On the Solvability of the Brinkman-Forchheimer-Extended Darcy Equation
The nonlinear Brinkman-Forchheimer-extended Darcy equation is used to model some porous medium flow in chemical reactors of packed bed type. The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with nonconstant porosity and for small data. Furthermore, the finite element approximations to the flow profiles in the fixed bed reactor are presented for several Reynolds numbers at the non-Darcy’s range.
2010 Mathematics Subject Classification (MSC): 76D03, 35Q35
Brinkman-Forchheimer Equation, Packed Bed Reactors, Existence and Uniqueness of Solution
In this section we introduce the mathematical model describing incompressible isothermal flow in porous medium without reaction. The considered equations for the velocity and pressure fields are for flows in fluid saturated porous media. Most of research results for flows in porous media are based on the Darcy equation which is considered to be a suitable model at a small range of Reynolds numbers. However, there are restrictions of Darcy equation for modeling some porous medium flows, e.g. in closely packed medium, saturated fluid flows at slow velocity but with relatively large Reynolds numbers. The flows in such closely packed medium behave nonlinearly and can not be modelled accurately by the Darcy equation which is linear. The deficiency can be circumvented with the Brinkman–Forchheimer-extended Darcy law for flows in closely packed media, which leads to the following model: Let , , represent the reactor channel. We denote its boundary by . The conservation of volume-averaged values of momentum and mass in the packed reactor reads as follows
where , denote the unknown velocity and pressure, respectively. The positive quantity stands for porosity which describes the proportion of the non-solid volume to the total volume of material and varies spatially in general. The expression represents the friction forces caused by the packing and will be specified later on. The right-hand side represents an outer force (e.g. gravitation), the constant fluid density and the constant kinematic viscosity of the fluid, respectively. The expression symbolizes the dyadic product of with itself.
The formula given by Ergun  will be used to model the influence of the packing on the flow inertia effects
Thereby stands for the diameter of pellets and denotes the Euclidean vector norm. The linear term in (2) accounts for the head loss according to Darcy and the quadratic term according to Forchheimer law, respectively. For the derivation of the equations, modelling and homogenization questions in porous media we refer to e.g. [2, 4]. To close the system (1) we prescribe Dirichlet boundary condition
has to be fulfilled on each connected component of the boundary . We remark that in the case of polygonally bounded domain the outer normal vector has jumps and thus the above integral should be replaced by a sum of integrals over each side of . The distribution of porosity is assumed to satisfy the following bounds
with some constants .
A comprehemsive account of fluid flows through porous media beyond the Darcy law’s valid regimes and classified by the Reynolds number, can be found in, e.g., . Also, see  for simulating pumped water levels in abstraction boreholes using such nonlinear Darcy-Forchheimer law, and , , and  for recent referenes on this model.
In the next section we use the porosity distribution which is estimated for packed beds consisting of spherical particles and takes the near wall channelling effect into account. This kind of porosity distribution obeys assumption (A1).
Let us introduce dimensionless quantities
whereby denotes the magnitude of some reference velocity. For simplicity of notation we omit the asterisks. Then, the reactor flow problem reads in dimensionless form as follows
and the Reynolds number is defined by
The existence and uniqueness of solution of the nonlinear model (5) with constant porosity and without the convective term has been established in . We will extend this result to the case when the porosity depends on the location and with the convective term in this work.
(5) becomes a Navier-Stokes problem if .
Notation Throughout the work we use the following notations for function spaces. For , and bounded subdomain let be the usual Sobolev space equipped with norm . If , we denote the Sobolev space by and use the standard abbreviations and for the norm and seminorm, respectively. We denote by the space of functions with compact support contained in . Furthermore, stands for the closure of with respect to the norm . The counterparts spaces consisting of vector valued functions will be denoted by bold faced symbols like or . The inner product over and will be denoted by and , respectively. In the case the domain index will be omitted. In the following we denote by the generic constant which is usually independent of the model parameters, otherwise dependences will be indicated.
2 Existence and uniqueness results
In the following the porosity is assumed to belong to . We start with the weak formulation of problem (5) and look for its solution in suitable Sobolev spaces.
2.1 Variational formulation
be the space consisting of functions with zero mean value. We define the spaces
Let us introduce the following bilinear forms
Furthermore, we define the semilinear form
and trilinear form
Multiplying momentum and mass balances in (5) by test functions and , respectively, and integrating by parts implies the weak formulation:
Find with such that
First, we recall the following result from :
The mapping is an isomorphism from onto itself and from onto itself. It holds for all
In the following the closed subspace of defined by
will be employed. Next, we establish and prove some properties of trilinear form and nonlinear form .
Let and with and . Then we have
Furthermore, the trilinear form and the nonlinear form are continuous, i.e.
and for and for a sequence with , we have also
Proof. We follow the proof of [7, Lemma 2.1, §2, Chapter IV] and adapt it to the trilinear form
which has the weighting factor . Hereby, symbols with subscripts denote components of bold faced vectors, e.g. . Let , and . Integrating by parts and employing density argument, we obtain immediately (9)
From Sobolev embedding (see ) and Hölder inequality follows
The continuity of follows from Hölder inequality and Sobolev embedding (see )
In the next stage we consider the difficulties caused by prescribing the inhomogeneous Dirichlet boundary condition. Analogous difficulties are already encountered in the analysis of Navier–Stokes problem. We will carry out the study of three dimensional case. The extension in two dimensions can be constructed analogously. Since , we can extend inside of in the form of
with some . The operator curl is defined then as
We note that in the two dimensional case the vector potential can be replaced by a scalar function and the operator curl is then redefined as . Our aim is to adapt the extension of Hopf (see ) to our model. We recall that for any parameter there exists a scalar function such that
For the construction of see also [7, Lemma 2.4,
§2, Chapter IV].
Let us define
In the following lemma we establish bounds which are crucial for proving existence of velocity.
The function satisfies the following conditions
and for any there exists sufficiently small such that
we obtain from Cauchy-Schwarz and triangle inequalities
Applying Hardy inequality (see )
and using Sobolev embedding , estimate (18) becomes
holds. Therefore the proof of estimate (15) is completed. Now, we take a look at the trilinear convective term
The first term of above difference becomes small due to [7, Lemma 2.3, §2, Chapter IV], and it satisfies
as long as is chosen sufficiently small. Using Hölder inequality, Sobolev embedding yields
which together with (19) implies for sufficiently small the bound
From (20) and (21) follows the desired estimate (16).
While the general framework for linear and non-symmetric saddle point problems can be found in , our problem requires more attention due to its nonlinear character. Setting , the weak formulation (8) is equivalent to the following problem
Find such that
Let us define the nonlinear mapping with
whereby defines the inner product in via
Then, the variational problem (22) reads in the space of -weighted divergence free functions as follows
Find such that
2.2 Solvability of nonlinear saddle point problem
We start our study of the nonlinear operator problem (24) with the following lemma.
The mapping defined in (23) is continuous and there exists such that
Proof. Let be a sequence in with . Then, applying Cauchy–Schwarz inequality and (16), we obtain for any
The boundedness of in , (12), the Poincaré inequality, and the above inequality imply that
we state that is continuous. Now, we note that for any we have
From the Poincaré inequality, we infer the estimate
Choosing such that
leads to the desired assertion (25).
The following lemma plays a key role in the existence proof.
Let be finite-dimensional Hilbert space with inner product inducing a norm , and be a continuous mapping such that
Then there exists , with , such that
Proof. See .
Now we are able to prove the main result concerning existence of velocity.
The problem (24) has at least one solution .
Proof. We construct the approximate sequence of Galerkin solutions. Since the space is separable, there exists a sequence of linearly independent elements . Let be the finite dimensional subspace of with
and endowed with the scalar product of . Let , be a Galerkin solution of (24) defined by
This gives together with (27) the uniform boundedness in
therefore there exists and a subsequence ( we write for the convenience instead of ) such that
Furthermore, the compactness of embedding implies
Taking the limit in (29) with we get
Proof. See [7, Theorem 1.4, §1, Chapter IV].
We end up this subsection by proving the existence of the pressure.
Let be solution of problem (24). Then, there exists unique pressure .
with a positive constant . Setting and applying the isomorphism in Theorem 2, we obtain the estimate
. From the above estimate we conclude the inf-sup condition (31).
2.3 Uniqueness of weak solution
We exploit a priori estimates in order to prove uniqueness of weak velocity and pressure.
If , are sufficiently small, then the solution of (24) is unique.