Propulsion in a viscoelastic fluid

Flagella beating in complex fluids are significantly influenced by viscoelastic stresses. Relevant examples include the ciliary transport of respiratory airway mucus and the motion of spermatozoa in the mucus-filled female reproductive tract. We consider the simplest model of such propulsion and transport in a complex fluid, a waving sheet of small amplitude free to move in a polymeric fluid with a single relaxation time. We show that, compared to self-propulsion in a Newtonian fluid occurring at a velocity U_N, the sheet swims (or transports fluid) with velocity U / U_N = [1+De^2 (eta_s)/(eta) ]/[1+De^2], where eta_s is the viscosity of the Newtonian solvent, eta is the zero-shear-rate viscosity of the polymeric fluid, and De is the Deborah number for the wave motion, product of the wave frequency by the fluid relaxation time. Similar expressions are derived for the rate of work of the sheet and the mechanical efficiency of the motion. These results are shown to be independent of the particular nonlinear constitutive equations chosen for the fluid, and are valid for both waves of tangential and normal motion. The generalization to more than one relaxation time is also provided. In stark contrast with the Newtonian case, these calculations suggest that transport and locomotion in a non-Newtonian fluid can be conveniently tuned without having to modify the waving gait of the sheet but instead by passively modulating the material properties of the liquid.Comment: 21 pages, 1 figur

Resonant oscillations of a plate in an electrically conducting rotating Johnson-Segalman fluid

AbstractAn analysis of hydromagnetic flow is examined in a semi-infinite expanse of electrically conducting rotating Johnson-Segalman fluid bounded by nonconducting plate in the presence of a transverse magnetic field and the governing equations are modeled first time. The structure of the velocity distribution and the associated hydromagnetic boundary layers are investigated including the case of resonant oscillations. It is shown that unlike the hydrodynamic situation for the case of resonance, the hydromagnetic steady solution satisfies the boundary condition at infinity. The inherent difficulty involved in the hydrodynamic resonance case has been resolved in the presence analysis

On a modified non-singular log-conformation formulation for Johnson-Segalman viscoelastic fluids

International audienceA modified log-conformation formulation of viscoelastic fluid flows is presented in this paper. This new formulation is non-singular for vanishing Weissenberg numbers and allows a direct steady numerical resolution by a Newton method. Moreover, an exact computation of all the terms of the linearized problem is provided. The use of an exact divergence-free finite element method for velocity-pressure approximation and a discontinuous Galerkin upwinding treatment for stresses leads to a robust discretization. A demonstration is provided by the computation of steady solutions at high Weissenberg numbers for the difficult benchmark case of the lid driven cavity flow. Numerical results are in good agreement, qualitatively with experiment measurements on real viscoelastic flows, and quantitatively with computations performed by others authors. The numerical algorithm is both robust and very efficient, as it requires a low mesh-invariant number of linear systems resolution to obtain solutions at high Weissenberg number. An adaptive mesh procedure is also presented: it alows representing accurately both boundary layers and the main and secondary vorties

Nonlinear Fluid Dynamics Description of non-Newtonian Fluids

Nonlinear hydrodynamic equations for visco-elastic media are discussed. We start from the recently derived fully hydrodynamic nonlinear description of permanent elasticity that utilizes the (Eulerian) strain tensor. The reversible quadratic nonlinearities in the strain tensor dynamics are of the 'lower convected' type, unambiguously. Replacing the (often neglected) strain diffusion by a relaxation of the strain as a minimal ingredient, a generalized hydrodynamic description of viscoelasticity is obtained. This can be used to get a nonlinear dynamic equation for the stress tensor (sometimes called constitutive equation) in terms of a power series in the variables. The form of this equation and in particular the form of the nonlinear convective term is not universal but depends on various material parameters. A comparison with existing phenomenological models is given. In particular we discuss how these ad-hoc models fit into the hydrodynamic description and where the various non-Newtonian contributions are coming from.Comment: Acta Rheologic

Computational analysis of non-isothermal flow of non-Newtonian fluids

The dynamics of complex fluids under various conditions is a model problem in bio-fluidics and in process industries. We investigate a class of such fluids and flows under conditions of heat and/or mass transfer. Experiments have shown that under certain flow conditions, some complex fluids (e.g. worm-like micellar solutions and some polymeric fluids) exhibit flow instabilities such as the emergence of regions of different shear rates (shear bands) within the flow field. It has also been observed that the reacting mixture in reaction injection molding of polymeric foams undergoes self-expansion with evolution of heat due to exothermic chemical reaction. These experimental observations form the foundation of this thesis. We explore the heat and mass transfer effects in various relevant flow problems of complex fluids. In each case, we construct adequate mathematical models capable of describing the experimentally observed flow phenomena. The mathematical models are inherently intractable to analytical treatment, being nonlinear coupled systems of time dependent partial differential equations. We therefore develop computational solutions for the model problems. Depending on geometrical or mathematical complexity, finite difference or finite volume methods will be adopted. We present the results from our numerical simulations via graphical illustrations and validate them (qualitatively) against' similar' results in the literature; the quotes being necessary in keeping in mind the novelties introduced in our investigations which are otherwise absent in the existing literature. In the case where experimental data is available, we validate our numerical simulations against such experimental results

Hydrodynamic stress on fractal aggregates of spheres

We calculate the average hydrodynamic stress on fractal aggregates of spheres using Stokesian dynamics. We find that for fractal aggregates of force-free particles, the stress does not grow as the cube of the radius of gyration, but rather as the number of particles in the aggregate. This behavior is only found for random aggregates of force-free particles held together by hydrodynamic lubrication forces. The stress on aggregates of particles rigidly connected by interparticle forces grows as the radius of gyration cubed. We explain this behavior by examining the transmission of the tension along connecting lines in an aggregate and use the concept of a persistance length in order to characterize this stress transmission within an aggregate

Global well-posedness for two-dimensional flows of viscoelastic rate-type fluids with stress diffusion

We consider the system of partial differential equations governing two-dimensional flows of a robust class of viscoelastic rate-type fluids with stress diffusion, involving a general objective derivative. The studied system generalizes the incompressible Navier--Stokes equations for the fluid velocity $v$ and pressure $p$ by the presence of an additional term in the constitutive equation for the Cauchy stress expressed in terms of a positive definite tensor $B$. The tensor $B$ evolves according to a diffusive variant of an equation that can be viewed as a combination of corresponding counterparts of Oldroyd-B and Giesekus models. Considering spatially periodic problem, we prove that for arbitrary initial data and forcing in appropriate $L^2$ spaces, there exists a unique globally defined weak solution to the equations of motion, and more regular initial data and forcing launch a more regular solution with \bs B positive definite everywhere