Gas displacement of viscoplastic liquids in capillary tubes

The displacement of viscoplastic liquids in capillary tubes by gas injection is examined. The viscoplasticity alters the flow kinematics and changes dramatically the amount of mass left attached at the tube wall as compared to the Newtonian case, studied experimentally by G.I. Taylor in 1961 [G.I. Taylor, Deposition of a viscous fluid on the wall of a tube, J. Fluid Mech. 10 (1961) 161–165]. Experiments with Carbopol aqueous solutions were performed for different flow rates. A recently proposed viscosity function for viscoplastic liquids was fitted to the rheological data of the Carbopol solutions. A new dimensionless rheological property – the jump number – arises in the dimensionless version of this viscosity function. The results show the effect of the viscoplastic character of the liquid on the free surface shape and on the thickness of the film of liquid left attached to the wall. This thickness decreases with the jump number and increases with the flow rate. It is also observed that there is a critical dimensionless flow rate below which the displacement is apparently perfect, i.e. there is no observable liquid left attached to the wall. This behavior is shown to be directly related to the fully developed flow far ahead the air–liquid interface.     

Nonisothermal structural regime of motion of a nonlinear viscoplastic medium in a plane channel

The motions of Newtonian and non-Newtonian media with consideration of the heat of friction and temperature dependence of the rheological characteristics of the medium have been examined by many authors [1–6]. A detailed bibliography of these works is given in [3, 5], In the dissertation of one of the authors* an attempt was made to investigate heat transfer during movement of a nonlinear viscoplastic medium with consideration of the temperature dependence of plastic viscosity at a constant value of the ultimate shear stress to. The last assumption is not always justified, since more often cases are found when not only plastic viscosity but also the ultimate shear stress changes with change in temperature. In this article we will examine the effect of the heat of internal friction with consideration of the temperature dependence of plastic viscosity and ultimate shear stress on the hydraulic characteristics of a laminar regime of motion in a plane channel of nonlinear viscoplastic media whose rheological behavior is described by the generalized Casson equation [5]. The upper and lower estimates are found for nonlinear differential equations describing the indicated processes.R. M. Sattarov, Certain isothermal and nonisothermal motions of viscoplastic media, Candidate's Dissertation, Institute of Heat and Mass Transfer, Academy of Sciences of the Belorussian SSR, Minsk (1973).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 162–166, March–April, 1977.In conclusion the authors thank V. I. Anikin for help in the calculations.     

Static wall layers in plane channel displacement flows

We study steady and pulsating displacement flows of a Bingham fluid by a Newtonian fluid, along a plane channel. For sufficiently large yield stress a static residual wall layer can result during the displacement. The flow is parameterised by the Reynolds number (Re), the Bingham number (B) and the viscosity ratio (M). Perhaps intuitively, thicker layers are found with larger M and at lower Re. The residual layer is formed on the advective timescale of the displacement but drains on a slower timescale governed by M. For larger M truly stationary layers are only found for large t when the layer has thinned sufficiently to become static. Increased Re results in increased energy production locally around the finger. For large enough Re the energy production can play a significant role in yielding the fluid. As the energy production rate increases it also becomes focused around the corner or shoulder region of the front, and spreads axially along the initial part of the residual layer. This causes fluid to yield increasingly far behind the front and allows for the layer to thin. As B increases the static layer tends to decrease (see also [1], [2]). At small Re the static layer thickness appears to be independent of M. At large Re the layer thickness is dependent on M and decreases asymptotically to a constant value as B → ∞.For pulsating displacement flow rates, Q(t) = 2(1 + Asin  ωt) : A ∈ [0, 1) we study two ranges: ωRe ? 2π and ωRe ? 2π. For the viscous regime (ωRe ? 2π) a pseudo-steady 1D model predicts that the residual layer should remain static for 3(1 + Asin  ωt) < MB. In practice we find that partial mobilisation of the residual layer occurs even when this inequality is satisfied, but not if MB becomes significantly larger than 3(1 + A). For ωRe ? 2π we mobilise the layer for significantly larger values of MB and at smaller A, than in the viscous regime. This effect is traced to the occurrence of out-of-phase velocity fluctuations in the displacing fluid within a wall layer close to the interface.     

Immiscible displacement in vertically fractured reservoirs

A dual-porosity model is defined for saturated, two-phase, compressible, immiscible flow in a vertically fractured reservoir or aquifer. This model allows detailed simulation of the matrix-fracture interaction as well as the matrix flow itself. This is accomplished by directly coupling the matrix and fracture systems along the vertical faces of the matrix blocks, incorporating gravitational effects directly, and simulating flow inside the block. Thus fluid segregation due to gravitational effects and heterogeneities can be simulated. We show that our model can be derived via homogenization techniques. The model (in incompressible form for simplicity of exposition) is then approximated by a computationally efficient finite difference scheme. Calculations are presented to show the convergence of the scheme and to indicate the behavior of the model as a function of several parameters.     

Horizontal capillary flow of a Newtonian liquid in a narrow gap between a plane wall and a sinusoidal wall

An experimental and theoretical study of the capillary flow of a Newtonian liquid (mineral oil) in a Hele-Shaw cell in which the gap varies sinusoidally in one coordinate direction, and flow takes place in the direction of constant channel cross-sectional area is reported. The geometric non-uniformity of the gap is observed to produce interface fingering. Finger length is observed to increase with decreasing spacing between plates of fixed shape, and with increasing gross penetration distance. In the regime of interest, finger length is observed to increase slowly with increasing interface advancement, motivating a quasi-steady model in which gross interface advancement is predicted by a Lucas–Washburn model and interface fingering is predicted by a Hele-Shaw model of steady flow. The steady interface velocity in the Hele-Shaw model is set equal to the instantaneous interface velocity predicted by the Lucas–Washburn model. Fingering predicted by the quasi-steady model matches the experimentally observed trends with regards to plate spacing and gross penetration distance.     

Immiscible fluid displacement in nonlinear seepage flow

The existing theory of immiscible fluid flow is extended to the seepage with nonzero initial shear stress of viscoplastic media. An analog of the Buckley-Leverett frontal-displacement theory is constructed.The authors are grateful to G. I. Barenblatt and V. M. Ryzhik for frequent stimulating discussions of their work.     

Deformation of a Bingham viscoplastic fluid in a plane confuser

A review is given to and comprehensive numerical-analytic study is carried out of the problem of steady Bingham viscoplastic flow in a plane confuser. The solution is constructed in the first approximation with the yield stress as a small parameter and the solution of the Jeffery-Hamel problem (steady radial motion of an incompressible viscous material in a plane confuser) as the zero-order approximation. The numerical analysis is based on the modified accelerated-convergence method proposed earlier by the authors. The bifurcations of the deformation pattern occurring when the parameters reach some critical values are discussed and commented on. The asymptotic boundaries of the rigid zones that appear at infinity upon perturbation of the yield stress are determined __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 3–45, April 2006.     

Unsteady shear flow of a viscoplastic material

A viscoplastic, or yield-stress, liquid occupies the space between two infinite parallel plates. Initially the whole system is at rest. The lower plate is suddenly jerked into motion with given speed or shear stress, while the upper plate is kept fixed. The flow consists of two regions; (1) a lower sheared region bounded above by the yield surface, (2) an upper unyielded region bounded below by the yield surface. The yield surface propagates to the upper plate as time proceeds. We first consider the equivalent one plate problem of flow over a jerked plate, and find similarity solutions and small time asymptotic solutions for prescribed shear and speed cases respectively. These solutions are used as initial solutions for the two plate case. The motion of the yield surface and the time taken for the entire material to yield are investigated. The problems considered here are two dimensional representations of some control devices, for example the light duty clutch, which consists of two corotating, coaxial discs separated by a layer of electrorheological material. In this application it is useful to know the time taken for all the material to yield.     

Combined convection of a viscoplastic liquid in a plane vertical layer

The combined (free and induced) convection of a viscoplastic Shvedov-Bingham liquid in a plane vertical layer is considered. The influence of the temperature dependence of the yield shear stress on the conditions of occurrence of the flow and the stationary convection regime in the case of heating from the side is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–113, November–December, 1979.     

Natural convection produced by unsteady heating of a viscoplastic liquid in a plane vertical layer

A study is made of the unsteady heating of a viscoplastic liquid in the space between two infinite vertical plates when one is thermally insulated and a constant, uniformly distributed heat flux is supplied to the other after a certain initial time. It is assumed that at the initial time the liquid is uniformly heated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 148–150, July–August, 1982.     

Characteristics of a viscoplastic material in the couette flow

A model governing a steady flow of a viscoplastic material between coaxial cylinders is proposed. Nonlinear velocity sensitivity typical of superplastic materials is taken into account. An algorithm of calculating the characteristics of the material is developed. The algorithm is based on the experimental data on moments and angular velocities of the rotating coaxial cylinders. The stability of the algorithm to errors in the initial data is estimated.     

Damage in a viscoplastic material part I: Cavity growth

The exact velocity, stress and strain rate fields around a spheroidal cavity in an infinite linear viscoplastic compressible matrix are derived analytically by the ‘three function approach’. The perturbation of the velocity field due to the cavity is the superposition of three independent modes, inducing homothetic growth, pure distortion and both volume and shape changes, respectively. This solution is then used to investigate the velocity field around a spheroidal cavity in a nonlinear viscous compressible material by means of a variational principle. The behaviour of such damaged linear and nonlinear materials will be discussed in a forthcoming companion paper.The importance of the reference strain, while studying void growth in a compressible material, is emphasized. If the axial strain is chosen as a reference, void growth is found to be enhanced at low triaxiality ratios, but lowered at high triaxiality ratios in a compressible matrix relative to an incompressible one. Finally, the transition from a power law to a linear law with intercept, at increasing strain rates, is shown to reduce damage growth rate.     

Hydraulic impact of “exponential” and nonlinearly viscoplastic media in pipes made of a viscoplastic material

Differential equations are derived and the hydraulic impact process for “exponential” and nonlinearly viscoplastic media in pipes made of a viscoelastic material is analyzed. Hydraulic impact problems for actual media in pipes has been repeatedly treated in the literature [1–4]. The hydraulic impact of a viscous and linearly viscoplastic media in pipes made of an elastic and viscoelastic material was studied in this work. It is well known [5] that many media in the region of low and moderate shear rates reveal a nonlinearity of the flow curve (oil, drilling fluids, polymer solutions and melts, loaded fuels, fuel mixtures, blood, etc.). It should be noted that flexible pipes made of natural materials (pipe boreholes made of polymer materials, membranes of blood vessels, etc.) are described by complicated rheological equations of state for viscoelastic media. Thus a calculation of the influence of nonlinearity of these media and of the viscoelastic properties of the pipe material on the hydraulic impact process is of theoretical and practical interest in many engineering problems.     

Further remarks on numerical investigation on gas displacement of a shear-thinning liquid and a visco-plastic material in capillary tubes

In this note, we present some results concerning the gas displacement of power-law liquids and visco-plastic Papanastasiou’s materials improving the understanding of the problem considered in Sousa et al. (2007) [1]. Specifically, we present: the fraction of mass attached to the wall for a viscosity-thickening power-law fluid, different transition patterns between by pass and full-recirculating flow regimes, expressions for the critical (in the sense proposed by Soares and Thompson (2009) [2]) fraction of residual mass as a function of the rheological parameter of interest, and fields of yielded and unyielded zones for the visco-plastic material.     

Mechanical characterisation of a viscoplastic material sensitive to hydrostatic pressure

The present paper deals with the characterisation of the static mechanical behaviour of an energetic material all along its lifespan. The material behaviour is viscoplastic, damageable and sensitive to hydrostatic pressure. For such materials, existing models have generally been developed in the framework of transient dynamic behaviour. These models are not suitable for a static study. Therefore a specific experimental protocol and an associated model are developed. Characterisation is derived from both uniaxial compressive, tensile tests and triaxial tests. Plastic behaviour is described by means of a parabolic yield criterion and a new hardening law. Non-associated plastic flow rule and isotropic damage complete the model. The performance of the model is illustrated through the simulation of various loading paths.     

On flow and stability of a Newtonian fluid past a rotating plane

An exact solution is given for the steady flow of a Newtonian fluid occupying the halfspace past the plane z=0 uniformly rotating about a fixed normal axis (Oz). This solution is obtained in a velocity field of the form considered by Berker [2] and can be deduced as a limiting case, as h+, of the solution to the problem relative to the strip 0zh imposing at z=h either the adherence boundary conditions or the free surface conditions. Furthermore, the stability of this flow, subject to periodic disturbances of finite amplitude, is studied using the energy method and the result is compared with those corresponding to stability of flows in the strip 0zh.

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Sommario In questa nota si mostra che-oltre alla calssica soluzione di von Karman [1] — esiste, per opportuni valori del gradiente di pressione all'infinito, una soluzione esatta per il moto stazionario di un fluido Newtoniano posto nel semispazio limitato dal piano z=0 uniformemente rotante attorno ad un asse ad esso perpendicolare (Oz). Tale soluzione, ottenuta sulla scia del lavoro di Berker [2], si può dedurre anche come limite, per h+, della soluzione del problema relativo alla striscia 0zh quando sul piano z=h si assegnano o le condizioni di aderenza o le condizioni di frontiera libera. Si studia poi la stabilità di tale moto rispetto a perturbazioni spazialmente periodiche di ampiezza finita col metodo dell'energia e si confronta il risultato ottenuto con quelli relativi alla stabilità dei moti nella striscia 0zh.

     

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.     

The problem of stopping the flow of a viscoplastic medium in a channel

The problem of stopping the flow of a viscoplastic medium is considered. Asymptotic estimates are obtained for stopping time. The results of numerical experiments performed for square and rectangular pipes are in good agreement with the estimates proposed.