As a possible model for fluid turbulence, a Reiner–Rivlin-type equation is used to study Poiseuille–Couette flow of a viscous fluid in a rotating cylindrical pipe. The equations of motion are derived in cylindrical coordinates, and small-amplitude perturbations are considered in full generality, involving all three velocity components. A new matrix-based numerical technique is proposed for the linearized problem, from which the stability is determined using a generalized eigenvalue approach. New results are obtained in this cylindrical geometry, which confirm and generalize the predictions of previous recent studies. A possible mechanism for the transition to turbulent flow is discussed.

We model the axisymmetric unidirectional flow of a Herschel–Bulkley fluid with rheological parameters that depend linearly on pressure. Adopting an appropriate scaling, we formulate the mathematical problem in cylindrical geometry exploiting an integral formulation for the momentum equation in the unyielded part. We prove that, under suitable assumptions on the data of the problem, explicit solutions can be determined. In particular, we determine the position of the yield surface together with the pressure and velocity profiles. With the aid of some plots, we finally discuss the dependence of the solution on the physical parameters of the problem.

The motion and rotation of an ellipsoidal particle inside square tubes and rectangular tubes with the confinement ratio R/a∈(1.0,4.0) are studied by the lattice Boltzmann method (LBM), where R and a are the radius of the tube and the semi-major axis length of the ellipsoid, respectively. The Reynolds numbers (Re) up to 50 are considered. For the prolate ellipsoid inside square and rectangular tubes, three typical stable motion modes which depend on R/a are identified, namely, the kayaking mode, the tumbling mode, and the log-rolling mode are identified for the prolate spheroid. The diagonal plane strongly attracts the particle in square tubes with 1.2≤R/a<3.0. To explore the mechanism, some constrained cases are simulated. It is found that the tumbling mode in the diagonal plane is stable because the fluid force acting on the particle tends to diminish the small displacement and will bring it back to the plane. Inside rectangular tubes the particle will migrate to a middle plane between short walls instead of the diagonal plane. Through the comparisons between the initial unstable equilibrium motion state and terminal stable mode, it is seems that the particle tend to adopt the mode with smaller kinetic energy.

Numerical instability may occur when simulating high Reynolds number flows by the lattice Boltzmann method (LBM). The multiple-relaxation-time (MRT) model of the LBM can improve the accuracy and stability, but is still subject to numerical instability when simulating flows with large single-grid Reynolds number (Reynolds number/grid number). The viscosity counteracting approach proposed recently is a method of enhancing the stability of the LBM. However, its effectiveness was only verified in the single-relaxation-time model of the LBM (SRT-LBM). This paper aims to propose the viscosity counteracting approach for the multiple-relaxation-time model (MRT-LBM) and analyze its numerical characteristics. The verification is conducted by simulating some benchmark cases: the two-dimensional (2D) lid-driven cavity flow, Poiseuille flow, Taylor-Green vortex flow and Couette flow, and three-dimensional (3D) rectangular jet. Qualitative and Quantitative comparisons show that the viscosity counteracting approach for the MRT-LBM has better accuracy and stability than that for the SRT-LBM.

The equilibrium position of a deformable drop in Couette and Poiseuille flows is investigated numerically by solving the full Navier-Stokes equations using a finite difference/front tracking method. The objective of this work is to study the motion of a non-neutrally buoyant drop in Couette and Poiseuille flows with different density ratios at finite Reynolds numbers. Couette flow: The equilibrium position of the lighter drops is higher than the heavier drops at each particle Reynolds number. Also, the equilibrium position height increases with increasing the Reynolds number at a fixed density ratio. At this equilibrium distance from the wall, the migration velocity is zero, while the velocity of the drop in the flow direction and rotational velocity of the drop is finite. It is observed that the equilibrium position is independent of the initial position of the drops and depends on the density ratio and the shear Reynolds number. Poiseuille flow: When the drop is slightly buoyant, it moves to an equilibrium position between the wall and the centerline. The equilibrium position is close to the centerline if the drop lags the fluid but close to the wall if the drop leads the fluid. As the Reynolds number increases, the equilibrium position of lighter drops moves slightly closer to the wall and the equilibrium position of heavier drops moves towards the centerline.

This work develops an analytical model of shear senses within an inclined ductile simple shear zone with parallel rigid boundaries and incompressible Newtonian viscous rheology. Taking account of gravity that tends to drive the material downdip and a possible pressure gradient that drives it upward along the shear zone, it is shown that (i) contradictory shear senses develop within two sub-zones even as a result of a single simple shear deformation; (ii) the highest velocity and least shear strain develop along the contact between the two sub-zones of reverse shear; (iii) for a uniform shear sense of the boundaries, a zone of reverse shear may develop within the top of the shear zone if the pressure gradient dominates the gravity component; otherwise it forms near the bottom boundary; (iv-a) a ‘pivot’ defined by the intersection between the velocity profile and the initial marker position distinguishes two sub-zones of opposite movement directions (not shear sense); (iv-b) a pivot inside any non-horizontal shear zone indicates a part of the zone that extrudes while the other subducts simultaneously; (v) the same shear sense develops: (v-a) when under a uniform shear of the boundaries, the shear zone remains horizontal and the pressure gradient vanishes; or alternatively (v-b) if the shear zone is inclined but the gravity component counterbalances the pressure gradient. Zones with shear sense reversal need to be reinterpreted since a pro-sheared sub-zone can retro-shear if the flow parameters change their magnitudes even though the same shear sense along the boundaries is maintained.

We consider a dense granular shear flow in a two-dimensional system. Granular systems(composed of a large number of macroscopic particles) are far from equilibrium due toinelastic collisions between particles: an external driving is needed to maintain themotion of particles. Theoretical description of driven granular media is especiallychallenging for dense granular flows. This paper focuses on a gravity-driven densegranular Poiseuille flow in a channel. A special focus here is on the intriguingphenomenon of fluid-solid coexistence: a solid plug in the center of the system,surrounded by fluid layers. To find and analyze various flow regimes, a multi-scaleapproach is taken. On macro scale, granular hydrodynamics is employed. On micro scale,event-driven molecular dynamics simulations are performed. The entire phase diagram ofparameters is explored, in order to determine which flow regime occurs in various regionsin the parameter space.

In this paper we construct a model to describe someaspects of the deformation of the central region of the human lung considered as acontinuous elastically deformable medium. To achieve this purpose, we studythe interaction between the pipes composing the tree and the fluid that goes through it. We use a stationary model to determine the deformed radius of each branch. Then, we solve a constrained minimization problem, so as to minimize the viscous (dissipated) energy in the tree. The key feature of our approach is the useof a fixed point theorem in order to find the optimal flow associatedto a deformed tree. We also give some numerical results withinteresting consequences on human lung deformation during expiration, particularlyconcerning the localization of the equal pressure point (EPP).

In this paper we elaborate a model to describe some aspects of the human lung considered as a continuous, deformable, medium. To that purpose, westudy the asymptotic behavior of aspring-mass system with dissipation. The key feature of our approach is the nature of this dissipation phenomena, which is related here to the flow of a viscous fluid through a dyadic tree of pipes (the branches), each exit of which being connected to an air pocket (alvelola) delimited by two successive masses.The first part focuses on the relation between fluxes and pressures at the outlets of a dyadic tree, assuming the flow within the tree obeys Poiseuille-like laws. In a second part, which contains the main convergence result, we intertwine the outlets of the tree with a spring-mass array. Letting again the number of generations (and therefore the number of masses) go to infinity, we show that the solutions to the finite dimensional problems converge in a weak sense to the solution of a wave-like partial differential equation with a non-local dissipative term.