as weld lines, burrs, or insufficient filling can occur. Numerical methods need to be employed to improve product quality by means of predicting and simulating the injection molding process. In the current work, a highly viscous incompressible non-isothermal two-phase flow is simulated, which takes place during the cavity filling. The injected melt exhibits a shear-thinning behavior, which is described by the Carreau-WLF model. Besides that, a novel discretization method is used in the context of 4D simplex space-time grids [2]. This method allows for local temporal refinement in the vicinity of, e.g., the evolving front of the melt [10]. Utilizing such an adaptive refinement can lead to locally improved numerical accuracy while maintaining the highest possible computational efficiency in the remaining of the domain. For demonstration purposes, a set of 2D and 3D benchmark cases, that involve the filling of various cavities with a distributor, are presented.Comment: 14 pages, 11 Figures, 4 Table

A note on the breathing mode of an elastic sphere in Newtonian and complex fluids

Experiments on the acoustic vibrations of elastic nanostructures in fluid media have been used to study the mechanical properties of materials, as well as for mechanical and biological sensing. The medium surrounding the nanostructure is typically modeled as a Newtonian fluid. A recent experiment however suggested that high-frequency longitudinal vibration of bipyramidal nanoparticles could trigger a viscoelastic response in water-glycerol mixtures [M. Pelton et al., "Viscoelastic flows in simple liquids generated by vibrating nanostructures," Phys. Rev. Lett. 111, 244502 (2013)]. Motivated by these experimental studies, we first revisit a classical continuum mechanics problem of the purely radial vibration of an elastic sphere, also called the breathing mode, in a compressible viscous fluid, and then extend our analysis to a viscoelastic medium using the Maxwell fluid model. The effects of fluid compressibility and viscoelasticity are discussed. Although in the case of longitudinal vibration of bipyramidal nanoparticles, the effects of fluid compressibility were shown to be negligible, we demonstrate that it plays a significant role in the breathing mode of an elastic sphere. On the other hand, despite the different vibration modes, the breathing mode of a sphere triggers a viscoelastic response in water-glycerol mixtures similar to that triggered by the longitudinal vibration of bipyramidal nanoparticles. We also comment on the effect of fluid viscoelasticity on the idea of destroying virus particles by acoustic resonance

Phase-field simulations of viscous fingering in shear-thinning fluids

A phase-field model for the Hele-Shaw flow of non-Newtonian fluids is developed. It extends a previous model for Newtonian fluids to a wide range of shear-dependent fluids. The model is applied to perform simulations of viscous fingering in shear- thinning fluids, and it is found to be capable of describing the complete crossover from the Newtonian regime at low shear rate to the strongly shear-thinning regime at high shear rate. The width selection of a single steady-state finger is studied in detail for a 2-plateaux shear-thinning law (Carreau law) in both its weakly and strongly shear-thinning limits, and the results are related to previous analyses. In the strongly shear-thinning regime a rescaling is found for power-law (Ostwald-de-Waehle) fluids that allows for a direct comparison between simulations and experiments without any adjustable parameters, and good agreement is obtained

A model for deformable roll coating with negative gaps and incompressible compliant layers

A soft elastohydrodynamic lubrication model is formulated for deformable roll coating involving two contra-rotating rolls, one rigid and the other covered with a compliant layer. Included is a finite-strip model (FSM) for the deformation of the layer and a lubrication model with suitable boundary conditions for the motion of the fluid. The scope of the analysis is restricted to Newtonian fluids, linear elasticity/viscoelasticity and equal roll speeds, with application to the industrially relevant highly loaded or 'negative gap' regime. Predictions are presented for coated film thickness, interroll thickness, meniscus location, pressure and layer deformation as the control parameters - load (gap), elasticity, layer thickness and capillary number, Ca - are varied. There are four main results: \ud (i) Hookean spring models are shown to be unable to model effectively the deformation of a compliant layer when Poisson's ratio nu --> 0.5. In particular, they fall to predict the swelling of the layer at the edge of the contact region which increases as v - 0.5; they also fail to locate accurately the position of the meniscus, X-M, and to identify the presence, close to the meniscus, of a 'nib' (constriction in gap thickness) and associated magnification of the sub-ambient pressure loop. (ii) Scaling arguments suggest that layer thickness and elasticity may have similar effects on the field variables. It is shown that for positive gaps this is true, whereas for negative gaps they have similar effects on the pressure profile and flow rate yet quite different effects on layer swelling (deformation at the edge of the contact region) and different effects on X-M. (iii) For negative gaps and Ca similar to O(1), the effect of varying either viscosity or speed and hence Ca is to significantly alter both the coating thickness and X-M. This is contrary to the case of fixed-gap rigid roll coating. (iv) Comparison between theoretical predictions and experimental data shows quantitive agreement in the case of X-M and qualitive agreement for flow rate. It is shown that this difference in the latter case may be due to viscoelastic effects in the compliant layer

Shape Optimization of hemolysis for shear thinning flows in moving domains

We consider the $3$D problem of hemolysis minimization in blood flows, namely the minimization of red blood cells damage, through the shape optimization of moving domains. Such a geometry is adopted to take into account the modeling of rotating systems and blood pumps. The blood flow is described by generalized Navier-Stokes equations, in the particular case of shear thinning flows. The velocity and stress fields are then used as data for a transport equation governing the hemolysis index, aimed to measure the red blood cells damage rate. For a sequence of converging moving domains, we show that a sequence of associated solutions to blood equations converges to a solution of the problem written on the limit moving domain. Thus, we extended the result given in (Soko\l{}owski, Stebel, 2014, in \textit{Evol. Eq. Control Theory}) for $q \geq 11/5$, to the range $6/5< q < 11/5$, where $q$ is the exponent of the rheological law. We then show that the sequence of hemolysis index solutions also converges to the limit solution. This shape continuity properties allows us to show the existence of minimal shapes for a class of functionals depending on the hemolysis index