a model for lime consolidation of porous solids
TRANSCRIPT
A model for lime consolidation of porous solids∗
Bettina Detmann † Chiara Gavioli ‡ Pavel Krejcı § Jan Lamac ¶
Yuliya Namlyeyeva ‖
Abstract
We propose a mathematical model describing the process of filling the pores of abuilding material with lime water solution with the goal to improve the consistency of theporous solid. Chemical reactions produce calcium carbonate which glues the solid particlestogether at some distance from the boundary and strengthens the whole structure. Themodel consists of a 3D convection-diffusion system with a nonlinear boundary conditionfor the liquid and for calcium hydroxide, coupled with the mass balance equations for thechemical reaction. The main result consists in proving that the system has a solution foreach initial data from a physically relevant class. A 1D numerical test shows a qualitativeagreement with experimental observations.
Keywords: porous media, reaction-diffusion, consolidation
2020 Mathematics Subject Classification: 35K51, 80A32, 92E20
Introduction
In this paper we investigate mathematically a process which is used by the building industry inorder to protect and conserve cultural goods and other structure works. Such structures whichare subject to weathering can be strengthened by filling the pores by a water-lime-mixture.The mixture penetrates into the pore structure of the stone and the calcium hydroxide reactswith carbon dioxide and builds calcium carbonate and water. A solid layer is built in the pore
∗This work was supported by the GACR Grant No. 20-14736S; the European Regional Development FundProject No. CZ.02.1.01/0.0/0.0/16 019/0000778; and the Austrian Science Fund (FWF) Project V662.
†University of Duisburg-Essen, Faculty of Engineering, Department of Civil Engineering, D-45117 Essen,Germany, E-mail: [email protected].
‡Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10, A-1040 Vienna (Aus-tria), E-mail: [email protected].
§Faculty of Civil Engineering, Czech Technical University, Thakurova 7, CZ-16629 Praha 6, Czech Republic,E-mail: [email protected].
¶Faculty of Civil Engineering, Czech Technical University, Thakurova 7, CZ-16629 Praha 6, Czech Republic,E-mail: [email protected].
‖Faculty of Civil Engineering, Czech Technical University, Thakurova 7, CZ-16629 Praha 6, Czech Republic,E-mail: [email protected].
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space which strengthens the material. The main problem is, however, that the consolidatedlayer is in practice rather thin and is located too close to the active boundary.
In order to avoid possible ambiguity, it is necessary to explain that the term ‘consolidation’ isto be interpreted here as a process of formation of calcium carbonate which has the propertyof binding the particles together. It might also be called ‘cementation’ or ‘compaction’.
In the literature there are a lot of works dealing with those problems. Different practicalstrategies for the wetting and drying regime which lead to a more uniform distribution of theconsolidant are compared in [24]. A mathematical model is proposed in [30, 31], and [12],where the authors derive governing equations for moisture, heat, and air flow through concrete.A numerical procedure based on the finite element method is developed there to solve theset of equations and to investigate the influence of relative humidity and temperature. It isshown that the amount of calcium carbonate formed in a unit of time depends on the degreeof carbonation, i. e., the availability of calcium hydroxide, the temperature, the carbon dioxideconcentration and the relative humidity in the pore structure of the concrete. An extension ofthe aforementioned papers by studying the hygro-thermal behavior of concrete in the specialsituation of high temperatures can be found in [16].
In the present case chemical reactions take place. Various approaches exist which describesuch processes by models which stem from different backgrounds (e. g., from mixture theoryor empirical models). An overview on the development of theories especially for porous mediaincluding chemical reactions is given in [13]. The interactions between the constituents of aporous medium are not necessarily of chemical nature which would lead to a chemical transfor-mation of one set of chemical substances to another. Simpler is the mass exchange between theconstituents by physical processes like adsorption. Adsorption-diffusion processes have beenstudied by B. Albers (the former name of B. Detmann) e. g. in [1]. Other works on sorption inporous solids including molecular condensation are [4] or [5]. In these works the diffusivities ofwater and carbon dioxide are assumed to be strongly dependent on pore humidity, temperatureand also on the degree of hydration of concrete. The authors realized that the porosity becomesnon-uniform in time. This is an observation which is interesting also in the present case be-cause the structure of the channels clearly changes with the progress of the reaction. A surveyof consolidation techniques for historical materials is published in [26]. The influence of theparticle size on the efficiency of the consolidation process is investigated in [34]. Experimentaldetermination of the penetration depth is the subject of [6, 7, 8]. Different variants of theconsolidants is studied in [14, 21, 27, 32], and an experimental work on mechanical interactionbetween the consolidant and the matrix material is carried out in [20].
A further work dealing with chemical reactions and diffusion in concrete based on the mixturetheory for fluids introduced by Truesdell and coworkers is by A. J. Vromans et al. [36]. Themodel describes the corrosion of concrete with sulfuric acid which means a transformation ofslaked lime and sulfuric acid into gypsum releasing water. It is a similar reaction we are lookingat. A similar topic is dealt with in [35], where it is shown how the carbonation process in limemortar is influenced by the diffusion of carbon dioxide into the mortar pore system by thekinetics of the lime carbonation reaction and by the drying and wetting process in the mortar.
Experimental results of CaCO3 precipitation kinetics can be found in [29]. The porosity changesduring the reaction. This was studied by Houst and Wittmann, who also investigated the
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influence of the water content on the diffusivity of CO2 and O2 through hydrated cementpaste [18]. An investigation of the physico-chemical characteristics of ancient mortars withcomparison to a reaction-diffusion model by Zouridakis et al. is presented in [37]. A slightlydifferent reaction involving also sulfur is mathematically studied in [9] by Bohm et al. There thecorrosion in a sewer pipe is modeled as a moving-boundary system. A strategy for predictingthe penetration of carbonation reaction fronts in concrete was proposed by Muntean et al. in[22]. A simple 1D mathematical model for the treatment of sandstone neglecting the effects ofchemical reactions is proposed in [11] and further refined in [10].
We model the consolidation process as a convection-diffusion system coupled with chemicalreaction in a 3D porous solid. The physical observation that only water can be evacuated fromthe porous body, while lime remains inside, requires a nonstandard boundary condition onthe active part of the boundary. We choose a simple one-sided condition for the lime exchangebetween the interior and the exterior. The main result of the paper consists in proving rigorouslythat the resulting initial-boundary value problem for the PDE system in 3D has a solutionsatisfying natural physical constraints, including the boundedness of the concentrations provedby means of time discrete Moser iterations. We also show the result of numerical simulation ina simplified 1D situation.
The structure of the present paper is the following. In Section 1, we explain the modelinghypotheses and derive the corresponding system of balance equations with nonlinear boundaryconditions. In Section 2, we give a rigorous formulation of the initial-boundary value problem,specify the mathematical hypotheses, and state the main result in Theorem 2.2. The solution isconstructed by a time-discretization scheme proposed in Section 3. The estimates independentof the time step size derived for this time-discrete system constitute the substantial step in theproof of Theorem 2.2, which is obtained in Section 4 by passing to the limit as the time steptends to zero. A numerical test for a reduced 1D system is carried out in Section 5 to illustratea qualitative agreement of the mathematical result with experimental observations.
1 The model
We imagine a porous medium (sandstone, for example) the structure of which is to be strength-ened by letting calcium hydroxide particles driven by water flow penetrate into the pores. Incontact with the air present in the pores, the calcium hydroxide reacts with the carbon dioxidecontained in the air and produces a precipitate (calcium carbonate) which is not water-soluble,remains in the pores, and glues the sandstone particles together. Unlike, e. g., in [17, 33], wedo not consider the porosity as one of the state variables. The porosity evolution law is re-placed with the assumption that the permeability decreases as a result of the calcium carbonatedeposit in the pores. The chemical reaction is assumed to be irreversible and we write it asCa(OH)2 + CO2 → CaCO3 +H2O .
Notation:
cW ... mass source rate of H2O produced by the chemical reaction
cH ... mass source rate of Ca(OH)2 produced by the chemical reaction
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cP ... mass source rate of CaCO3 produced by the chemical reaction
cG ... mass source rate of CO2 produced by the chemical reaction
mW ... molar mass of H2O
mH ... molar mass of Ca(OH)2
mP ... molar mass of CaCO3
mG ... molar mass of CO2
ρW ... mass density of H2O
ρH ... mass density of Ca(OH)2
p ... capillary pressure
s ... water volume saturation
h ... relative concentration of Ca(OH)2
p∂Ω ... outer pressure
h∂Ω ... outer concentration of Ca(OH)2
v ... transport velocity vector
k(cP ) ... permeability of the porous solid
q ... liquid mass flux
qH ... mass flux of Ca(OH)2
γ ... speed of the chemical reaction
κ ... diffusivity of Ca(OH)2
n ... unit outward normal vector
σ(x) ... transport velocity interaction kernel
α(x) ... boundary permeability for water
β(x) ... boundary permeability for the inflow of Ca(OH)2
Mass balance of the chemical reaction:
cP
mP=
cW
mW= − cH
mH= − cG
mG.
Water mass balance in an arbitrary subdomain V of the porous body:
d
dt
∫V
ρW s dx+
∫∂V
q · n dS =
∫V
cW dx.
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Calcium hydroxide mass balance in an arbitrary subdomain V of the porous body:
d
dt
∫V
ρHh dx+
∫∂V
qH · n dS =
∫V
cH dx.
Water mass balance in differential form:
ρW s+ div q = cW .
Calcium hydroxide mass balance in differential form:
ρH h+ div qH = cH .
The water mass flux is assumed to obey the Darcy law:
q = −k(cP )∇p,
with permeability coefficient k(cP ) which is assumed to decrease as the amount cP of CaCO3
given by the formula
cP (x, t) =
∫ t
0
cP (x, t′) dt′
increases and fills the pores.
The flux of Ca(OH)2 consists of transport and diffusion terms:
qH = ρHhv − κs∇h.
The mobility coefficient κs in the diffusion term is assumed to be proportional to s : If thereis no water in the pores, no diffusion takes place.
We assume that the transport of Ca(OH)2 at the point x ∈ Ω is driven by the water flux ina small neighborhood of x . In mathematical terms, we assume that there exists a nonnegativefunction σ with support in a small neighborhood of the origin such that the transport velocityv can be defined as
v(x, t) =1
ρH
∫Ω
σ(x− y)q(y, t) dy.
The main reason for this assumption is a mathematical one. The strong nonlinear couplingbetween s and h makes it difficult to control the bounds for the unknowns in the approximationscheme. We believe that such a regularization of the transport velocity makes physically senseas well.
The wetting-dewetting curve is described by an increasing function f :
p = f(s).
We focus on modeling the chemical reactions. Capillary hysteresis, deformations of the solidmatrix, and thermal effects are therefore neglected here. We plan to include them followingthe ideas of [2] in a subsequent study.
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The dynamics of the chemical reaction is modeled according to the so-called law of mass action,which states that the rate of the chemical reaction is directly proportional to the product ofthe concentrations of the reactants. We assume it in the form
cP = γmPhs(1− s). (1.1)
Its meaning is that no reaction can take place if either no Ca(OH)2 is available (that is, h = 0),or no water is available (that is, s = 0), or no CO2 is available (that is, s = 1), accordingto the hypothesis that the chemical reaction takes dominantly place on the contact betweenwater and air. In order to reduce the complexity of the problem, we assume directly that theavailable quantity of CO2 is proportional to the air content.
On the boundary ∂Ω we prescribe the normal fluxes. For the normal component of q , weassume that it is proportional to the difference between the pressures p inside and p∂Ω outsidethe body. For the flux of Ca(OH)2 , we assume that it can point only inward proportionallyto the difference of concentrations and to s , and no outward flux is possible. Inward flux takesplace only if the outer concentration h∂Ω is bigger than the inner concentration h :
q · n = α(x)(p− p∂Ω)
qH · n = −β(x)s(h∂Ω − h)+
on ∂Ω.
2 Mathematical problem
Let Ω ⊂ R3 be a bounded Lipschitzian domain. We consider the Hilbert triplet V ⊂ H ≡H ′ ⊂ V ′ with compact embeddings and with H = L2(Ω), V = W 1,2(Ω). For two unknownfunctions s(x, t), h(x, t) defined for (x, t) ∈ Ω× (0, T ) the resulting PDE system reads∫
Ω
(ρW stφ(x) + k(cP )f ′(s)∇s · ∇φ(x)) dx+
∫∂Ω
α(x)(f(s)− f(s∂Ω))φ(x) dS(x)
= γmW
∫Ω
hs(1− s)φ(x) dx, (2.1)∫Ω
(ρHhtψ(x) + (κs∇h− ρHhv) · ∇ψ(x)) dx−∫∂Ω
β(x)s(h∂Ω − h)+ψ(x) dS(x)
= −γmH
∫Ω
hs(1− s)ψ(x) dx. (2.2)
for all test functions φ, ψ ∈ V , where s∂Ω := f−1(p∂Ω), and with initial conditions
s(x, 0) = s0(x), h(x, 0) = h0(x) for x ∈ Ω. (2.3)
Hypothesis 2.1. The data have the properties
(i) s∂Ω ∈ L∞(∂Ω×(0, T )) , s0 ∈ L∞(Ω)∩W 1,2(Ω) are given such that s∂Ωt ∈ L2(∂Ω×(0, T )) ,
0 < s[ ≤ s∂Ω(x, t) ≤ 1 for a. e. (x, t) ∈ ∂Ω×(0, T ) , 0 < s[ ≤ s0(x) ≤ 1 for a. e. x ∈ Ω ;
(ii) h∂Ω ∈ L∞(∂Ω × (0, T )) , h0 ∈ L∞(Ω) are given such that 0 ≤ h∂Ω(x, t) ≤ h] for a. e.(x, t) ∈ ∂Ω× (0, T ) , 0 ≤ h0(x) ≤ h] for some h] > 0 and for a. e. x ∈ Ω ;
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(iii) f : [0, 1]→ R is continuously differentiable, 0 < f [ ≤ f ′(s) ≤ f ] for 0 ≤ s ≤ 1 ;
(iv) k is continuously differentiable and nonincreasing, 0 < k[ ≤ k(r) ≤ k] for r ≥ 0 ;
(v) σ : R3 → [0,∞) is continuous with compact support,∫R3 σ(x) dx = 1 ;
(vi) α, β ∈ L∞(∂Ω) , α(x) ≥ 0 , β(x) ≥ 0 on ∂Ω ,∫∂Ωα(x) dS(x) > 0 ,
∫∂Ωβ(x) dS(x) > 0 .
The meaning of Hypothesis 2.1 (vi) is that the boundary ∂Ω is inhomogeneous, with differentpermeabilities at different parts of the boundary. The transport of water (supply of Ca(OH)2 )through the boundary takes place only on parts of ∂Ω where α > 0 (β > 0, respectively).
The remaining sections are devoted to the proof of the following result.
Theorem 2.2. Let Hypothesis 2.1 hold. Then system (2.1)–(2.2) with initial conditions (2.3)admits a solution (s, h) such that st ∈ L2(Ω×(0, T )) , ∇s ∈ L∞(0, T ;L2(Ω;R3)) , ∇h ∈ L2(Ω×(0, T );R3) , ht ∈ L2(0, T ;W−1,2(Ω)) , h ∈ L∞(Ω× (0, T )) , s(x, t) ∈ [0, 1] a. e., h(x, t) ≥ 0 a. e.
We omit the positive physical constants which are not relevant for the analysis. The strategyof the proof is based on choosing a cut-off parameter R > 0, replacing h in the nonlinear termswith QR(h) = minh+, R , 1−s with QR(1−s), and v in (2.2) with vR := (QR(|v|)/|v|)v . Wealso extend the values of the function f outside the interval [0, 1] by introducing the functionf by the formula
f(s) =
f(0) + f ′(0)s for s < 0,f(s) for s ∈ [0, 1],f(1) + f ′(1)(s− 1) for s > 1,
and consider the system∫Ω
(stφ(x) + k(cP )f ′(s)∇s · ∇φ(x)) dx+
∫∂Ω
α(x)(f(s)− f(s∂Ω))φ(x) dS(x)
=
∫Ω
QR(h)sQR(1− s)φ(x) dx, (2.4)∫Ω
(htψ(x) + (s∇h− hvR) · ∇ψ(x)) dx−∫∂Ω
β(x)s(h∂Ω − h)+ψ(x) dS(x)
= −∫
Ω
hs(1− s)ψ(x) dx (2.5)
for all φ, ψ ∈ V . We first construct and solve in Section 3 a time-discrete approximating systemof (2.4)–(2.5), and derive estimates independent of the time step. In Section 4, we let the timestep tend to 0 and prove that the limit is a solution (s, h) to (2.4)–(2.5). We also prove thatthis solution has the property that s ∈ [0, 1], h is positive and bounded, and v is bounded,so that for R sufficiently large, the truncations are never active and the solution thus satisfies(2.1)–(2.2) as well.
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3 Time discretization
For proving Theorem 2.2, we first choose n ∈ N and replace (2.4)–(2.5) with the followingtime-discrete system with time step τ = T
n:∫
Ω
(1
τ(sj − sj−1)φ(x) + k(cPj−1)f ′(sj)∇sj · ∇φ(x)
)dx+
∫∂Ω
α(x)(f(sj)− f(s∂Ωj ))φ(x) dS(x)
=
∫Ω
QR(hj−1)sjQR(1− sj)φ(x) dx, (3.1)∫Ω
(1
τ(hj−hj−1)ψ(x) + (sj−1∇hj−hjvRj−1) · ∇ψ(x)
)dx−
∫∂Ω
β(x)sj−1(h∂Ωj − hj)+ψ(x) dS(x)
= −∫
Ω
hjsj−1(1− sj−1)ψ(x) dx, (3.2)
for j = 1, . . . , n with initial conditions s0 = s0 , h0 = h0 , and with cPi = 0 for i ≤ 0, with
qj(x) = −k(cPj (x))∇f(sj(x)) for x ∈ Ω, (3.3)
vj(x) =
∫Ω
σ(x− y)qj(y) dy for x ∈ Ω, (3.4)
s∂Ωj (x) =
1
τ
∫ tj
tj−1
s∂Ω(x, t) dt for x ∈ ∂Ω, (3.5)
h∂Ωj (x) =
1
τ
∫ tj
tj−1
h∂Ω(x, t) dt for x ∈ ∂Ω, (3.6)
where tj = jτ for j = 0, 1, . . . , n . Moreover, we define inductively
cPj − cPj−1 = τhjsj(1− sj) for j = 1, . . . , n. (3.7)
We now prove the existence of solutions to (3.1)–(3.2) and derive a series of estimates whichwill allow us to pass to the limit as n→∞ . We denote by C any positive constant dependingpossibly on the data and independent of n .
For ε > 0 we denote by Hε : R→ R the function
Hε(r) =
0 for r ≤ 0,
rε
for r ∈ (0, ε),
1 for r ≥ ε
(3.8)
as a Lipschitz continuous regularization of the Heaviside function, and by Hε its antiderivative
Hε(r) =
0 for r ≤ 0,
r2
2εfor r ∈ (0, ε),
r − ε2
for r ≥ ε
(3.9)
as a continuously differentiable approximation of the “positive part” function. Note that wehave rHε(r) ≤ 2Hε(r) for all r ∈ R .
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Lemma 3.1. Let Hypothesis 2.1 hold. Then for all n sufficiently large there exists a solutionhj, sj of the time-discrete system (3.1)–(3.2) with initial conditions s0 = s0 , h0 = h0 , cPi = 0for i ≤ 0 , which satisfies the bounds:
s[ ≤ sj(x) ≤ 1 a. e. for j = 0, 1, . . . , n. (3.10)
hj(x) ≥ 0 a. e. for j = 0, 1, . . . , n. (3.11)
Proof. To prove the existence, we proceed by induction. Assume that the solution to (3.1)-(3.2) is available for i = 1, . . . , j − 1 with the properties (3.10)–(3.11). Then Eq. (3.1) for theunknown s := sj is of the form∫
Ω
(a0(x, s)φ(x) + a1(x)f ′(s)∇s · ∇φ(x)) dx+
∫∂Ω
α(x)(f(s)− a2(x))φ(x) dS(x)
=
∫Ω
a3(x)φ(x) dx,
(3.12)
where
a0(x, s) =1
τs(x)−QR(hj−1(x))s(x)QR(1− s(x))
and ak , k = 1, 2, 3, are given functions which are known from the previous step j − 1. Forn > TR2 the function s 7→ a0(x, s) is increasing. Hence, (3.12) is a monotone elliptic problem,and a unique solution exists by virtue of the Browder-Minty Theorem, see [28, Theorem 10.49].Similarly, Eq. (3.2) is for the unknown function h := hj of the form∫
Ω
(a4(x)hψ(x) + (a5(x)∇h− a6(x)h) · ∇ψ(x)) dx−∫∂Ω
β(x)a7(x)(a8(x)− h)+ψ(x) dS(x)
=
∫Ω
a9(x)ψ(x) dx, (3.13)
which we can solve in an elementary way in two steps. First, we consider the PDE∫Ω
(a4(x)hψ(x) + (a5(x)∇h− a6(x)w) · ∇ψ(x)) dx−∫∂Ω
β(x)a7(x)(a8(x)− h)+ψ(x) dS(x)
=
∫Ω
a9(x)ψ(x) dx (3.14)
with a given function w ∈ L2(Ω). Here again the functions ak , k = 4, . . . , 9, are known.We find a solution h to (3.14) once more by the Browder-Minty Theorem. Since a4(x) ≥ 1
τ,
a5(x) = sj−1(x) ≥ s[ , and a6(x) = vRj−1 ∈ [−R,R] , we see that for n > TR2/2s[ , the mappingwhich with w associates h is a contraction on L2(Ω), and the solution to (3.13) is obtainedfrom the Banach Contraction Principle.
To derive the bounds for the solution, we first test (3.1) by φ = Hε(sj − 1) (or any otherfunction of the form g(sj − 1) with g Lipschitz continuous, nondecreasing, and such thatg(s) = 0 for s ≤ 0). The right-hand side identically vanishes, whereas the boundary term and∇sj · ∇Hε(sj − 1) are nonnegative, which yields that∫
Ω
(sj − sj−1)Hε(sj − 1) dx ≤ 0.
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From the convexity of Hε we obtain that (sj − sj−1)Hε(sj − 1) ≥ Hε(sj − 1) − Hε(sj−1 − 1),hence, ∫
Ω
Hε(sj − 1) dx ≤∫
Ω
Hε(sj−1 − 1) dx.
We have by hypothesis sj−1 ≤ 1 a. e., and by induction we get
sj(x) ≤ 1 a. e. for j = 0, 1, . . . , n. (3.15)
We further test (3.1) by φ = −Hε(s[−sj). Then both the boundary term and the elliptic term
give a nonnegative contribution, and using again the convexity of Hε we have
1
τ
∫Ω
(Hε(s[ − sj)− Hε(s
[ − sj−1)) dx ≤∫
Ω
−sjHε(s[ − sj)QR(hj−1)QR(1− sj) dx
≤∫
Ω
(s[ − sj)Hε(s[ − sj)QR(hj−1)QR(1− sj) dx
≤ 2R2
∫Ω
Hε(s[ − sj) dx,
and from the induction hypothesis we get for n > 2TR2 that
sj(x) ≥ s[ a. e. for j = 0, 1, . . . , n. (3.16)
We have in particular QR(1− sj) = 1− sj for R ≥ 1 as well as f = f .
Test (3.2) by ψ = −Hε(−hj). Then
1
τ
∫Ω
(Hε(−hj)− Hε(−hj−1)) dx+
∫Ω
s[H ′ε(−hj)|∇hj|2 dx ≤∫
Ω
hjH′ε(−hj)vRj−1 · ∇hj dx
≤ s[
2
∫Ω
H ′ε(−hj)|∇hj|2 dx+1
2s[
∫Ω
h2j |vRj−1|2H ′ε(−hj) dx
We have 0 ≤ h2jH′ε(−hj) ≤ ε and limε→0 Hε(−hj) = (−hj)+ , hence, passing to the limit as
ε→ 0, by induction we get (3.11).
Lemma 3.2. Let Hypothesis 2.1 hold. Then hj satisfies the following estimate∫Ω
hj(x) dx ≤ C (3.17)
independently of j = 0, 1, . . . , n .
Proof. Test (3.2) by ψ = 1. Note that the boundary term is bounded above by a multiple ofh] and the right-hand side is negative or zero, so that
1
τ
∫Ω
(hj − hj−1) dx ≤ C,
that is, ∫Ω
hj dx ≤ τC +
∫Ω
hj−1 dx.
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Summing up over j = 1, . . . , j∗ we get∫Ω
hj∗(x) dx ≤∫
Ω
h0(x) dx+ TC,
for an arbitrary j∗ , which yields (3.17).
The main issue will be a uniform upper bound for hj which will be obtained by a time-discrete variant of the Moser-Alikakos iteration technique presented in [3]. We start from somepreliminary integral estimates of sj .
Lemma 3.3. Let Hypothesis 2.1 hold. Then sj satisfy the bounds
τ
n∑j=1
∫Ω
|∇sj(x)|2 dx+ τn∑j=1
∫∂Ω
α(x)s2j(x) dS(x) ≤ C, (3.18)
1
τ
n∑j=1
∫Ω
|sj − sj−1|2 dx+ maxj=1,...,n
∫Ω
|∇sj(x)|2 dx ≤ C
(1 + τ
n∑j=0
∫Ω
h2j(x) dx
). (3.19)
Proof. Test (3.1) by φ = sj . From (3.17) we then obtain:
1
2τ
∫Ω
(s2j − s2
j−1) dx+f [
2
∫∂Ω
α(x)s2j(x) dS(x) + k[f [
∫Ω
|∇sj(x)|2 dx
≤ C
∫∂Ω
α(x) dS(x) +
∫Ω
hj−1 dx ≤ C.
Taking the sum with respect to j = 1, . . . , n yields
τn∑j=1
∫Ω
|∇sj(x)|2 dx+ τn∑j=1
∫∂Ω
α(x)s2j(x) dS(x) ≤ C +
∫Ω
s20 dx ≤ C,
which is precisely (3.18).
Let us prove (3.19) now. Test (3.1) by φ = f(sj)− f(sj−1). Then
f [
τ
∫Ω
|sj − sj−1|2 dx+1
2
∫Ω
(k(cPj−1)|∇f(sj)|2 − k(cPj−2)|∇f(sj−1)|2
)dx
+1
2
∫∂Ω
α(x)(f 2(sj)− f 2(sj−1)) dS(x)
≤ 1
2
∫Ω
(k(cPj−1)− k(cPj−2))|∇f(sj−1)|2 dx+
∫∂Ω
α(x)(f(sj)f(s∂Ωj )− f(sj−1)f(s∂Ω
j−1)) dS(x)
+
(∫∂Ω
α(x)f 2(sj−1) dS(x)
)1/2 (∫∂Ω
α(x)|f(s∂Ωj−1)− f(s∂Ω
j )|2 dS(x)
)1/2
+
∫Ω
hj−1(f(sj)− f(sj−1)) dx. (3.20)
11
The function k is nonincreasing and the sequence cPj is nondecreasing, so the first integral inthe right-hand side of (3.20) is negative. By Hypotheses 2.1 (i),(iii) and (3.5) we further have
1
τ
n∑j=1
∫∂Ω
α(x)|f(s∂Ωj )− f(s∂Ω
j−1)|2 dS(x) ≤ C,
and from (3.18) it follows that
τn∑j=1
∫∂Ω
α(x)f 2(sj−1) dS(x) ≤ C,
hence, by Holder’s inequality for sums,
n∑j=1
(∫∂Ω
α(x)f 2(sj−1) dS(x)
)1/2 (∫∂Ω
α(x)|f(s∂Ωj−1)− f(s∂Ω
j )|2 dS(x)
)1/2
≤ C.
Applying Young’s inequality to the last integral in the right hand side we similarly have∫Ω
hj−1(f(sj)− f(sj−1)) dx ≤ Cτ
∫Ω
h2j−1 dx+
f [
2τ
∫Ω
|sj − sj−1|2 dx.
Hence, taking the sum with respect to j and using again (3.18), we get (3.19).
Remark 3.4. As a consequence of the definition of vj and vRj , we get for all j = 1, . . . , n that
sup essx∈Ω
|vRj (x)| ≤ sup essx∈Ω
|vj(x)| ≤ C
(1 +
(∫Ω
|∇sj(x)|2 dx
)1/2). (3.21)
Lemma 3.5. Let Hypothesis 2.1 hold. Then hj satisfies the bound
maxj=1,...,n
∫Ω
|hj(x)|2 dx+ τn∑j=1
∫Ω
|∇hj(x)|2 dx ≤ C. (3.22)
Proof. Test (3.2) by ψ = hj . Note that the boundary term is bounded above by a multiple of(h])2 . Then
1
2τ
∫Ω
(h2j − h2
j−1) dx+ s[∫
Ω
|∇hj(x)|2 dx ≤ C +Kj (3.23)
with Kj :=∫
Ωhjv
Rj−1 · ∇hj dx . The evaluation of this integral constitutes the most delicate
part of the argument. For simplicity, we denote by | · |r the norm in Lr(Ω) for 1 ≤ r ≤ ∞ .We first notice that by Holder’s inequality and (3.21) we have
Kj ≤ C(1 + |hj|2|∇sj−1|2|∇hj|2).
Let us recall the Gagliardo-Nirenberg inequality for functions u ∈ W 1,p(Ω) on bounded Lips-chitzian domains Ω ⊂ RN in the form
|u|q ≤ C(|u|s + |u|1−νs |∇u|νp
)(3.24)
12
which goes back to [15, 25] and holds for every s ≤ p ≤ q such that 1q≥ 1
p− 1
N, where
ν =
1s− 1
q
1N
+ 1s− 1
p
. (3.25)
In our case we have|hj|2 ≤ C
(|hj|1 + |hj|1−ν1 |∇hj|ν2
)(3.26)
with ν = 3/5. Hence, by (3.11) and (3.17),
Kj ≤ C(
1 + |∇sj−1|2|∇hj|8/52
).
Using Holder’s inequality once again we obtain
τ
n∑j=1
Kj ≤ C
1 +
(τ
n∑j=1
|∇sj−1|52
)1/5(τ
n∑j=1
|∇hj|22
)4/5 .
We have
τn∑j=1
|∇sj−1|52 ≤ τ maxj=0,...,n
|∇sj|32n∑j=0
|∇sj|22,
and (3.18)–(3.19) together with (3.26) yield
τn∑j=1
|∇sj−1|52 ≤ C
1 +
(τ
n∑j=1
|hj|22
)3/2 ≤ C
1 +
(τ
n∑j=1
|∇hj|22
)9/10 ,
so that
τn∑j=1
Kj ≤ C
1 +
(τ
n∑j=1
|∇hj|22
)49/50 ,
and we conclude by summing up over j in (3.23) that (3.22) is true.
Corollary 3.6. As an immediate consequence of (3.22) and of (3.19), (3.21) we obtain
1
τ
n∑j=1
∫Ω
|sj − sj−1|2 dx+ maxj=1,...,n
∫Ω
|∇sj(x)|2 dx ≤ C, (3.27)
maxj=1,...,n
sup essx∈Ω
|vj(x)| ≤ C
(1 + max
j=1,...,n
(∫Ω
|∇sj(x)|2 dx
) 12
)≤ C (3.28)
with a constant C independent of R and n .
Corollary 3.7. The following estimate is a direct consequence of the inequality (3.22)
n∑j=1
∫Ω
|hj − hj−1|2 dx ≤ C. (3.29)
13
Proof. To get it we test (3.2) by ψ = hj − hj−1 . On the left-hand side we keep the term
1
τ
∫Ω
|hj − hj−1|2 dx
and move all the other terms to the right-hand side. Thanks to (3.10) and (3.28), the right-hand side contains only quadratic terms in hj , hj−1 , ∇hj , ∇hj−1 . The boundary term can beestimated using the trace theorem, so that we get an inequality of the form
1
τ
∫Ω
|hj − hj−1|2 dx ≤ C
(1 +
∫Ω
(|h2j |+ |hj−1|2 + |∇hj|2 + |∇hj−1|2
)dx
), (3.30)
and it suffices to apply (3.22).
The next lemma shows global boundedness of hj by means of the Moser-Alikakos iterationtechnique.
Lemma 3.8. Let Hypothesis 2.1 hold. Then hj satisfies the bound:
maxj=1,...,n
sup essx∈Ω
|hj(x)| ≤ C. (3.31)
Proof. Consider a convex increasing function g : [0,∞)→ [0,∞) with linear growth, and test(3.2) by ψ = g(hj). We define
G(h) =
∫ h
0
g(u) du, Γ(h) =
∫ h
0
√g′(u) du.
This yields
1
τ
∫Ω
(G(hj)−G(hj−1)) dx+ s[∫
Ω
|∇Γ(hj)|2 dx ≤ Cg(C) + C
∫Ω
hjg′(hj)|∇hj| dx,
hence,
maxj=1,...,n
∫Ω
G(hj) dx+ τn∑j=1
∫Ω
|∇Γ(hj)|2 dx ≤ G(C) + Cg(C) + Cτn∑j=1
∫Ω
h2jg′(hj) dx (3.32)
with a constant C independent of n and of the choice of the function g . We now make aparticular choice g = gM,k depending on two parameters M > 1 and k > 0, namely
gM,k(h) =
1
2k+1h2k+1 for 0 ≤ h ≤M,
12k+1
M2k+1 +M2k(h−M) for h > M.
Then
g′M,k(h) = minh,M2k =
h2k for 0 ≤ h ≤M,
M2k for h > M,
GM,k(h) =
1
(2k+2)(2k+1)h2k+2 for 0 ≤ h ≤M,
1(2k+2)(2k+1)
M2k+2 + 12k+1
M2k+1(h−M) + 12M2k(h−M)2 for h > M,
ΓM,k(h) =
1
k+1hk+1 for 0 ≤ h ≤M,
1k+1
Mk+1 +Mk(h−M) for h > M.
14
Note that for all h ≥ 0, M > 0 and k > 0 we have
GM,k(h) ≤ Γ2M,k(h) ≤ 4GM,k(h), h2g′M,k(h) ≤ (k + 1)2Γ2
M,k(h), hgM,k(h) ≤ (k + 1)Γ2M,k(h).
(3.33)It thus follows from (3.32) that
maxj=1,...,n
∫Ω
Γ2M,k(hj) dx+ τ
n∑j=1
∫Ω
|∇ΓM,k(hj)|2 dx
≤ C
((k + 1)2Γ2
M,k(C) + τ
n∑j=1
∫Ω
h2jg′M,k(hj) dx
)(3.34)
with a constant C independent of k and M .
We now apply again the Gagliardo-Nirenberg inequality (3.24) in the form
|u|q ≤ C(|u|2 + |u|1−ν2 |∇u|ν2
)to the function u = ΓM,k(hj), with q = 10/3 and ν = 3/5. From (3.33)–(3.34) it follows that
1
k + 1
(τ
n∑j=1
∣∣∣hj√g′M,k(hj)∣∣∣qq
)1/q
≤
(τ
n∑j=1
|ΓM,k(hj)|qq
)1/q
≤ C max
(k + 1)2Γ2
M,k(C), τn∑j=1
∫Ω
h2jg′M,k(hj) dx
1/2
. (3.35)
Let us start with k = 0. The right-hand side of (3.35) is bounded independently of M as aconsequence of (3.22). We can therefore let M →∞ in the left-hand side of (3.35) and obtain
τn∑j=1
|hj|qq <∞.
We continue by induction and put ωi := (q/2)i for i ∈ N . Assuming that
τ
n∑j=1
|hj|2ωi2ωi
<∞
for some i ∈ N (which we have just checked for i = 1) we can estimate the right-hand sideof (3.35) for k = ωi − 1 independently of M , and letting M → ∞ in the left-hand side weconclude that
1
ωi
(τ
n∑j=1
|hj|2ωi+1
2ωi+1
)ωi/2ωi+1
≤ C max
ω2i Γ
2M,ωi−1(C), τ
n∑j=1
|hj|2ωi2ωi
1/2
≤ C max
C2ωi , τ
n∑j=1
|hj|2ωi2ωi
1/2
.
15
which implies that(τ
n∑j=1
|hj|2ωi+1
2ωi+1
)1/2ωi+1
≤ (Cωi)1/ωi max
C,(τ
n∑j=1
|hj|2ωi2ωi
)1/2ωi
(3.36)
with a constant C > 0 independent of n and i . For i ∈ N set
Xi := max
C,(τ
n∑j=1
|hj|2ωi2ωi
)1/2ωi
,
and Λi = logXi . From (3.36) it follows that Λi+1 ≤ 1ωi
log(Cωi) + Λi . Summing up over i ∈ Nwe obtain (
τn∑j=1
|hj|2ωi2ωi
)1/2ωi
≤ C (3.37)
with a constant C independent of i . The statement now follows in a standard way. For ε > 0and j = 1, . . . , n put Ωj,ε := x ∈ Ω : |hj(x)| ≥ C + ε , where C is the constant from (3.37).Then ∫
Ω
|hj(x)|2ωi dx ≥ |Ωj,ε|(C + ε)2ωi ,
so that
C2ωi ≥ τn∑j=1
∫Ω
|hj(x)|2ωi dx ≥
(τ
n∑j=1
|Ωj,ε|
)(C + ε)2ωi .
Letting i→∞ we thus obtain
τn∑j=1
|Ωj,ε| ≤ limi→∞
(C
C + ε
)2ωi
= 0.
Passing to the limit as ε→ 0, we obtain (3.31).
4 Limit as n→∞
We now construct piecewise linear and piecewise constant interpolations of the sequences sj ,hj constructed in Section 3. Since we plan to let the discretization parameter n tend to
∞ , we denote them by s(n)j , h(n)
j to emphasize the dependence on n . For x ∈ Ω andt ∈ ((j − 1)τ, jτ ] , j = 1, . . . , n set
s(n)(x, t) = s(n)j (x),
s(n)(x, t) = s(n)j−1(x),
s(n)(x, t) = s(n)j−1(x) +
t− (j − 1)τ
τ(s
(n)j (x)− s(n)
j−1(x)),
(4.1)
16
and similarly for h(n), h(n), h(n), v(n), cP,(n), s∂Ω,(n), h∂Ω,(n) etc.
By virtue of the above estimates we can choose R sufficiently large, so that the truncations arenever active, and we can rewrite the system (3.1)–(3.2) in the form∫
Ω
(s(n)t φ(x) + k(cP,(n))f ′(s(n))∇s(n) · ∇φ(x)) dx+
∫∂Ω
α(x)(f(s(n))− f(s∂Ω,(n)))φ(x) dS(x)
=
∫Ω
h(n)s(n)(1− s(n))φ(x) dx, (4.2)∫Ω
(h(n)t ψ(x) + (s(n)∇h(n) − h(n)v(n)) · ∇ψ(x)) dx−
∫∂Ω
β(x)s(h∂Ω,(n) − h(n))+ψ(x) dS(x)
= −∫
Ω
h(n)s(n)(1− s(n))ψ(x) dx. (4.3)
The estimates derived in Section 3 imply the following bounds independent of n :
• h(n) are bounded in L2(0, T ;V );
• s(n) are bounded in L∞(0, T ;V );
• s(n)t are bounded in L2(Ω× (0, T )),
• h(n)t are bounded in L2(0, T ;V ′);
• h(n), s(n) are bounded in L∞(Ω× (0, T )).
The bound for h(n)t in L2(0, T ;V ′) follows by comparison in (4.3). Indeed, choosing arbitrary
test functions ψ ∈ V and ζ ∈ L2(0, T ) in (4.3), we obtain from the above estimates that theinequality ∫
Ω
h(n)t (x, t)ψ(x)ζ(t) dx ≤ C(1 + |∇h(n)(t)|2)|ψ|V |ζ(t)|
holds for a. e. t ∈ (0, T ). Integrating over (0, T ) and owing to estimate (3.22), we obtain theassertion.
By the Aubin-Lions Compactness Lemma ([19, Theorem 5.1]) we can find a subsequence (stilllabeled by (n) for simplicity) and functions s, h such that
• h(n) → h , s(n) → s strongly in Lp(Ω× (0, T )) for every 1 ≤ p <∞ ;
• ∇h(n) → ∇h weakly in L2(Ω× (0, T );R3);
• ∇s(n) → ∇s weakly-* in L∞(0, T ;L2(Ω;R3)).
In fact, the Aubin-Lions Lemma guarantees only compactness in L2(Ω× (0, T )). Compactnessin Lp(Ω×(0, T )) for p > 2 follows from the fact that the functions are bounded in L∞(Ω×(0, T ))by a constant K > 0, so that, for example,∫ T
0
∫Ω
|s(n)(x, t)− s(x, t)|p dx dt ≤ (2K)p−2
∫ T
0
∫Ω
|s(n)(x, t)− s(x, t)|2 dx dt.
17
Note that for t ∈ ((j − 1)τ, jτ ] we have
|s(n)(x, t)− s(n)(x, t)| ≤ |s(n)j (x)− s(n)
j−1(x)|,
hence, ∫ T
0
∫Ω
|s(n)(x, t)− s(n)(x, t)|2 dx dt ≤ τ
n∑j=1
∫Ω
|s(n)j (x)− s(n)
j−1(x)|2 dx ≤ Cτ 2
by virtue of (3.19) and (3.22). Similarly,∫ T
0
∫Ω
|h(n)(x, t)− h(n)(x, t)|2 dx dt ≤ τn∑j=1
∫Ω
|h(n)j (x)− h(n)
j−1(x)|2 dx ≤ Cτ
by virtue of (3.29). The same estimates hold for the differences s(n) − s(n) , h(n) − h(n) . Weconclude that
• h(n) → h , s(n) → s strongly in Lp(Ω× (0, T )) for every 1 ≤ p <∞ ;
• h(n) → h , s(n) → s strongly in Lp(Ω× (0, T )) for every 1 ≤ p <∞ ;
• ∇h(n) → ∇h weakly in L2(Ω× (0, T );R3);
• ∇s(n) → ∇s weakly-* in L∞(0, T ;L2(Ω;R3)).
As a by-product of the arguments in [23, Proof of Theorem 4.2, p. 84], we can derive the traceembedding formula ∫
∂Ω
|u|2 dS(x) ≤ C(|u|22 + |u|2|∇u|2)
which holds for every function u ∈ V . Consequently, we obtain strong convergence also in theboundary terms
• h(n)∣∣∂Ω→ h
∣∣∂Ω
, s(n)∣∣∂Ω→ s
∣∣∂Ω
strongly in L2(∂Ω× (0, T )).
We can therefore pass to the limit as n→∞ in all terms in (4.2)–(4.3) and check that s, h aresolutions of (2.1)–(2.2) modulo the physical constants provided R is chosen bigger than theconstants C in (3.28) and (3.31).
5 Numerical test
In order to illustrate the behavior of the solution, we propose a simplified 1D model withΩ = [0, 1] described by the system
ρW st(x, t)− ksxx(x, t) = γmWhs(1− s), (5.1)
ρHht(x, t)− (κshx − ρHhv)x = −γmHhs(1− s), (5.2)
18
0 0.2 0.4 0.6 0.8 1
x
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3cP(x,t)
t = 0.25 T t = 0.50 T t = 0.75 T t = 1.00 T
0 0.2 0.4 0.6 0.8 1
x
0
0.2
0.4
0.6
0.8
1s(x,t)
t = 0.25 T t = 0.50 T t = 0.75 T t = 1.00 T
0 0.2 0.4 0.6 0.8 1
x
-0.2
0
0.2
0.4
0.6
0.8
1h(x,t)
t = 0.25 T t = 0.50 T t = 0.75 T t = 1.00 T
Figure 1: Numerical simulations for the system (5.1)–(5.6).
for x ∈ (0, L) and t ∈ (0, T ), with boundary conditions
ksx(0, t) = α(s(0, t)− s(0, t)), (5.3)
κshx(0, t)− ρHh(0, t)v(0, t) = −βs(0, t)(h(0, t)− h(0, t))+, (5.4)
ksx(1, t) = −αs(1, t), (5.5)
κshx(1, t)− ρHh(1, t)v(1, t) = 0, (5.6)
with some constants α > 0, β > 0. The data are chosen so as to model the following situation:We start with initial conditions s0 = s[ , h0 = 0, and in the time interval [0, T/4], we chooses = 1 and h = 1. This corresponds to the process of filling the structure with lime watersolution until the time t = T/4. Then, at time t = T/4 we start the process of drying byswitching s to s[ and h to 0. With these boundary data, we let the process run in thetime interval [T/4, T ] . Figure 1 shows the spatial distributions across the profile x ∈ [0, 1] atsuccessive times t = T/4, T/2, 3T/4, T . We have chosen a finer mesh size near the origin, wherethe solution exhibits higher gradients. High concentration of CaCO3 near the active boundaryx = 0 exactly corresponds to the measurements shown, e. g., in [8, 24]. The parameters of ourmodel cannot be easily taken from the available measurements, and a complicated identificationprocedure would be necessary. This is beyond the scope of this paper, whose purpose is topresent a model to be validated by numerical simulations. For this qualitative study we havetherefore chosen fictitious parameters with simple numerical values ρW = ρH = mW = mH =α = β = 1, s[ = 0, k = 2 · 10−4, κ = 10−3 and γ = 10−2 . The final time T is determined bythe number of time steps which are necessary to reach approximate equilibrium. In fact, thequestion of asymptotic stabilization for large times will be a subject of a subsequent study.
Acknowledgments
The authors wish to thank Zuzana Slızkova and Milos Drdacky for stimulating discussions ontechnical aspects of the problem.
19
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