Generation of an average flow by a pulsating stream near a curved free surface

The generation of an average flow near a curved free surface under the action of small-amplitude harmonic translational vibrations is studied. It is found that, in contrast to the case of a flat non-deformable free surface, an average flow is generated in the viscous boundary layer. This flow, propagating in the bulk fluid, can be described by the steady-state Navier-Stokes equations with effective boundary conditions for the shear stresses. It is shown that, in contrast to the flow generation near a solid surface, the flow generated near the free surface depends on the fluid viscosity and the curvature of the surface.     

Squeezing a viscoelastic liquid from a wedge: an exact solution

The paper reports an exact solution for the squeezing flow from a wedge of a general viscoelastic liquid. To obtain numerical values for the field variables, a network model that allows stress overshoot and shear-thinning in the start-up of a shear flow is adopted. It is found that both these features are important in this transient flow; stress overshoot is responsible for a stiffer response of the fluid (compared to the inelastic case) at moderate time —at large time, shear-thinning dominates and the fluid behaves like an inelastic fluid. On the other hand, the Oldroyd-B fluid always predicts a softer response than the Newtonian one. Furthermore, there is a limiting Weissenberg number above which one component of the stresses of the Oldroyd-B fluid increases unboundedly with time. This limiting Weissenberg number is approximately $\text{sol2}\text{3}$.     

A finite element formulation for laminar flow of a fluid with microstructure

A finite element formulation for the steady laminar flow of an incompressible fluid with microstructure has been developed. The particular fluids considered are commonly known as micropolar fluids, in which case suspended particulate microstructures are modelled by an ‘extended’ continuum formulation. The particle microspin is a new kinematic variable which is independent of the classical vorticity vector and thereby allows relative rotation between particles and the surrounding fluid. This formulation also gives rise to couple stresses in addition to classical force or traction stresses. The finite element formulation utilizes a variational approach and imposes conservation of mass through a penalty function. A general boundary condition for microspin has been incorporated whereby microspin at a solid boundary is constrained to be proportional to the fluid vorticity. The proportionality constant in this case can vary from zero to unity. Sample solutions are presented for fully developed flow through a straight tube and compared with an analytical solution. Results are also generated for flow through a constricted tube and compared with a Newtonian fluid solution.     

Extensional flow behavior of polymer solutions and particle suspensions in a spinning motion

Mechanical spinning of fluid filaments was used to generate an extensional flow, in which rheological measurements were obtained for a Newtonian fluid, two aqueous polymer solutions, and two fluid suspensions of rod-shaped particles. The tensile stress was determined by measuring the tensile force of the fluid filament while the kinematics were determined from photographic measurement of the filament profile and the assumption of a flat velocity profile. The measured tensile stresses for the Newtonian fluid matched predicted stresses, thereby confirming the validity of the experimental technique.The spinning behavior of each polymer solution could be correlated as stress versus extension rate. The apparent “spinning viscosity” increased with increasing rate of extension, in contrast to shear-thinning behavior in viscometric flow. For the fluid suspensions, the presence of rod-shaped particles increased the apparent viscosity far more in extensional flow than in shear. Tensile stresses calculated from a theoretical formula for suspensions proposed by Batchelor agreed rather well with experiment. Some general criteria for the interpretation of the spinning experiment are proposed, and some microrheological implications of the present findings are discussed.     

Numerical simulation of the divergent flow regime in a circular contraction flow of a viscoelastic fluid

A numerical method is used to study the influence of inertia and elongational properties on the vortex growth in the flow of a viscoelastic fluid through a four-to-one contraction. It appears that the vortex growth regime and the divergent flow regime, which are observed in experiments for some fluids, can be found for a choice of the material parameters where both the elongational stresses and the inertia forces are large for the flow rate considered. After studying the type and the vorticity it is concluded that the appearance of a divergent flow regime is likely to be a critical phenomenon (i.e., a change of type for a critical velocity) and that large elongational stresses are essential as well.     

Effect of couple stresses on the flow in a constricted annulus

The flow of an incompressible couple stress fluid in an annulus with local constriction at the outer wall is considered. This configuration is intended as a simple model for studying blood flow in a stenosed artery when a catheter is inserted into it. The effects couple stress fluid parameters α and σ, height of the constriction (ε), and ratio of radii (k) on the impedance and wall shear stresses are studied graphically. Graphical results show that the resistance to the flow as well as the wall shear stress increases as the ratio of the radii increases and decreases as the couple stress fluid parameters increases.     

Unsteady motion of a non-linear viscoelastic fluid

In this paper, we study the unsteady flow of a generalized second grade fluid. Specifically, we solve numerically the linear momentum equations for the flow of this viscoelastic shear-thinning (shear-thickening) fluid surrounding a solid cylindrical rod that is suddenly set into longitudinal and torsional motion. The equations are made dimensionless. The results are presented for the shear stresses at the wall, related to the drag force; these are physical quantities of interest, especially in oil-drilling applications.     

Do general viscoelastic stresses for the flow of an upper convected Maxwell fluid satisfy the momentum equation?

Given a general velocity field consistent with the stagnation point flow, can the viscoelastic stresses arising in the flow of an upper convected Maxwell fluid found by solving the constitutive equation also satisfy the momentum equation? Consideration is given to the study of the stress tensor arising in the steady flow of an upper convected Maxwell (UCM) fluid with a velocity field consistent with the stagnation point flow. By the method of characteristics, exact solutions to the partial differential equations arising in the approximating model of the viscoelastic stresses in the flow of an upper convected Maxwell (UCM) fluid are obtained for the three components of the stress tensor, for reasonably general velocity fields. We are able to account for the effects of variable boundary data at the inflow by considering the viscoelastic stresses over two spatial variables. Furthermore, we assume a relatively general velocity field. As a special case, some results present in the recent literature are obtained; it is known that these special case solutions do not satisfy the momentum equation. In the general case we consider, we find that the general solution will not satisfy the momentum equation except in a limited restricted case. We discuss how this shortcoming might be rectified by use of a more general velocity field.     

Onset and dynamic shallow flow of a viscoplastic fluid on a plane slope

The shallow flow of a viscoplastic fluid on a plane slope is investigated. The material constitutive law may include two plasticity (flow/no-flow) criteria: Von-Mises (Bingham fluid) and Drucker–Prager (Mohr–Coulomb). Coulomb frictional conditions on the bottom are included, which implies that the shear stresses are small and the extensional and in-plane shear stress becomes important. A stress analysis is used to deduce a Saint-Venant type asymptotic model for small thickness aspect ratio. The 2D (asymptotic) constitutive law, which relates the average plane stresses to the horizontal rate of deformation, is obtained from the initial (3D) viscoplastic model.The “safety factor” (limit load) is introduced to model the link between the yield limit (material resistance) and the external forces distribution which could generate or not the shallow flow of the viscoplastic fluid. The DVDS method, developed in [I.R. Ionescu, E. Oudet, Discontinuous velocity domain splitting method in limit load analysis, Int. J. Solids Struct., doi:10.1016/j.ijsolstr.2010.02.012], is used to evaluate the safety factor and to find the onset of an avalanche flow.A mixed finite element and finite volume strategy is developed. Specifically, the variational inequality for the velocity field is discretized using the finite element method while a finite volume method is adopted for the hyperbolic equation related to the thickness variable. To solve the velocity problem, a decomposition–coordination formulation coupled with the augmented lagrangian method, is adapted here for the asymptotic model. The finite volume method makes use of an upwind strategy in the choice of the flux.Several boundary value problems, modeling shallow dense avalanches, for different visoplastic laws are selected to illustrate the predictive capabilities of the model.