N. Phan-Thien

The role of hydrodynamic interaction in the locomotion of microorganisms

A general Boundary Element Method is presented and benchmarked with existing Slender Body Theory results and reflection solutions for the motion of spheres and slender bodies near plane boundaries. This method is used to model the swimming of a microorganism with a spherical cell body, propelled by a single rotating flagellum. The swimming of such an organism near a plane boundary, midway between two plane boundaries or in the vicinity of another similar organism, is investigated. It is found that only a small increase (less than 10%) results in the mean swimming speed of an organism swimming near and parallel to another identical organism. Similarly, only a minor propulsive advantage (again, less than 10% increase in mean swimming speed) is predicted when an organism swims very close and parallel to plane boundaries (such as a microscopic plate and (or) a coverslip, for example). This is explained in terms of the flagellar propulsive advantage derived from an increase in the ratio of the normal to tangential resistance coefficients of a slender body being offset by the apparently equally significant increase in the cell body drag. For an organism swimming normal to and toward a plane boundary, however, it is predicted that (assuming it is rotating its flagellum, relative to its cell body, with a constant angular frequency) the resulting swimming speed decreases asymptotically as the organism approaches the boundary.

Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method

Abstract In this paper, a general class of viscoelastic model is used to investigate numerically the pattern and strength of the secondary flows in rectangular pipes as well as the influence of material parameters on them. To solve the coupled governing equation system, an implicit finite volume method based on the SIMPLEST algorithm, which is applicable for both time-dependent and steady-state flow computations, has been developed and extended for viscoelastic flow computations by applying the decoupled techniques. The main feature of the method is to split the solution process into a series of steps in which the continuity of the flow field is enforced by solving a Poissons equation for the pressure, and at the end of the steps, both the pressure and velocity fields are made to satisfy one and the same momentum equation. For viscoelastic flow computations, artificial diffusion terms are introduced on both sides of the discretized constitutive equations to improve numerical stability. It is found that there are in total two vortices in each quadrant of the pipe at different aspect ratios (from 1 to 16), and at each ratio the pattern of secondary flows takes the same form for different material parameters, but their strength is very sensitive to the viscoelastic material parameters. Numerical results indicate that the presence of secondary flow strongly depends on the primary flow rate and the elasticity of the fluid, namely, the first and the second normal stress differences as well as their functional departure from the constant multiple viscosity.

Three dimensional numerical simulations of viscoelastic flows through planar contractions

Abstract We present in this paper a fully three dimensional (3D) convergent numerical study of planar viscoelastic contraction flows. A 3D finite volume method (FVM) with the primary variable elastic viscous split stress (EVSS) formulation is employed, and a very efficient 3D block solver coupled with block correction is developed to speed up the convergence rate. Full 3D simulations of viscoelastic flows in 4:1 planar abrupt contractions are carried out using experimental conditions. Upstream vortex patterns comparable with the existing flow visualisation observations are captured using the upper convected Maxwell model for a Boger fluid and the Phan-Thien–Tanner model for a shear thinning fluid. Comprehensive comparisons between numerical simulation results and data measured in the dynamic fields in a 4:1 planar abrupt contraction are made, and the results indicate that the experimental measurements can be quantitatively reproduced if the fluid is well characterised by an appropriate viscoelastic model. It is confirmed numerically that the shear thinning of the fluid reduces the intensity of the singularity of viscoelastic flow near the re-entrant corner. With the Oldroyd-B model, by extensive computations on successively refined meshes with the minimum dimensionless size being 0.16–0.014 on the contraction plane in 2D configuration, it is revealed that, although the asymptotic flow behavior near the re-entrant corner and the build-up of the overall pressure and extensional stresses as well as the kinematic behavior along the centreline are insensitive to mesh refinement, completely different vortex activities may be predicted if the mesh is not sufficiently fine. It is verified numerically that, depending on the flow inertia and rheological properties of fluids, both the lip vortex mechanism and the corner vortex mechanism may be responsible for the vortex activities of viscoelastic fluids in 4:1 planar contraction flow, and the elasticity number E and Mach number M of the flow can be used to determine the vortex mechanism approximately. It is clear that the development process of the vortex activities could be underestimated with 2D simplification, and overpredicted with the creeping flow assumption, particularly when Re>0.5. Therefore, in planar contraction flow analyses, numerical artifacts may be produced with a coarse mesh, and 2D flow simulation is only a good approximation to the fully 3D flow if the upstream aspect ratio W/H in the experiment is at least 10.

Galerkin/least-square finite-element methods for steady viscoelastic flows

The elastic viscous split stress formulation (EVSS) and the discrete EVSS formulation (DEVSS) are effective in stabilizing numerical simulations of viscoelastic flows and have been widely used. Following the concept of Galerkin least-square perturbations proposed by Hughes et al. [Comput. Meth. Appl. Mech. Eng. 73 (1989) 173–189] and Franca et al. [SIAM J. Numer. Anal. 28(6) (1991) 1680–1697; Comput . Meth. Appl. Mech. Eng. 99 (1992) 209–233; Ibid. 104 (1993) 31–48] we are able to give the DEVSS formulation a new explanation as a perturbation to the Galerkin method based on the strain-rate residual, and furthermore, introduce another stabilized formulation, here named as MIX1, based on the incompressibility residual of the finite element discretizations. The three formulations (EVSS, DEVSS, MIX1), combined with a h–p type finite element algorithm that employs the SUPG technique to solve the viscoelastic constitutive equations are then tested on three benchmark problems: the flow of the upper-convected Maxwell fluid between eccentric cylinders, the flow of the Maxwell fluid around a sphere in a tube and the flow of the Maxwell and Oldroyd-B fluids around a cylinder in a channel. The results are checked with previous published works; good agreement is observed. Our numerical experiments convincingly demonstrate that the MIX1 is an accurate algorithm and convergent in terms of the p-extension, it has the same level of stability and robustness as the DEVSS method and is superior to the EVSS method in some respects. More important is that with MIX1 method one needs not solve for the strain-rate tensor as in EVSS and DEVSS methods, therefore, the CPU time consumption in the MIX1 method especially when using a coupled iteration scheme can be radically reduced. The success of the MIX1 method presents a challenge to the widely accepted concept of making the momentum equation explicitly elliptic.

A direct simulation of fibre suspensions

Abstract A numerical method to simulate fibre suspensions in shear flow is reported, which takes into account short range interaction via lubrication forces and long range interaction via slender body approximation, together with an appropriate Ewald summation technique. The numerical results are averaged to produce macroscopic properties of the suspension, including the Folgar–Tucker diffusion constant, the structure functions, and the reduced viscosity. In the semi-concentrated to concentrated regime, the fibres no longer follow Jefferys orbits, they align mostly with the shear direction. Numerical data on the diffusivity constant, the structure functions, and the reduced viscosity agree reasonably well with available experimental data.

Linear viscoelastic properties of bovine brain tissue in shear

We report the results from a series of rheological tests of fresh bovine brain tissue. Using a standard Bohlin VOR shear rheometer, shear relaxation and oscillating strain sweep experiments were performed on disks of brain tissue 30 mm in diameter, with a thickness of 1.5-2 mm. The strain sweep experiment showed that the viscoelastic strain limit is of the order of 0.1% strain. Shear relaxation data do not indicate the presence of a long-term elastic modulus, indicating fluid-like behavior. A relaxation spectrum was calculated by inverting the experimental data and used to predict oscillatory response, which agreed well with measured data.

Differential multiphase models for polydispersed suspensions and particulate solids

Abstract We report some novel applications of the differential scheme to construct the effective viscosity and moduli of suspensions of spheres of diverse sizes. Exact closed form expression for the effective viscosity of droplets of diverse sizes is derived, which includes the generalised Einstein formula of Brinkman and Roscoe. Exact closed form solutions for the effective moduli of a particulate solid consisting of rigid spheres of diverse sizes embedded in an elastic solid shows that the Poissons ratio of the composite should approach the value 1 5 at high volume fraction. The effect of a limited range of sizes of the included phase and locally-inhomogeneous distribution of inclusions is taken into account by considering a three-component system, with a fictitious inclusion phase made up of the matrix (solvent) material, which limits the space available for the real inclusions.

A boundary-element analysis of flagellar propulsion

The swimming of a flagellar micro-organism by the propagation of helical waves along its flagellum is analysed by a boundary-element method. The method is not restricted to any particular geometry of the organism nor does it assume a specific wave motion for the flagellum. However, only results for an organism with a spherical or ellipsoidal cell body and a helically beating flagellum are presented here. With regard to the flagellum, it is concluded that the optimum helical wave (amplitude α and wavenumber k ) has α k ≈ 1 (pitch angle of 45°) and that for the optimum flagellar length L / A = 10 ( L being the flagellar length, A being the radius of the assumed spherical cell body) the optimum number of wavelengths N λ is about 1.5. Furthermore there appears to be no optimal value for the flagellar radius a , with the thinner flagella being favoured. These conclusions show excellent quantitative agreement with those of slender-body theory. For the case of an ellipsoidal cell body, the optimum aspect ratios B / A and C / A of the ellipsoid are about 0.7 and 0.3 respectively; A , B and C are the principal radii of the ellipsoid. These and all of the above conclusions show good qualitative agreement with experimental observations of efficiently swimming micro-organisms.

Stress relaxation and oscillatory tests to distinguish between doughs prepared from wheat flours of different varietal origin

ABSTRACT The relaxation properties of flour-water-salt doughs prepared from four different flour types (weak, medium, strong, and extra strong) at different water absorption levels from 58 to 66% with protein contents of 10.0, 10.9, 13.2, and 11.8%, respectively, were studied by imposing varying strain amplitudes of 0.1–29%. Oscillatory tests in the linear viscoelastic region of the 66% absorption strong and weak dough cannot distinguish between the two types of dough. The inability to differentiate between dough types also applied to oscillatory tests on 58% absorption weak and 66% absorption strong doughs. However, the relaxation modulus of dough (extending over time) behaved quite distinctively at high strains, where dough samples experience large deformations. At strain amplitudes of ≤0.1% (i.e., in the linear viscoelastic region), different dough types behaved similarly. Likewise, the relaxation modulus completely relaxed at sufficiently long times. The magnitude of the modulus at intermediate- and h...

An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG

Abstract We report an adaptive viscoelastic stress splitting (AVSS) scheme, which was found to be extremely robust in the numerical simulation of viscoelastic flow involving steep stress boundary layers. The scheme is different from the elastic viscous split stress (EVSS) in that the local Newtonian component is allowed to depend adaptively on the magnitude of the local elastic stress. Two decoupled versions of the scheme were implemented for the Upper Convected Maxwell (UCM) model: the streamline integration (AVSS/SI), and the streamline upwind Petrov-Galerkin (AVSS/SUPG) methods of integrating the stress. The implementations were benchmarked against the known analytic Poiseuille solution, and no upper limiting Weissenberg number was found (the computation was stopped at Weissenberg number of O (10 4 )). The flow past a sphere in a tube was solved next, giving convergent results up to a Weissenberg number of 3.2 with the AVSS/SI and 1.55 with the AVSS/SUPG (both were decoupled schemes; without the adaptive scheme, the limiting Weissenberg number for the decoupled streamline integration method was about 0.3). These results show that the decoupled AVSS is more stable than the corresponding EVSS, and the SI is more robust than SUPG in solving the constitutive equation of hyperbolic type. In addition, we found a very long stress wake behind the sphere, and a weak vortex in the rear stagnation region at a Weissenberg number above W i of about 1.6, which does not seem to increase in size or strength with increasing W i .