Decay of Potential Vortex and Diffusion of Temperature in a Generalized Oldroyd-B Fluid through a Porous Medium

Based on a modified Darcy law, the decay of potential vortex and diffusion of temperature in a generalized Oldroyd-B fluid with fractional derivatives through a porous medium is studied. Exact solutions of the velocity and temperature fields are obtained in terms of the generalized Mittag-Leffler function by using the Hankel transform and discrete Laplace transform of the sequential fractional derivatives. One of the solutions is the sum of the Newtonian solutions and the non-Newtonian contributions. As limiting cases of the present solutions, the corresponding solutions of the fractional Maxwell fluid and classical Maxwell fluids are given. The influences of the fractional parameters, material parameters, and the porous space on the decay of the vortex are interpreted by graphical results

An extensible lattice Boltzmann method for viscoelastic flows: complex and moving boundaries in Oldroyd-B fluids

Most biological fluids are viscoelastic, meaning that they have elastic properties in addition to the dissipative properties found in Newtonian fluids. Computational models can help us understand viscoelastic flow, but are often limited in how they deal with complex flow geometries and suspended particles. Here, we present a lattice Boltzmann solver for Oldroyd-B fluids that can handle arbitrarily-shaped fixed and moving boundary conditions, which makes it ideally suited for the simulation of confined colloidal suspensions. We validate our method using several standard rheological setups, and additionally study a single sedimenting colloid, also finding good agreement with literature. Our approach can readily be extended to constitutive equations other than Oldroyd-B. This flexibility and the handling of complex boundaries holds promise for the study of microswimmers in viscoelastic fluids.Comment: 13 pages, 11 figure

Exact solutions for the unsteady rotational flow of a generalized second grade fluid through a circular cylinder

Here the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized second grade fluid within an infinite circular cylinder are determined by means of the Laplace and finite Hankel transforms. At time t = 0 the fluid is at rest and the motion is produced by the rotation of the cylinder around its axis with a time dependent angular velocity Ωt. The solutions that have been obtained are presented under series form in terms of the generalized G-functions. The similar solutions for the ordinary second grade and Newtonian fluids, performing the same motion, are obtained as special cases of our general solution

Taylor–Couette flow of a generalized second grade fluid due to a constant couple

The velocity field and the adequate shear stress, corresponding to the flow of a generalized second grade fluid in an annular region between two infinite coaxial cylinders, are determined by means of Laplace and finite Hankel transforms. The motion is produced by the inner cylinder which is rotating about its axis due to a constant torque f per unit length. The solutions that have been obtained satisfy all imposed initial and boundary conditions. For β → 1 or β → 1 and α1 → 0, the corresponding solutions for an ordinary second grade fluid, respectively, for the Newtonian fluid, performing the same motion, are obtained as limiting cases

Hybrid Lattice Boltzmann/Finite Difference simulations of viscoelastic multicomponent flows in confined geometries

We propose numerical simulations of viscoelastic fluids based on a hybrid algorithm combining Lattice-Boltzmann models (LBM) and Finite Differences (FD) schemes, the former used to model the macroscopic hydrodynamic equations, and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible non-linear elastic dumbbells with Peterlin's closure (FENE-P). The numerical model is first benchmarked by characterizing the rheological behaviour of dilute homogeneous solutions in various configurations, including steady shear, elongational flows, transient shear and oscillatory flows. As an upgrade of complexity, we study the model in presence of non-ideal multicomponent interfaces, where immiscibility is introduced in the LBM description using the "Shan-Chen" model. The problem of a confined viscoelastic (Newtonian) droplet in a Newtonian (viscoelastic) matrix under simple shear is investigated and numerical results are compared with the predictions of various theoretical models. The proposed numerical simulations explore problems where the capabilities of LBM were never quantified before.Comment: 32 Pages, 11 Figure

Numerical Simulation of Electroosmotic Flow of Viscoelastic Fluid in Microchannel

Electroosmotic flow (EOF) has been widely used in various biochemical microfluidic applications, many of which often involve the use of viscoelastic non-Newtonian fluids. Due to the existence of the elastic effect, the viscoelastic EOF develops into chaotic flow under extremely low Reynolds numbers, which is known as elastic turbulence. The mechanism of elastic turbulence in electroosmotic flow remains unclear. Numerical simulation plays an important role in understanding the mechanisms of elastic turbulence. This dissertation is aimed to study the EOF of viscoelastic fluids in constriction microchannels under various direct current (DC) and alternating current (AC) electric fields. First, the EOF of viscoelastic fluid in a straight contraction microchannel is investigated. The influences of the polymer concentration and the applied DC electric field on the elastic instabilities are analyzed. The flow fluctuations and secondary upstream vortices before the entrance of the microchannel are found to be related to the induced elastic stress within the microchannel. The polymer concentration shows a more significant influence on the elastic instability. A flow map in polymer concentration and electric field domain is formed as guidance for further studies. Then, the study is extended to the viscoelastic EOF in a microchannel with 90◦ bends under the combination of DC and AC electric fields. The elastic turbulence is identified from the fluctuation of the velocity field and upstream vortices. The energy spectra of the velocity fluctuation show power-law decay over a wide range of frequencies, which is a typical characteristic of elastic turbulence. The 90◦ bends show influence on the dye concentration profile in cross sections of the microchannel. A more even dye concentration distribution is obtained with an increasing number of 90◦ bends. Moreover, the opening angle of the particle trace at the exit of the contraction microchannel show dependency on the frequency of the AC electric field, which is related to the characteristic frequency of the viscoelastic EOF. The study is then focused on the influence of the frequency of the AC electric field on the viscoelastic EOF. Short contraction microchannels are adopted for the frequency study. The peak in the energy spectra of the velocity fluctuation under DC electric field indicates the characteristic frequency of the viscoelastic EOF. Under AC electric field, the highest amplitude of the energy spectra is obtained when the frequency of AC electric field is close to the characteristic frequency. The same trend is also observed in the statistical results of the average velocity. However, when the frequency is relatively high, both the amplitude of the energy spectra and the average velocity decrease to a level even lower than under a DC electric field, which indicates the existence of an optimal frequency of the AC electric field in order to achieve the highest flow rate