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We present a new regularized Oldroyd-B model in three dimensions which satisfies an energy estimate analogous to that of the standard model, and maintains the positive semi-definiteness of the conformation tensor. This results in the unique existence and stability of global solutions in a periodic domain. To be precise, given an initial velocity u 0 and initial conformation tensor σ 0 , both with components in H 2 , we obtain a velocity u and conformation tensor σ both with components in C ([ 0 , T ] ; H 2) for all T > 0. Assuming better regularity for the initial data allows us to obtain better regularity for the solutions. We treat both the diffusive and non-diffusive cases of the model. Notably, the regularization in the equation for the conformation tensor in our new model has been applied only to the velocity, rather than to the conformation tensor, unlike other available regularization techniques [5]. This is desired since the stress, and thus the conformation tensor, is typically less regular than the velocity for the creeping flow of non-Newtonian fluids. In [9] the existence and regularity of solutions to the non-regularized two dimensional diffusive Oldroyd-B model was established. However, the proof cannot be generalized to three dimensions nor to the non-diffusive case. The proposed regularization overcomes these obstacles. Moreover, we show that the solutions in the diffusive case are stable in the H 2 norm. In the non-diffusive case, we are able to establish that the solutions are stable in the L 2 norm. Furthermore, we show that as the diffusivity parameter goes to zero, our solutions converge in the L 2 norm to the non-diffusive solution. [ABSTRACT FROM AUTHOR]

In this work, the decoupled finite particle coupled with smoothed particle hydrodynamics (DFP-SPH) method is firstly developed to simulate the viscoelastic fluid flow with free surface. A particle shifting technology is introduced to eliminate the tensile instability and nonuniformity of fluid particles. The motivations and embodiments include: (a) the partial derivatives of physical quantities of velocity, density and viscoelastic stress are calculated based on multivariate Taylor series and dominated contributions from diagonal matrices, which avoid the inversion of encountered ill-conditioned matrices potentially and improve the precision of numerical approximation to a certain extent; (b) a particle shifting technology is added to remove the tensile instability and nonuniformity of fluid particles in the frame of proposed DFP-SPH, and the determination of free surface particles is also proceeded in nature; (c) the dynamic boundary particles possessing computational simplicity are introduced and adopted in the proposed method. A moderate uniformization is proposed to eliminate the spatial effect of nonphysical boundary layers. Subsequently, the accuracy, consistency and convergence of present method are investigated on some benchmarks of Newtonian and Oldroyd-B fluids with analytical solutions. The validity of particle shifting technology in DFP-SPH is verified and multiple inter miscible viscoelastic drops impacting on the rigid wall are simulated and analyzed numerically for the capacity of proposed method. The procedure of jet bulking is well demonstrated through using Newtonian and Oldroyd-B models at different Weissenberg numbers, which possesses flexible free surfaces and tangled interfaces of multifluid layers. Numerical results show that the coupled particle model has a good performance on the viscoelastic fluid flow with free surface. [ABSTRACT FROM AUTHOR]

We consider the system of equations describing the flow of incompressible fluids in bounded domain. In the considered setting, the Cauchy stress tensor is a monotone mapping and has asymptotically (s − 1) -growth with the parameter s depending on the spatial and time variable. We do not assume any smoothness of s with respect to time variable and assume the log-Hölder continuity with respect to spatial variable. Such a setting is a natural choice if the material properties are instantaneous, e.g. changed by the switched electric field. We establish the long time and the large data existence of weak solution provided that s ≥ 3 d + 2 d + 2. [ABSTRACT FROM AUTHOR]

Non-Newtonian fluids are present in most manufacturing industry processes. Computational models used to describe the rheological behavior of such materials can be costly due to the non-linear behavior of the viscosity. This work presents a parametrized projection-based model reduction approach to address time-dependent generalized Newtonian fluids in a computational framework. We develop our discrete formulation using three ingredients: an offline-online setting for the model reduction based on a proper orthogonal and Tucker decompositions, the finite element method for the discretization of the spatial domain and finite differences for the temporal integration, and the variational multiscale approach as a stabilization technique for both the full and reduced order models. We also evaluate the reduced method with some numerical tests, where the first part involves testing the accuracy of the model reduction method for time as a single parameter of the dynamic reduced problem. The second part involves the solution of parametrized time-dependent reduced problems with the Reynolds number and the power-law index of the fluid as the varying parameters. [ABSTRACT FROM AUTHOR]

[Display omitted] • A portable magnetic resonance instrument is used to measure the flow behaviour indices of non-Newtonian fluids. • Velocity induced phase interference leads to a rheological dependence of signal magnitude. • Measured values of the flow behavior indices are in close agreement with literature values. • Various procedures for calibration and data analysis outlined and compared. In this work, we expand on past portable magnetic resonance flow methods and propose a novel method for characterizing circular (laminar) Couette flow of non-Newtonian fluids. Symmetry of the flow system combined with a constant magnetic field gradient leads to phase interference, affecting the signal magnitude, and net phase cancellation when averaging across the excited slice, preventing the use of phase-sensitive methods. Therefore, we utilize the dependence of signal magnitude at variable echo times and shear rates to characterize rheological properties. Theoretical equations governing the velocity distributions of fluids that obey a simple power-law model are used to obtain integral expressions for signal magnitude. Integral expressions can be simplified by approximating a thin excited slice or complete excitation of the Couette cell depending on experimental parameters. With simple data acquisition and analysis procedures employed, our measurements of the flow behavior indices of non-Newtonian xanthan gum dispersions are in close agreement with conventional rheological magnetic resonance measurements. [ABSTRACT FROM AUTHOR]

• Stabilised finite element method for incompressible non-Newtonian fluid flow. • Full consistency for low-order pairs by using only first-order derivatives. • Linearisation and decoupling of velocity-pressure-viscosity system via extrapolation. • Adaptive timestepping via predictor-corrector approach or fractional step θ -scheme. • Fast physics-based preconditioned Krylov subspace solver. Various materials and solid-fluid mixtures of engineering and biomedical interest can be modelled as generalised Newtonian fluids, as their apparent viscosity depends locally on the flow field. Despite the particular features of such models, it is common practice to combine them with numerical techniques originally conceived for Newtonian fluids, which can bring several issues such as spurious pressure boundary layers, unsuitable natural boundary conditions and coupling terms spoiling the efficiency of nonlinear solvers and preconditioners. In this work, we present a finite element framework dealing with such issues while maintaining low computational cost and simple implementation. The building blocks of our algorithm are (i) an equal-order stabilisation method preserving consistency even for lowest-order discretisations, (ii) robust extrapolation of velocities in the time-dependent case to decouple the rheological law from the overall system, (iii) adaptive time step selection and (iv) a fast physics-based preconditioned Krylov subspace solver, to tackle the relevant range of discretisation parameters including highly varying viscosity. Selected numerical experiments are provided demonstrating the potential of our approach in terms of robustness, accuracy and efficiency for problems of practical interest. [ABSTRACT FROM AUTHOR]

• Development of a refined two-scale model which numerically resolves fluid flow within a fracture in a poroelastic medium. • Complex fluid behaviour can be handled, as demonstrated for a Carreau fluid. • Pressure waves can be captured through inclusion of fluid inertia. • Study of the effect of mesh refinement on propagation velocity and possible pressure oscillations. A refined model is presented which numerically resolves the fluid flow within a fracture inside a saturated poroelastic material. At the discontinuity, the mass balance of the fluid is solved using the velocity profile inside the fracture, with the velocity profile being determined numerically from the momentum balance in the integration points at the discontinuity. The resolution of the mass balance and the velocity profile in the integration points at the discontinuity is coupled to surrounding poroelastic material model in a two-scale approach. The resulting monolithic scheme allows for complex fluid behaviour to be included, which is demonstrated via the inclusion of the fluid inertia and the use of a Carreau fluid. The governing equations are discretised using T-splines, while the fracture is represented by spline-based interface elements, and an implicit time discretisation scheme is used. Mesh refinement studies are carried out for a typical case, which contains a pressurised and propagating fracture. A coarse macro-scale mesh is sufficient to obtain the correct propagation velocities, but a finer mesh is needed to prevent pressure oscillations. Time step refinement studies show the capabilities of the model to capture pressure waves within the fracture. Finally, it is shown that if inertial effects are limited and a Newtonian fluid is used, a fracture scale discretisation using a single element is sufficient. However, for a Carreau fluid or when the problem is inertia-dominated, smaller elements are needed to correctly represent the velocity profile. [ABSTRACT FROM AUTHOR]

[Display omitted] • Cerebrospinal fluid flow is driven by the arterial pulsation. • Net flow velocity in numerical simulations agrees with that in animal experiments. • The net flow velocity is theoretically given by the lubrication theory in analogy with Taylor's swimming sheet model. • Lubrication theory rationalizes the net flow veolocity. The lubrication theory used is analogous to Taylor's swimming sheet model. Cerebrospinal fluid (CSF) flow in the perivascular space (PVS), which surrounds the arteries in the brain, is of paramount importance in the removal of metabolic waste. Despite a number of experimental and numerical studies regarding CSF flow, the underlying mechanics of CSF flow are still debated, especially regarding whether an arterial pulsation can indeed produce net CSF flow velocity. Furthermore, the relationship between CSF flow and arterial wall pulsation has not been fully defined. To clarify these questions, we numerically investigated the CSF flow in the PVS in an axisymmetric channel with a pulsating boundary, where CSF is modeled as an incompressible, Newtonian viscous fluid in non-porous space. Our numerical results show that the net CSF flow velocity driven by the arterial pulsation is consistent with that of previous animal experiments. However, the peak oscillatory velocity is two orders of magnitude larger than the net velocity. Interestingly, the net CSF flow velocity collapses on the analytical solution derived from the lubrication theory in analogy with Taylor's swimming sheet model. [ABSTRACT FROM AUTHOR]

• Mathematical framework and well-conditioned computational tools for resolvent analysis. • A feedback interconnection of the frequency response operator with its adjoint. • Application to channel flows of Newtonian and viscoelastic fluids. Modal and nonmodal analyses of fluid flows provide fundamental insight into the early stages of transition to turbulence. Eigenvalues of the dynamical generator govern temporal growth or decay of individual modes, while singular values of the frequency response operator quantify the amplification of disturbances for linearly stable flows. In this paper, we develop well-conditioned ultraspherical and spectral integration methods for frequency response analysis of channel flows of Newtonian and viscoelastic fluids. Even if a discretization method is well-conditioned, we demonstrate that calculations can be erroneous if singular values are computed as the eigenvalues of a cascade connection of the frequency response operator and its adjoint. To address this issue, we utilize a feedback interconnection of the frequency response operator with its adjoint to avoid computation of inverses and facilitate robust singular value decomposition. Specifically, in contrast to conventional spectral collocation methods, the proposed method (i) produces reliable results in channel flows of viscoelastic fluids at high Weissenberg numbers (∼500); and (ii) does not require a staggered grid for the equations in primitive variables. [ABSTRACT FROM AUTHOR]