SIMULATION OF A FLOW THROUGH A RIGID POROUS MEDIUM SUBJECTED TO A CONSTRAINT

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1 SIMULATION O A LOW THROUGH A RIGID POROUS MEDIUM SUBJECTED TO A CONSTRAINT R. M. Saldanha da Gama a, J. J. Pedrosa ilho a, and M. L. Martins-Costa b a Universidade do Estado do Rio de Janeiro Departamento de Engenharia Mecânica EN Rua São rancisco Xavier, 54, CEP , Rio de Janeiro, RJ, Brasil rsggama@gmail.com, zejulio@terra.com.br b Universidade ederal luminense Departamento de Engenharia Mecânica TEM/PGMEC Laboratório de Mecânica Teórica e Aplicada Rua Passo da Pátria, 156, CEP , Niterói, RJ, Brasil ABSTRACT This work uses a previuosly proposed mathematical model to study the illing up o an unsaturated rigid porous medium by a liquid identiying the transition rom unsaturated to saturated low. This model accounts or the physical upper bound o the luid raction that depends on the volume o the pores and employs a mixture theory to describe the low. The mathematical description o the phenomenon leads to a nonlinear hyperbolic system. In order to solve this system, the space o admissible solutions must be enlarged to admit discontinuous solutions that may be shock waves. The complete solution o a Riemann problem associated to the system o conservation laws satisying the constraint given by the saturation upper bound is presented. Some meaningul results are presented and discussed. Received: October 5, 013 Revised: November 7, 013 Accepted: December 30, 013 Keywords: low through unsaturated porous media, transition unsaturated/saturated low, constrained unknown, Riemann problem, shock waves. NOMENCLATURE Subscripts b body orce per unit mass, m/s c constant L porous slab thickness, m m momentum supply on luid constituent, N/m p pressure, N/m p dimensionless pressure t time, s T partial stress tensor, N/m u dimensionless luid constituent velocity on x direction v luid constituent velocity in the mixture, m/s v luid constituent velocity on x direction, m/s x Cartesian coordinate, m Greek symbols ε porous matrix porosity θ dimensionless time ρ luid constituent mass density kg/m 3 ρ luid mass density kg/m 3 ϕ luid raction ψ saturation L let state R right state * intermediate state 0 initial state or luid raction INTRODUCTION This work employs a physically realistic mathematical model to represent the illing up o an unsaturated rigid porous matrix by a luid, identiying the transition rom unsaturated to saturated low, by imposing a constraint an upper bound) on the saturation proposed by Saldanha da Gama et al. 01). According to a comprehensive review by Alazmi and Vaai 000), an adequate description o this transition remains an open subject. The mechanical modeling uses a mixture theory approach to model a porous medium bounded by an impermeable wall. Distinct physical applications may give rise to constrained hyperbolic systems. Among them twophase lows o gas and liquid models, compressible plasticity with shocks and models or traic o vehicles or crowds These may be described with the help o luid dynamic models with distinct degrees o complexity that may account or accidents on roads, traic jams or toll gates. Bouchut et al. 000) model 80 Engenharia Térmica Thermal Engineering), Vol. 1 No. December 013 p

2 Gama, et al. Simulation o a low through a Rigid the dynamics o gas occlusions on pipes by studying two-phase lows o gas and liquid models, or incompressible liquids, ranging rom nonconservative conservation laws with relaxation to a system o pressureless gases with concentration constraint and undetermined pressure. Després et al. 011) present a mathematical ramework or constrained weak solutions o hyperbolic equations, to model compressible plasticity with shocks. This developed weak ormalism allows accounting or both Tresca and Von Mises plasticity criteria. Concerning traic low models, Daganzo 1995) observed that models like the Euler equations or gas dynamics may lead to absurd behavior like vehicles going backwards. The Aw-Rascle model, proposed by Aw and Rascle 000), ensures that both density and velocity remain nonnegative and correcting the absurd behavior detected by Daganzo 1995). Berthelin et al. 008) studied the dynamics o traic jams by proposing a traic low model consisting o a pressureless gas dynamics system under a maximal constraint on the density so that the density constraint is always preserved. Aiming at predicting traic accidents, Herty and Schleper 011), consider dierent mathematical models that can describe macroscopically the traic low, accounting or the mathematical properties o a coupled macroscopic second-order traic model with dierent pressure laws on the connected roads. Colombo and Goatin 007) consider a single hyperbolic equation Cauchy problem) subjected to a local variable unilateral constraint on the lux in order to model a tollgate along a highway. The total number o vehicles is conserved and the traic speed is assumed to be a unction o traic density. This work could be compared with the model proposed by Saldanha da Gama et al. 01) in which a system o two partial dierential equations is considered. However, Colombo and Goatin 007) consider a variable unilateral constraint only on the boundary, while Saldanha da Gama et al. 01) consider a constraint in the whole domain or all time. In order to avoid solutions without physical meaning, Saldanha da Gama 1986) proposed a constitutive relation or the partial pressure accounting or a geometrical bound that results rom the luid incompressibility and the porous medium rigidity. Later, an improvement to this constitutive relation was proposed by Martins-Costa and Saldanha da Gama 011) in which, the unilateral geometrical constraint or the luid raction, instead o being assumed in the whole domain, is considered only in a convenient neighborhood o the porosity provided that the luid raction is smaller than the porosity). Since Martins-Costa and Saldanha da Gama 011) assured continuity or the pressure and or its irst derivative, the Riemann invariants associated to the problem could be analytically computed. It is important to note that while in Saldanha da Gama et al. 01) the porous medium can actually be saturated by the luid, the equation proposed by Saldanha da Gama 1986) only imposes a physical behavior or the luid preventing it to saturate the porous matrix. MECHANICAL MODEL The unsaturated porous medium is modeled using a mixture theory approach Atkin and Craine, 1976; Rajagopal and Tao, 1995) in which three overlapping continuous constituents are considered: a solid a rigid, homogeneous and isotropic porous matrix), a liquid an incompressible luid) and an inert gas, assumed with very low mass density; which was included to account or the compressibility o the system as a whole. Since the solid constituent is rigid and the gas constituent is inert it suices to solve mass and momentum equations or the liquid luid) constituent, which are given by: 1) where ρ, the luid constituent mass density, represents the local ratio between the luid constituent mass and the corresponding volume o mixture, m is a momentum supply acting on the luid constituent due to its interaction with the remaining constituents o the mixture and the partial stress tensor T is analogous to Cauchy stress tensor in Continuum Mechanics. These latter quantities require constitutive assumptions. The ratio between the luid raction ϕ and the porous matrix porosity ε is deined as the saturation ψ, so that ψ = ϕ ε = ρ ερ 0 < ψ 1 everywhere ) where ρ is the actual mass density o the luid in a Continuum Mechanics viewpoint). Constitutive relations are now presented or the partial stress tensor considering the normal luid stresses dominant over shear stresses and interphase tractions as proposed by Allen see Saldanha da Gama et al., 01 and reerences therein) and or the momentum source m as given by a term related to the luid constituent velocity, usually called Darcian and term a term related to the saturation gradient. See Saldanha da Gama et al., 01 and reerences therein.) 3) Engenharia Térmica Thermal Engineering), Vol. 1 No. December 013 p

3 Gama, et al. Simulation o a low through a Rigid where µ is the luid viscosity, K the porous matrix speciic permeability in a Continuum Mechanics viewpoint), D a diusion coeicient and p is a pressure assumed constant while the low is unsaturated). The Darcian term irst term o the momentum source) is neglected, since unsaturated lows through porous media are characterized by a strong dependence o the motion on the saturation. At this point the mechanical model can be built by combining the balance equations 1) with the constitutive relations 3) together with the assumption that the pressure p = ˆp ϕ may be redeined as ) p = p ϕ + µ D / K )ϕ. Also, assuming absence o body orces, that all quantities depend on the time t and on the position x only, and that v is the only nonvanishing component o the luid constituent velocity v, the problem may be written as φ t + x ϕv )= 0 ϕv )+ t x p +ϕv )= 0 The unknowns ϕ and v, depending on position x and on time t, represent, respectively, the luid raction and the luid constituent velocity in the description o the low through an unsaturated porous medium. Physically, there is no restriction on the velocity, which can assume any real value, but the luid raction ϕ must be positive valued and smaller than or equal to) the porosity ε, otherwise the problem is no longer physically consistent. Since the maximum amount o luid is bounded by the porous medium porosity, the luid raction is a constrained unknown given by 4) 0 < ϕ ε, x, t > 0 5) In addition, in this work, the ollowing relationship Saldanha da Gama et al., 01) is considered p = c ϕ, 0< ϕ ε 6) It is important to note that the ollowing relations must hold or a rigid and homogeneous porous medium: p = ˆpϕ) or 0 < ϕ < ε, characterizing unsaturated low and ˆpε) p < or ϕ = ε, characterizing saturated low Saldanha da Gama et al., 01). The governing equations may be rewritten in a dimensionless orm as ollows ϕ θ + x ϕu )= 0 θ ϕu )+ x p +ϕu )= 0 ϕ ε in which θ = t c, u = v c and 7) p = p c 8) PROBLEM ORMULATION AND SOLUTION The problem is deined in an ininite porous medium with constant porosity ε and an initial constant luid raction ϕ such that 0 < ϕ < ε. Supposing that the luid is initially at rest and, suddenly, a motion takes place in such a way that u = u L = constant or t = 0, < x < 0 u = u R = constant or t = 0, 0 < x < 9) This phenomenon is mathematically represented as ollows ϕ θ + x ϕu )= 0 θ ϕu )+ x p +ϕu )= 0, with p = ϕ or ϕ < ε ϕ ε ϕ,v )= ϕ,u 0 L,u R ) or t = 0, < x < 0 ) or t = 0, 0 < x < 10) It is important to note that problem 10) is called a Riemann problem. The solution in a generalized sense) o this Riemann Problem is reached by connecting the let state and the right state to an intermediate state reached by means o continuous and/or discontinuous unctional relationships. When the inequality u L 11) takes place, there is no continuous connection. In addition, in such cases, when the inequality ε ϕ 0 ε < u u 1) L R 8 Engenharia Térmica Thermal Engineering), Vol. 1 No. December 013 p

4 Gama, et al. Simulation o a low through a Rigid holds, the porous medium is saturated. In such cases, the intermediate luid raction becomes equal to the porosity, while p is no longer given by the constitutive equation p = ϕ. In these cases, p is obtained directly rom the balance equations. Denoting by ϕ,u ) the intermediate state, it comes that the intermediate pressure is given by = ϕ = ϕ u 0 L u R + 4 u L u R ) provided ϕ is less than ε. In these cases, the intermediate pressure p is also given byϕ. On the other hand, when the above equation gives rise to ϕ greater than or equal to the porosity, in order that the problem remains physically consistent,, the intermediate pressure is given by igure 1 shows the relationship between the ratio o dimensionless pressure and porosity / ε and the dierence o dimensionless velocities ul ur or two values o the ratio between the initial luid raction and the porosity / ε, characterizing two initial value problems. Two distinct sets o curves are represented in igure 1 showing two sets o curves. The upper lines represent the relationship between / ε and u L u R or / ε = 0.5 while the lower ones were obtained or / ε = 0.3. Also, the thicker lines correspond, or both considered cases to the lines that represent the behavior o the ratio / ε with the variation o the dimensionless velocity dierence ul ur. The black dot indicates the transition rom unsaturated to saturated low. At the let side o each black dot, the inerior lines hold. At the right side, where the low becomes saturated and the constraint must be imposed, the superior lines hold. u L u R = ε 14) while the intermediate velocity u * is given by u = u L + u R ϕ * ϕ * ϕ 0 ϕ * ϕ 15) 0 The speeds o propagation can be computed rom the jump conditions being given by s 1 = u L u * speed o shock 1) s = u R u * speed o shock ) 16) At this point, assuming u L, the solution o the Riemann problem represented by equation 10), in a generalized sense, may be represented by the lowing equations,u L ) i < x θ < ϕ u ϕ u 0 L * * ϕ,u )= ϕ *,u * )i ϕ u ϕ u 0 L * * < x θ < ϕ u ϕ u 0 R * * 17),u R ) i ϕ u ϕ u 0 R * * < x θ < igure 1. Relationship between the ratio / ε and u L u or two values o the ratio / ε. R It is important to note that whenever the constraint is not satisied the results are no longer physically reasonable which is indicated by does not hold. igure and 3 present the exact solution o the Riemann problem represented by equation 10), presented in equation 17). igure. Solution o Riemann problem 9) luid raction as a unction o the ratio x /θ. Engenharia Térmica Thermal Engineering), Vol. 1 No. December 013 p

5 Gama, et al. Simulation o a low through a Rigid Since u L, the solution is shock 1 / shock, and the speeds o propagation given by equation 15) are also depicted in igures and 3. As indicated in Equation 17), the luid raction equals, except or the region s 1 < x /θ < s, in which it equals ϕ *, as depicted in igure. In igure 3, the initial condition stated in equation 9) is presented, considering u L, and the value or intermediate velocity u * is given by equation 15). igure 3. Solution o Riemann problem 9) Dimensionless velocity as a unction o the ratio x /θ. INAL REMARKS This work presented an exact solution or lows through unsaturated porous media, identiying the transition rom unsaturated to saturated low. The mechanical model included a constraint that must be satisied to build physically realistic generalized solutions, valid or any initial data. The complete solution o a constrained nonlinear hyperbolic problem with shock waves an associated Riemann problem containing a restriction an upper bound or the luid raction, represented by the porosity), was presented as well as its application to lows through porous media. ACKNOWLEDGEMENTS Journal on Applied Mathematics, Vol. 60, No. 3, pp Berthelin,., Degond, P., Delitala, M., and Rascle, M., 008, A Model or the ormation and Evolution o Traic Jams, Archive or Rational Mechanics and Analysis, Vol. 187, pp Bouchut,., Brenier, Y., Cortes, J. and Ripoll, J.., 000, A Hierarchy o Models or Two-Phase lows, Journal Nonlinear Science, Vol. 10, pp Colombo, R. M. and Goatin, P., 007, A Well Posed Conservation Law with a Variable Unilateral Constraint, J. Dierential Equations, Vol. 34, pp Daganzo, C., 1995, Requiem or Second Order luid Approximations o Traic low, Transportation Research Part B: Methodological, Vol. 9B, pp Herty, M., and Schleper, V., 011, Traic low with Unobservant Drivers, Journal o Applied Mathematics and Mechanics, Vol. 91, No. 10, pp Martins-Costa, M. L. and Saldanha da Gama, R. M., 011, A New Constitutive Equation or Unsaturated lows o Incompressible Liquids through Rigid Porous Media, Journal Porous Media, Vol. 14, pp Rajagopal, K. R., and Tao, L., 1995, Mechanics o Mixtures, Series Advances in Math. Appl. Sci., Vol. 35, World Scientiic, Singapore Saldanha da Gama, R. M., 1986, A New Mathematical Modelling or Describing the Unsaturated low o an Incompressible Liquid through a Rigid Porous Medium, International Journal o Non-Linear Mechanics, Vol. 40, pp Saldanha da Gama, R. M., Pedrosa ilho, J. J., and Martins-Costa, M. L., 01, Modeling the Saturation Process o lows through Rigid Porous Media by the Solution o a Nonlinear Hyperbolic System with One Constrained Unknown, Journal o Applied Mathematics and Mechanics, Vol. 9, No. 11-1, pp The authors R. M. Saldanha da Gama and M. L. Martins-Costa grateully acknowledge the inancial support provided by the Brazilian agency CNPq. REERENCES Alazmi, B., and Vaai, K, 000, Analysis o Variants within the Porous Media Transport Models, Journal Heat Transer, Vol. 1, pp Atkin, R. J., and Craine, R. E., 1976, Continuum Theories o Mixtures. Basic Theory and Historical Development, Quarterly Journal o Mechanics and Applied Mathematics, Vol. 9, pp Aw, A., and Rascle, M., 000, Resurrection o Second Order Models o Traic low, SIAM 84 Engenharia Térmica Thermal Engineering), Vol. 1 No. December 013 p