Mathematical Models for Removal of Pharmaceutical Pollutants in Rehabilitated Treatment Plants

Abstract

This paper aims to investigate appropriate mathematical models devoted to the optimization of some cleaning processes related to pharmaceutical contaminant removal. In our recent works, we found the rehabilitation of the existing cleaning plants as a viable solution for the removal of this type of micropollutants from waters by introducing efficient techniques such as adsorption on granulated active carbon filters and micro-, nano-, or ultrafiltration. To have these processes under better control and to assure the transfer from small- to large-scale treatment stations, specific mathematical models are necessary. Starting from Navier–Stokes equations and imposing proper boundary conditions, some mathematical physics problems are obtained for which adequate solving methods via variational methods and surjectivity results are proposed. The importance of these solution characterizations consists in their continuation in adequate numerical methods and the possibility to visualize the result by using a CFD program.

1. Introduction

Pharmaceutical pollutants (PhP) represent an enormous worldwide problem, rising from one year to another due to the increasing consumption of medicines. Many recent studies debate on a lot of issues such as their trajectories, fate, harmful effects on humans and on the entire environment all over the world, impact and finding new remedial solutions [1]. Despite this multitude of works and effervescent activities, many topics are still open, like documented and comprehensive inventories for PhP in water sources in each country, to establish adequate sampling methodologies, control parameters and indicators, and the standardization of the acceptance limits for PhP concentrations in waters, to perform appropriate researches in order to quantify the effects of the presence of these contaminants in drinkable water on humans and, in general, from aquatic systems on flora and fauna. One of the current problems raised in this regard is to search for appropriate cleaning or treatment methods to remove this kind of micropollutants which exhibit special characteristics, while the usual existent plants are improper for them [2].

In our recent studies, we reported on elaborating innovative technological solutions for the removal of pharmaceutical contaminants from water sources, and we found the rehabilitation of the existing cleaning/treatment plants as a viable depollution alternative [3]. In this cited paper, it was obtained that such a rehabilitation of potable or wastewater treatment plants can be achieved by completing the existing technologies with complementary steps like adsorption on activated carbon, micro-, ultra-, and nanofiltration, or reverse osmosis. Such an improvement by using these special types of membranes has been designed in paper [4], where a calculation scheme and other technical details have been developed. To optimize the filtration of PhP through these media made of solid or liquid membranes, we are interested in modeling various real phenomena, to draw solutions for some mathematical physics problems in order to visualize, in the end, what should be performed to keep the processes under control.

The mathematical modeling of the movement through porous membranes is not a new subject, nor is its usage in depollution studies, but its novelty here consists in setting the problem for pharmaceutical microcontaminants that exhibit special properties and characteristics. Among the important studies in this field of filtration through porous membranes, we can mention the comprehensive paper [5], where a set of equations for fluid flow through synthetic membranes with extended applicability was obtained.

Concerning the fluid passage through porous media, the characterizing models conduct to some mathematical physics problems with partial differential equations involving the p-Laplacian operator [6,7,8,9,10,11,12,13,14].

Related to the complexity of the formulation of such a problem when chemical compounds are involved, one can mention reference [15], where mathematical models for reactive flows through porous media are largely analyzed; moreover, interesting movements residing in the evolution equation for the existence domains and stability problems appear in the paper [16].

Our attention was focused on the structure of the membrane that should retain a special kind of particle in which we are interested, since our goal is to improve the PhP retention efficiency of the membranes used in the cleaning process. On the other hand, we approached the engineering facade of this problem in paper [4], where the novelty is to involve some methods and models to improve the membrane capability which is conferred by the structure of the pores that are able to retain such kind of micropollutants. However, finding new mathematical models able to characterize the flow through porous membranes is an extensively used theme, being widely discussed in [5] and deeply analyzed in papers [17,18]. The issue of pore scaling is also discussed in paper [19], and an extended study on the pore characteristics for the membranes used can be found in [20]; moreover, a formulation of the problem can be consulted in [21], where the accent falls on different kinds of stability of the solution. Studies on chemical interactions with porous medium with a mathematical model developed for convection–diffusion were reported in [22] as also in paper [23], where reactive transport through porous membranes was developed.

For membranes modeling and also for flow through them, the p-Laplacian is the most indicated procedure, as we can see in many studies like [14,24], while, in relation to the fractional porous medium equation, one can mention articles [25,26,27].

Models for flow through porous media involving transport problems in particle–fluid systems found their solutions in terms of the stream function for axisymmetric flow governed by Navier–Stokes equations represented by a special kind of function, as in paper [28]. For such a type of flow, the stream function is obtained in an interesting and complete way via variational inequalities, as the applications presented in the book [29] prove. Even though a series of problems in this subject have been solved over a period of several decades, innovative mathematical models for passage through porous membranes remain an important current topic, as the papers [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45] demonstrate. By analyzing these recent works, one can remark that the approach that is proposed in this paper is novel, and this article offers a complete scheme for obtaining a concrete mathematical model for PhP retention in porous membranes in order to improve the efficiency of rehabilitated treatment plants.

The multitude of problems that might appear when we try to find an appropriate model reflecting the real-world conditions for these processes involves a series of properties like membrane structure and formation, the structure and size of pores, and fouling and transport phenomena, such as diffusion, convection, and adsorption, as is described in paper [31]. In the case of our problem, a special care should be given to pore sizes that might be selected in accordance with molecular dimensions of the compounds found in analysis bulletins, as explained in more detail in the work [4]. Among the above-mentioned process phenomena, one may stress that adsorption is the main step responsible for efficiency while the others can be neglected.

In this paper, starting from the fundamental concepts of fluid flow through porous media models reported in the works [5,46,47,48], we propose their adaptation for our special types of membranes used in the rehabilitation of the existent treatment plants; in this way, we arrive at some mathematical physics problems for which adequate solving methods are described. In this context, we highlight some surjectivity and variational methods which can facilitate the passage to appropriate numerical methods, and then one can verify the found model via CFD program. The novelty of the study consists in the usage of membranes modeling for pharmaceutical pollutants removal from waters and also the applications of the above-mentioned mathematical procedures.

2. Materials and Methods

Since we found the rehabilitation of the existing cleaning/treatment plants as a viable solution for PhP removal from water sources by using a special kind of membranes [3], we are interested in determining a more precise image on the flow through them by obtaining an appropriate mathematical model. To find such a model, we start by writing the continuity equation for mass conservation and the Navier–Stokes equations for the transport of momentum under special conditions by taking into account the form and the dimension of the pore (knowing the type of pharmaceutical micropollutants from analysis bulletins, [4]), following the way proposed in paper [5].

Regarding the conditions that should be imposed, these are related to the membrane whose morphology follows the conditions to be stationary—which means that the motion of the membrane is negligible relative to the seepage velocity, and this should fulfill the isotropy assumption, i.e., this is a composite constructed by different types of porous structures which can be described as basic pore geometrical models. For the fluid passage through the membrane, this is considered as a single phase, presenting constant density and viscosity. The micro-flow, considering when the fluid traverses the pores of the membrane, should be laminar and with no-slip boundary conditions imposed at the fluid–solid interfaces. The condition set for macro-flow is to have small gradients for average velocity, like in paper [5].

2.1. Basic Formulation of the Problem

To find the searched mathematical problem, we write the equations governing the flow of the liquid infill in the membrane pores representing the continuity equation for mass conservation and the Navier–Stokes relation, respectively:

div v = 0,

(1)

$$\mathsf{\rho }{\displaystyle \frac{\partial v}{\partial t}}+\mathsf{\rho }v·\nabla v+\nabla p-\mathsf{\rho }g-\mathsf{\mu }∆v=0,$$

(2)

where

• v is the field of fluid velocity within the volume V_f of the fluid filling the void within representative unit cell (notion proposed in [49]),
• p is the fluid pressure,
• g is the gravitational body force per unit mass,
• $\mathsf{\rho }$ is the fluid mass density,
• μ is the fluid dynamic viscosity.

It is impossible to write the movement in each pore; therefore, one applies an averaging method through which all the elements involved in Equations (1) and (2) are averaged on V_c—a control volume (the total volume of the representative unit cell, actually), as was established in paper [50]. Thus, we have the fluid part V_f of V_c, on which we integrate the above two relations in order to average all their terms. The porosity, $\mathsf{ϵ}$, is defined by the following relation:

$$\mathsf{ϵ}={\displaystyle \frac{V_f}{V_c}}.$$

(3)

We consider the specific discharge q representing the average volume of the fluid within the pore and introduce them in Equations (1) and (2) to average them. Then, we have

$$q={\displaystyle \frac{1}{V_c}}{{\displaystyle \int \int \int }}_{V_f}vdV={\displaystyle \frac{\mathsf{ϵ}}{V_f}}{{\displaystyle \int \int \int }}_{V_f}vdV.$$

(4)

The form of the continuity equation under volumetric phase averaging becomes [50]

div q = 0.

(5)

In a similar manner, the volumetrically averaged Equation (2) is written as in paper [49], where the surface integral is evaluated by taking into account the real velocity gradients at the pore surface, thus providing a quite accurate description for the microstructure of pores. This can be modeled by using methods similar to those from work [49]. One can neglect the triple integral of the velocity dispersion, this being very small compared with the other terms, since we made the assumption that the velocity gradients are not high. In paper [49], the model of flow in a porous structure can be found by considering a laminar flow in the pore sections. Under such conditions, the surface integral is approximated and the equation for the transport of the momentum, which can be applied at any local porosity $\mathsf{ϵ}$ and for each microscopic characteristic length d in accordance with [51], can be written as follows:

$$\mathsf{\rho }{\displaystyle \frac{\partial q}{\partial t}}+\mathsf{\rho }q·\nabla \left({\displaystyle \frac{q}{\mathsf{ϵ}}}\right)+\mathsf{ϵ}\nabla p_f-\mathsf{ϵ}\mathsf{\rho }g-\mathsf{\mu }∆q+\mathsf{\mu }Fq=0,$$

(6)

where

• $p_f$ is the pressure of the fluid filling the void within representative unit cell;
• F is the microscopic shear factor.

2.2. Problems Involving p-Laplacian

As specified in the first section of this work, the filtration in the porous medium equation, in its most general expression, involves the p-Laplacian; hence, we have the equation of the form:

$${\displaystyle \frac{\partial u}{\partial t}}-{∆}_{p}A\left(u\right)=f,$$

(7)

where one may consider A(u) = u^m, having m > 1 for the fractional porous medium equation and 0 < m < 1 for the fractional fast diffusion equation [52]. By applying the method of implicit time discretization, as in paper [53], one obtains the nonlinear equation of elliptic type:

$$-h{∆}_{p}v+B\left(v\right)=f,$$

(8)

where h > 0 is a constant and B is the inverse function of A, which appears in the parabolic Equation (7) [52]. Regarding the spatial definition sets for both Equations (7) and (8), this is, in the general case, either R^N or Ω ⊂ R^N. For the problem considered in this paper, we take into account the second case with special conditions for Ω related to the real situation. We emphasize that the mathematical physics problem is completed in a Dirichlet problem with zero boundary conditions on ∂Ω. Details related to the p-Laplacian can be consulted in [54].

3. Results

In this section, a series of results giving/characterizing solutions for the problem

$$-h{∆}_{p}v+B\left(v\right)=f,\mathrm{on} \mathsf{\Omega },$$

(9)

v = 0 on ∂Ω

(10)

is presented. These propositions have been obtained by the author in papers [54,55], and they are proposed in the following to be applied in solving and/or characterizing the above formulated Dirichlet problem, to which the infiltration through porous membranes can be reduced via the previously mentioned method.

3.1. Solutions via Surjectivity Approaches

3.1.1. Solving the Problem

$$(\ast )\left\{\begin{array}{}\mathrm{(11)}& -\mathsf{\lambda }\hspace{0.17em}{\Delta }_{\mathrm{p}\hspace{0.33em}}u=f(\hspace{0.33em}\cdot \hspace{0.33em},\hspace{0.17em}u(\hspace{0.33em}\cdot \hspace{0.33em}))+h,\hspace{0.33em}x\in \Omega ,\hspace{0.33em}\mathsf{\lambda }\in \mathbf{R}\hfill \mathrm{(12)}& u|\partial \Omega =0\end{array}\right.$$

Proposition 1.

Let Ω be an open bounded set of C¹ class from R^N, N ≥ 2, p ∈ (1, +∞), h from $W^{-1,p\prime }$(Ω), and f: Ω × R → R a Carathéodory function with the following properties:

• 1⁰ f (x, − s) = − f (x, s) ∀s from R, ∀x from Ω,
• 2⁰ |f (x, s)| ≤ c₁ |s|^q−1 + β(x) ∀s from R, ∀x from Ω\A, μ(A) = 0,

where c₁ ≥ 0, q ∈ (1, p), β ∈ $L^{q\prime }$(Ω), ${\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{q\prime }}=1$.

Then, for any λ ≠ 0, the problem (∗) has a solution in $W_0^{1,p}$(Ω) in the sense of $W^{-1,p\prime }$(Ω) [54].

Proposition 2.

Let Ω be open bounded set of C¹ class from R^N, N ≥ 2, p ∈ (1, +∞), h from $W^{-1,p\prime }$(Ω), and f: Ω × R → R Carathéodory function having the following properties:

• 1⁰ f (x, − s) = − f (x, s) ∀x from Ω, ∀s from R,
• 2⁰ |f (x, s)| ≤ c₁ |s|^p−1 + β(x) ∀s from R, ∀x from Ω\A, μ(A) = 0,

where c₁ ≥ 0, β ∈ $L^{p\prime }$(Ω),${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p\prime }}$= 1.

Finally, let i: $W_0^{1,p}$(Ω) →$L^p$(Ω) be linear compact embedding. Then, for any λ, if

$$\left|\mathsf{\lambda }\right|>c_1{\mathsf{\lambda }}_1^{-1}, {\mathsf{\lambda }}_1: =\inf \left\{{\displaystyle \frac{\left|\right|u|{|}_{1,p}^p}{\left|\right|i(u)|{|}_{0,p}^p}}:u\in W_0^{1,p}(\Omega )\backslash \{0\}\right\},$$

(13)

the problem (∗) has a solution in $W_0^{1,p}$(Ω) in the sense of $W^{-1,p\prime }$(Ω) [54].

Remark 1.

For all the necessary elements—definitions and any other details—see paper [54].

Solution for Problem (9) & (10).

Since h > 0, one can divide (9) by h and one may apply Proposition 1 by taking λ = 1 (λ ≠ 0 there) and can take f ( · , u( · )) ≡ B(u) and f from (9) instead of h from the used result. The conditions required by this assertion are fulfilled. For the application of Proposition 2, one may adapt correspondingly the positive constant h.

3.1.2. Applications for Results of the Fredholm Alternative Type

For the same problem (∗), the following results have been proven in [54].

Proposition 3.

Let p be from (1, +∞) and λ ≠ 0. If

λ(−Δ_p u) = |u|^p−2 u

(14)

has not a nonzero solution in $W_0^{1,p}$(Ω), then, for any h from $W^{-1,p\prime }$(Ω), the equation

λ(−Δ_p u) = |u|^p−2 u + h

(15)

has a solution in $W_0^{1,p}$(Ω) in the sense of $W^{-1,p\prime }$(Ω) [54].

Proposition 4.

Let p be from (1, +∞) and i If

λ(−Δ_p u) = N_f u

(16)

has no nonzero solution in $W_0^{1,p}$(Ω) in the sense of $W^{-1,p\prime }$(Ω), then, for any h from $W^{-1,p\prime }$(Ω), the equation

λ(−Δ_p u) = f ( · , u( · )) + h, x ∈ Ω

(17)

has a solution in $W_0^{1,p}$(Ω) in the sense of $W^{-1,p\prime }$(Ω) [54].

Clarification. N_f: $L^p$(Ω) → $L^{p\prime }$(Ω),${\displaystyle \frac{1}{p}}$+${\displaystyle \frac{1}{p\prime }}$= 1, (N_f u)x = f (x, u(x)), the Nemytskii operator with f: Ω × R → R Carathéodory function which verifies

1⁰ |f (x, s)| ≤ c₁ |s|^p–1 + β(x) ∀s ∈ R, ∀x ∈ Ω\A, μ(A) = 0, where c₁ ≥ 0, β ∈ $L^{p\prime }$(Ω);

2⁰ f is odd and (p − 1) - homogeneous in the second variable.

Solution for Problem (9) & (10).

When the conditions required by the above two propositions are fulfilled, then we can consider such a characterization for the existence of the solution of the problem under study.

3.1.3. Applications for Results via Surjectivity at Different Homogeneity Degrees

In order to present the following two results, let us mention the operator

$$N: L^q\left(\mathsf{\Omega }\right) \to L^{q\prime }\left(\mathsf{\Omega }\right), \mathit{Nu} =|u^{q-2} u, {\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{q^{\prime }}}=1, \mathrm{with} q\in (1,p),$$

(18)

i is the canonical embedding and i′ is its adjoint.

Proposition 5.

Under the above conditions, for any λ ≠ 0 and for any h from $W^{-1,p\prime }$ (Ω), there exists u₀ in $W_0^{1,p}$(Ω), such that [54]

λ(−Δ_p)u₀ = (i′ o N o i)u₀ + h.

(19)

For the second proposition, replace the operator N from Proposition 5 with N_f, Nemytskii operator, i.e., take N: $L^q$(Ω) → $L^{q\prime }$(Ω), N = N_f with f: Ω × R → R odd Carathéodory function and (q − 1) - homogeneous in the second variable, and which verifies the following growth condition:

|f (x, s)| ≤ c₁ |s|^q−1 + β(x) ∀s in R, ∀x in Ω\A, μ(A) = 0,

(20)

where c₁ ≥ 0, β ∈ $L^{q\prime }$(Ω).

Proposition 6.

Under the above conditions, for any λ ≠ 0 and for any h in $W^{-1,p\prime }$ (Ω), there exists u₀ in $W_0^{1,p}$(Ω), such that [54]

λ(−Δ_p)u₀ = (i′ o N_f o i)u₀ + h.

(21)

Remark 2.

All other notions involved above can be found in [54].

Solution for Problem (9) & (10).

One can also formulate the problem in such a manner to superpose it on one of the situations provided by Propositions 5 and 6, and then, in case the asked conditions are fulfilled, one may prove the existence via these two results. Starting from the real conditions of the flow through mentioned porous membranes that we use in rehabilitation of the existing cleaning plants, we arrive in the end, after the mentioned preparation, at some functions fulfilling the conditions for all the presented results.

3.2. Other Characterization Types

3.2.1. Characterization of Weak Solutions Starting from Ekeland Variational Principle

Let Ω be an open bounded nonempty set in R^N, N > 1, f: Ω × R → R, and u₀ ∈ $W_0^{1,p}$(Ω). Consider the problem

$$(\ast \ast ) \left\{\begin{array}{}\mathrm{(22)}& -{\mathsf{\Delta }}_{\mathrm{p}}u=f(x,u),\hspace{0.17em}\hspace{0.33em}x\in \mathsf{\Omega }\mathrm{(23)}& u=0\hspace{0.17em}\hspace{0.33em}\mathrm{on}\hspace{0.33em}\partial \mathsf{\Omega }\hspace{0.17em},\hspace{1em}\end{array}\right.$$

and f: Ω × R → R a Carathéodory function with the growth condition

|f (x, s)| ≤ c|s|^p−1 + b(x),

(24)

where $c>0,\hspace{0.33em}2\le p\le {\displaystyle \frac{2\mathrm{N}}{N-2}}$ when N ≥ 3, 2 ≤ p < +∞ when N = 1, 2, and b ∈ $L^q$(Ω), ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.$

Proposition 7.

Let Ω be an open bounded of C¹ class set in R^N, N ≥ 3, f: Ω × R → R a Carathéodory function and u₁, u₂ from the $W_0^{1,p}$ (Ω) bounded weak subsolution and the weak supersolution of (∗∗), respectively, with u₁ (x) ≤ u₂ (x) a.e. on Ω. Suppose that f verifies (24) and there is ρ > 0 such that the function g: g(x, s) = f (x, s) + ρs is strictly increasing in s on [inf u₁ (Ω), sup u₂ (Ω)]. Then, there is a weak solution $\overline{u}$ of (∗) in $W_0^{1,p}$(Ω) with the property [54]

$$u_1\left(x\right)\le \overline{u}\left(x\right)\le u_2\left(x\right) a.e. on \Omega .$$

(25)

Clarification. For the definitions and other elements necessary for the last result, paper [54] can be consulted.

Solution for Problem (9) & (10).

In this case, the place of f from (22) is taken by

$$\left(x,u\right)\to {\displaystyle \frac{1}{h}}\left(-B\left(u\left(x\right)\right)+f\left(x\right)\right).$$

(26)

When the conditions of Proposition 7 are fulfilled, then the characterization of the solution of the studied problem is obtained.

3.2.2. Solutions Involving Critical Points for Nondifferentiable Functionals

To introduce the next results, let Ω be a bounded domain of R^N with the smooth boundary ∂Ω (topological boundary). Consider the nonlinear boundary value problem (∗∗) from (22)+(23), where f: Ω × R → R is a measurable function with subcritical growth, i.e.,

(I) |f (x, s)| ≤ a + b|s|^σ ∀s ∈ R, x ∈ Ω a.e.,

(27)

where a, b > 0, 0 ≤ σ < ${\displaystyle \frac{N+2}{N-2}}$ for N > 2 and σ ∈ [0, +∞) for N = 1 or N = 2.

Set as in [56]:

$$\underset{¯}{f}\left(x,t\right)=\underset{\mathrm{s}\to \mathrm{t}}{\underset{¯}{\lim }}f\left(x,s\right),\overline{f}\left(x,t\right)=\underset{\mathrm{s}\to \mathrm{t}}{\overline{\lim }}f\left(x,s\right).$$

(28)

Suppose

(II) $\underset{¯}{f}$, $\overline{f}$: Ω × R → R are measurable with respect to x.

Emphasize that (II) is verified in the following two cases:

1⁰ f is independent of x;

2⁰ f is Baire measurable and s → f (x, s) is decreasing ∀x ∈ Ω, in which case, we have

$$\overline{f}(x, t)=\max \{f (x, t+), f (x, t-\left)\right\},\underset{¯}{f}(x, t)=\min \{f (x, t+), f (x, t-\left)\right\}.$$

(29)

For the announced result, the following definition is necessary: u from $W_0^{1,p}$(Ω), p > 1 is solution of (∗∗) if u = 0 on ∂Ω in the sense of trace (see details in [54]) and

$$–{\mathsf{\Delta }}_{\mathrm{p}} u\left(x\right) \in \left[\underset{¯}{f}\right(x, u\left(x\right)),\overline{f}(x, u\left(x\right)\left)\right] \mathrm{in} \mathsf{\Omega } \mathrm{a}.\mathrm{e}.$$

(30)

Associate to the problem (∗∗) the locally Lipschitz functional Φ: $W_0^{1,p}$(Ω) → R,

$$\mathsf{\Phi }\left(u\right)={\displaystyle \frac{1}{p}}\left|\right|u|{|}_{1,\mathrm{p}}^{\mathrm{p}} - {\displaystyle \underset{\mathsf{\Omega }}{\int }F(x,u)}dx, u \in W_0^{1,p}\left(\mathsf{\Omega }\right),$$

(31)

Proposition 8.

If (I) and (II) are verified, every critical point of Φ is a solution for (∗∗).

Solution for Problem (9) & (10).

The conditions (I) and (II) are fulfilled for a wide set of functions having the form (26), hence, in this manner, another characterization of the solution with Proposition 8 can be proposed.

3.3. Weak Solutions Using a Perturbed Variational Principle

For the following statement, some clarifications are necessary. Let Ω be an open bounded set of C¹ class in R^N, N ≥ 3. Consider the above problem (∗∗), where f: Ω × R → R is a Carathéodory function with the growth condition

$$\left|f\left(x,s\right)\right| \le c{\left|s\right|}^{\mathrm{p}-1}+b\left(x\right),c>0, 2 \le p\le {\displaystyle \frac{2N}{N-2}}, b \in L^{p\prime }\left(\mathsf{\Omega }\right),{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p\prime }}=1.\hspace{0.17em}$$

(32)

The functional φ: $W_0^{1,p}$(Ω) → R,

$$\mathsf{\phi }\left(u\right)={\displaystyle \underset{\Omega }{\int }\left[\frac{1}{p}\left(|u{|}^{\mathrm{p}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\left|\frac{\partial u}{\partial x_{\mathrm{i}}}\right|}^{\mathrm{p}}}\right)-F(x,u(x))\right]\hspace{0.17em}dx}$$

(33)

with F(x, s): = ${\displaystyle \underset{0}{\overset{\hspace{0.17em}\hspace{0.33em}\mathrm{s}}{\int }}f(x,t)\hspace{0.17em}dt}$, is of Fréchet C¹ class, and its critical points are the weak solutions of the problem (∗∗).

Proposition 9.

Under the above assumptions and, in addition, the growth condition

$$F\left(x, s\right) \le c1 {\displaystyle \frac{s^p}{p}}+\mathsf{\alpha }\left(x\right)s,$$

(34)

with 0 < c₁ < λ₁, α ∈ $L^{q\prime }$(Ω) for some 2 ≤ q ≤ ${\displaystyle \frac{2N}{N-2}}$ and f (x, −s) = −f (x, s), ∀x from Ω, the following assertions hold:

(i) The set of functions h from $W^{-1,p\prime }$(Ω), having the property that the functional

$${\mathsf{\phi }}_h:W_0^{1,p}\left(\mathsf{\Omega }\right) \to R,{ \mathsf{\phi }}_h\left(\mathrm{u}\right)={\displaystyle \frac{1}{\mathrm{p}}} {\Vert \mathrm{u}\Vert }_p^p-{\displaystyle \underset{\mathsf{\Omega }}{\int }\left(\mathrm{F}\right(\mathrm{x},\mathrm{u}\left(\mathrm{x}\right))+\mathrm{h}\left(\mathrm{u}\right(\mathrm{x}\left)\right))\hspace{0.17em}\mathrm{dx}}$$

(35)

has in only one point an attained minimum, includes a G_δ set everywhere dense;

(ii) The set of functions h from $W^{-1,p\prime }$(Ω), having the property

$$the problem\left\{\hspace{0.17em}\begin{array}{c}-{\mathsf{\Delta }}_{\mathrm{p}}\mathrm{u}=\mathrm{f}(\mathrm{x},\mathrm{u})+\mathrm{h}\left(\mathrm{u}\right)\hspace{0.33em}\mathrm{in}\hspace{0.33em}\mathsf{\Omega }\\ \mathrm{u}=0\hspace{0.33em}\mathrm{on}\hspace{0.33em}\partial \mathsf{\Omega }\end{array}\right.\hspace{1em}has solutions,$$

includes a G_δ set everywhere dense;

(iii) Moreover, if s → f (x, s) is increasing, then the set of functions h from $W^{-1,p\prime }$(Ω), having the property

$$the problem\left\{\hspace{0.17em}\begin{array}{c}-{\mathsf{\Delta }}_{\mathrm{p}}\mathrm{u}=\mathrm{f}(\mathrm{x},\mathrm{u})+\mathrm{h}\left(\mathrm{u}\right)\hspace{0.33em}in\hspace{0.33em}\mathsf{\Omega }\\ \mathrm{u}=0\hspace{0.33em}on\hspace{0.33em}\partial \mathsf{\Omega }\end{array}\right.\hspace{1em}has a unique solution,$$

includes a G_δ set everywhere dense [54].

Clarification. For any notion, see the details and explanations given in paper [54].

Solution for Problem (9) & (10).

In this last case, the role of

$$\left(x,u\right)\to f\left(x,u\left(x\right)\right)+h\left(u\right)$$

(36)

is played by (26). So, when the conditions of Proposition 9 are fulfilled, then one finds this kind of description of the solution to the considered problem.

3.4. A Theorem Based on a Result of Moutain Pass Type

To enounce the last result applied in this work, we need the function f: $\overline{\mathsf{\Omega }}$ × R → R, which is of Carathéodory type and satisfies the growth condition

|f (x, s)| ≤ c|s|^p−1 + α(x), ∀x ∈ Ω\A with μ(A) = 0 ∀s ∈ R,

(37)

where c ≥ 0, α ∈ $L^{p\prime }$(Ω), with ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p\prime }}$ = 1 and 1 ≤ p ≤ ${\displaystyle \frac{2N}{N-2}}$ if N ≥ 3 and 1 ≤ p ≤ +∞ if N = 2, and let F(x, t) = ${{\displaystyle \int }}_0^{t}f\left(x,s\right)ds$.

Proposition 10.

Assume that f satisfies

• (f1) f (x, s) = o(|s|), s → 0, uniformly for x ∈ $\overline{\mathsf{\Omega }}$;
• (f2) There exist constants μ > p and r > 0, e.g., for |s| ≥ r,
  •            0 < μF(x, s) ≤ sf (x, s).

Then, the above problem (∗∗) possesses a nontrivial solution [55].

Solution for Problem (9) & (10).

One can remark that the conditions (f1) and (f2) are fulfilled for a great class of functions and that it is possible for the function f of the form (26) to be fulfilled; thus, another solving method for our problem under study is established.

4. Discussion

This paper discussed novel mathematical aspects of filtration through porous membranes. We studied these problems when we noticed the possibility of involving them in modeling a special kind of flow through some types of porous membranes used in the removal of pharmaceutical micropollutants. The issue of these contaminants was quite recently highlighted, and their enormous amount together with the inability of the usual cleaning plants to remove them from water sources opened an extensive field of research. In other works [3,4], we proposed as a viable solution (which is not the unique possibility!) the rehabilitation of the existing treatment plants by introducing some specific membranes that are able to retain the pharmaceutical contaminants. Hence, special dimensions and geometry of the pores are needed together with the construction of a mathematical model for a better control of this process. For this reason, we adapted firstly a given model from the work [5] for the flow through a porous membrane in order to comply with the requirements imposed by the PhP particles obtained from the analysis bulletins. The detailed description of pores will be the subject of future works together with the detailed description of the parabolic problem and the developed passage towards the Dirichlet mathematical physics characterization.

We developed the problem involving p-Laplacian, and, by implicit time discretization, the parabolic problem was changed to an elliptic one, arriving to a Dirichlet problem for p-Laplacian. The author applied ten existence results obtained in papers [54,55]. These approaching methods actually suggest the continuation in ten types of numerical methods proposed for future studies. A comparison and ranking of these ways should be achieved in relation to the pore conditions that will be introduced in the base model. A final verification will be performed by introducing the model into CFD program to visualize the flow through the proposed pores.

5. Conclusions

This work traces a new path to model the flow through some special types of porous membranes involved in the retention of pharmaceuticals from waste and drinkable waters in rehabilitated cleaning plants. The starting point of such a method is to take into account the modeling of the considered flow which has been adapted to the requirements imposed by the retention of this kind of micropollutant.

The next step was to formulate the flow model as a mathematical physics problem involving a parabolic partial differential equation with p-Laplacian, which was transformed, via the method of implicit time discretization, into a Dirichlet problem for an elliptic equation with p-Laplacian.

A series of ten results previously found by the author has been proposed to solve and/or characterize the solution of the Dirichlet problem obtained. This is an intermediate step on the way to visualize a concrete solution to the studied problem since there exists the possibility to continue these variational and surjectivity theorems in appropriate numerical methods and also to apply the CFD program to see the trajectories of these pollutants.

The aim of this paper was to establish an original approach for the solution of the problem of PhP removal from water treatment plants. The presentation of this approach justifies the continuation of the research to determine the expressions of the free term of the movement equation in relation to the concrete data which we obtained from some rehabilitated plants, to particularize some value for p related to our phenomena, to develop the passage from parabolic to elliptic problem, and to finalize the existing results in corresponding numerical methods. We consider the application of these abstract results in solving the highlighted real-world problem of special importance, which increases their value since they have not been involved in such a concrete model until now.