Fluid flow elastic media

Bielski, W., Telega, JJ. and Wojnar, R. (1999) Macroscopic Equations for Nonstationary Flow of Stokesian Fluid through Porous Elastic Medium. Arch. Mech. 51, 243-274... 

Under very rapid mechanical actions or in observations with characteristic time t < to, the substance behaves as an ideal elastic medium. For t to the developing flow becomes stronger than the elastic deformation, and the substance can be treated as a simple Newtonian fluid. It is only if t is of the same order of magnitude as to that the elastic and viscous effects act simultaneously, and the complex nature of the deformation displays itself. 

Zhao (1994) presented a model of coupled coal deformation and methane migration based on a consolidation theory of elastic medium with Darcy fluid flow and the Terzaghi effective stress law, and its numerical solution technique and applications to practical problems. Works using similar approaches were also reported in Liang et al. (1995,1996), Sun and Xian (1999), Ding et al. 

It is shown that the development of the equations governing THM processes in elastic media with double porosity can be approached in a systematic manner where all the constitutive equations governing deformability, fluid flow and heat transfer are combined with the relevant conservation laws. The double porosity nature of the medium requires the introduction of dependent variables applicable to the deformable solid, and the fluid phases in the two void spaces. The governing partial differential equations are linear in view of the linearized forms of the constitutive assumptions invoked in the formulations. The linearity of these governing equations makes them amenable to solution through conventional mathematical techniques applicable to the study of initial boundary value problems in mathematical physics (Selvadurai, 2000). Such solutions should serve as benchmarks for appropriate computational developments. 

While anchors and gate-type impellers are known to produce poor axial circulation of the liquid in a vessel, in one study [Peters and Smith, 1967] it seems that the fluid visco-elasticity promotes axial flow. For instance, axial flow is reported to be almost 15 times greater in a visco-elastic solution as compared with a Newtonian medium. The shear rate profiles reported by these authors shown in Figure 8.11 clearly indicate that the liquid in the tank is virtually unaffected by the blade passage. 

Ojjela, O., Kumar, N.N., 2013. Numerical study of MHD flow and heat transfer through porous medium between two parallel plates with hall and ion slip effects. In Proceedings of the 2013 International Conference on Mechanics, Fluids, Heat, Elasticity, and Electromagnetic Fields, pp. 161-167. 

Subhas, A., Venna, P, 1998. Visco-elastic fluid flow and heat transfer in a porous medium over a stretching sheet. Int. J. Non Linear Mech. 33, 531-540. 

The theory of fluid flow, together with the theory of elasticity, makes up the field of continuum mechanics, which is the study of the mechanics of continuously distributed materials. Such materials may be either soKd or fluid, or may have intermediate viscoelastic properties. Since the concept of a continuous medium, or continuum, does not take into consideration the molecular structure of matter, it is inherently an idealization. However, as long as the smallest length scale in any problem under consideration is very much larger than the size of the molecules making up the medium and the mean free path within the medium, for mechanical purposes all mass may safely be assumed to be continuously distributed in space. As a result, the density of materials can be considered to be a continuous function of spatial position and time. 

Static leak-off experiments with borate-crosslinked and zirconate-cross-Unked hydroxypropylguar fluids showed practically the same leak-off coefficients [1883]. An investigation of the stress-sensitive properties showed that zirconate filter-cakes have viscoelastic properties, but borate filter-cakes are merely elastic. Noncrosslinked fluids show no filter-cake-type behavior for a large range of core permeabilities, but rather a viscous flow dependent on porous medium characteristics. 

Telega, J.J. and Bielski, W. (2002) Nonstationary Flow of Stokesian Fluid through Random Porous Medium with Elastic Skeleton. Poromechanics II, 569-574, Lisse Abbington Exton (PA) Tokyo... 

Analysis of time-dependent consolidation requires the solution of Biot s consolidation equations coupled to the equations describing flow. The transient hydro-mechanical coupling between pore pressure and volumetric strain for a linear elastic, mechanically isotropic porous medium and fully saturated with a single fluid phase (i.e. water), is given by the fluid continuity equation ... 

The given presentation of the mechanism of the interaction of polymer molecules with turbulent flow admits a peculiar theoretical examination. The presence of polymer addition besides the increase of longitudinal viscosity is resulted in the appearance of such rheological solution properties as elastic plasticity, pseudo-plasticity, anisotropy. In [3] the influence of different rheological fluid characteristics on the wall turbulence is theoretically analyzed within the limits of monoharmonic approximation, which affords to take into account turbulent blows-out. Different variants of rheological behaviour were considered. For all that we succeded to show, that the decrease of turbulent friction arose only in mediums, possessing... 

In the case when both the droplets and the suspending medium are viscoelastic liquids, Wu (1987) reported that drops can break up during extmsion even when A > 4. However, owing to the complex nature of the deformation during flow through an extruder, it was difficult to even speculate on the origin of this phenomenon. Van Oene (1978) studied the mechanisms of two-phase formation in a mixture of two viscoelastic fluids. He pointed out that, besides the viscosity ratio and the equilibrium interfacial tension of the two liquids, the elasticity of the liquids plays an important... 

Deformation of a single drop in a medium subjected to convergent flow was observed. Both liquids were of the Boger fluid type. For a given matrix, the drop deformability decreased with elasticity of the dispersed phase. For a given Mighii et al. 1997... 

The prepreg fibre bed is typically assumed to be an elastic porous medium with incompressible and inextensible fibres and fully saturated with the resin. The resin is assumed to flow in the pores between the fibres, and the fibre mass in the laminate remains constant during cure. The governing equations of the system must describe the behaviour of the composite constituents the fibre bed and the resin. Firstly, the equilibrium of forces on the representative element is considered. Secondly, the mass conservation for the representative element must be satisfied. For a porous medium saturated with a single phase fluid, the total stress tensor a,) is separated into two parts as (tensile stresses are considered positive) ... 

Barik, R.N., Dash, G.C., Rath, P.K., 2014. Homotopy perturbation method (HPM) solution for flow of a conducting visco-elastic fluid through a porous medium. Proc. Natl. Acad. Sci. India 84, 55-61. 

It appears, then, that the mechanical degradation process is intimately connected with the molecular structure of the macromolecule and the resulting fluid rheology that arises from this structure. For a flexible coil macromolecule, such as HPAM or polyethylene oxide, the polymer solutions are known to display viscoelastic behaviour (see Chapter 3) and thus a liquid relaxation time, may be defined as the time for the fluid to respond to the changing flow field in the porous medium. It may be computed from several possible models (Rouse, 1953 Warner, 1972 Durst et al, 1982 Haas and Durst, 1982 Bird et al. 1987). The finite extendible non-linear elastic (FENE) (Warner, 1972 Bird et al, 1987a Haas and Durst, 1982 Durst et al, 1982) dumbbell model of the polymer molecule may be used to find the relaxation time, tg, as it is known that this model provides a good description of HPAM flow in porous media (Durst et al, 1982 Haas and Durst, 1982) the expression for fe is ... 

In an early paper, Sadowski and Bird (1965) recognised that using a bulk viscosity function for the fluid together with a capillary-hydraulic radius model for the porous media in the manner described above did not take into account any time-dependent elastic phenomena. They suggested that, in a tortuous channel of a porous medium, elastic effects would not be seen provided that the fluid s relaxation time was small compared with the transit time through the contraction/expansion. The fluid would have enough time to readjust to the changing flow conditions. However, if the transit time is small compared with the fluid s relaxation time, then the elasticity of the fluid would have an effect. 

The general concensus Is that the steep Increase In pressure beyond a critical flow rate Is a direct consequence of viscoelasticity In relation to the unsteadiness of the flow field In a porous medium. During flow the molecules can become extended, and this Is particularly the case In converglng/dlverging extenslonal flows, such as the fluid experiences In pore throats In a reservoir rock. If the fluid relaxation time Is small with respect to the time It takes to pass through a contraction or expansion, the fluid will accommodate quickly and no elastic effects are observed. If, on the other hand, relaxation time Is relatively large, elastic effects become dominant, resulting In excessive pressure Increase. 

Elastic behavior is that of bodies that tend to restore their original shape after stress is discontinued viscosity is a property of bodies that irreversibly change their shape, i.e. flow, under stress. Imagine a fluid medium as composed of parallel layers (see Fig. 11.6) which slide past one another under shear stress viscosity is defined as the ratio of shear stress to differential displacement of successive layers (flow speed ) ... 

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