Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 74, pp. 1–16.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ESTIMATES OF CHARACTERISTICS OF A MICROPOLAR

FLOW PASSING THROUGH AN AXIALLY SYMMETRIC CELL

YULIA O. KOROLEVA, DARIA YU. KHANUKAEVA

Abstract. We study a model for the filtration of micropolar fluid in the
framework of a cell model technique. A porous medium is presented as an

assemblage of axially symmetric cells of an arbitrarily geometry. Each cell

consists of a solid core, porous layer and liquid shell. The influence of the
neighboring cells is taken into account via Cunningham’s-type boundary con-

dition. We derive a priori estimates for flow characteristics which show the

behavior of the velocity filed. The boundedness of velocity filed is justified by
the derived estimates.

1. Introduction

Modern technologies and devices utilize microscale processes and knowledge of
the material structure for their adequate description and modeling. Regarding
hydrodynamics, these two items are demanded for fluid flows in constrained con-
ditions. One of the examples of such flows is a filtration through membranes,
which represent porous media of an irregular structure. A widely spread method
of the membrane flow simulations is the cell model technique [13]. According to
this method the chaotic membrane structure is replaced by a set of identical cells.
Each cell may be constructed of solid or porous, or liquid core and liquid shell. In
the most general case the core can be a solid-porous composite. The influence of
the neighboring particles is taken into account by setting the appropriate bound-
ary conditions at the outer cell surfaces. So, the consideration of the flow in the
membrane is reduced to the solution of the flow problem in a single cell. Originally
the cell shape was cubical, and this led to a computational instability in the cor-
ners. A substantial progress occurred in applying the cell model when the shape
of the cell and its core were taken to be the same, namely, cylindrical and spher-
ical [14]-[21], with the inter-cell space being neglected. These papers dealt with
a simple cell with solid core and used various conditions at the outer cell surface.
Happel [14, 15] used vanishing of shear stress, Kuwabara [20] supposed vanishing
of rotation, Mehta-Morse [24] applied flow homogeneity originally introduced by
Cunningham [5], Kvashnin [21] exploited the symmetry of the velocity profile. An-
alytical solutions were obtained in all of the mentioned papers, and the specifics of
the cell geometry was substantially used. A spherical cell with concentric porous
spherical core was considered in [26]. The obtained explicit analytical solution was

2010 Mathematics Subject Classification. 76D03, 76D07, 76D10, 76S99.

Key words and phrases. Micropolar fluid flow; porous medium; weak solution.
c©2021. This work is licensed under a CC BY 4.0 license.

Submitted March 17, 2021. Published September 11, 2021.

1



2 YU. O. KOROLEVA, D. YU. KHANUKAEVA EJDE-2021/74

used for the calculation of the hydrodynamic permeability of a set of porous parti-
cles (membrane). The latter is an important macro characteristic of a membrane
measured in an experiment. The flow problem in a cell with partially porous core
consisting of a solid inner sphere, covered with a concentric porous hydrodynami-
cally uniform layer, was solved in [28]. Later, the developed method was applied
to the filtration modeling in fibrous membranes [29]. The membrane material was
simulated as an ensemble of identical cylindrical cells. The core of each cell con-
sisted of a solid inner cylinder, covered by a coaxial porous layer. The flows along
and across the cell axis were studied. The obtained solutions were composed to
simulate the chaotic orientation of fibers with respect to the flow direction. Along
with the aforementioned four boundary conditions at the outer cell surface various
conditions are known at the porous-liquid interface. A stress jump condition was
applied at this boundary and its influence on the membrane hydrodynamic per-
meability was analyzed in [6, 30]. The flow of a fluid with variable viscosity was
considered in [9] for composite cylindrical cells, and in [10] for composite spherical
cells.

Nevertheless, classical hydrodynamics does not take into account effects which
may occur at the micro-level because it treats fluids as an ensemble of material
points. As a result, one observes the loss of solution similarity for micro flow
domains or the unexpected behavior of viscosities in the vicinity of boundaries.
Numerous non-Newtonian liquid models confirm the fact that none of them is sat-
isfactory and adequate enough to describe all the known effects, despite each of
them works fine in the particular case it was designed for.

The simplest sophistication of the classical model of liquid is the micropolar
fluids. They treat liquid particles as infinitesimal rigid bodies, which can perform
independent translational and rotational motion. The original idea of the micro
structured continua belongs to the Cosserat brothers [4], but its active application
began after the development of the mathematical theory of simple micro fluids by
Eringen [7, 8]. Among basic applications of the micropolar fluids theory one can
mention flows of suspensions, lubricants, blood, synovial liquid, drilling solutions.
A review of the known analytical solutions and applications of the micropolar fluids
was given by Khanukaeva and Filippov [16].

Field equations for such liquids include the continuity equation, momentum equa-
tion, and moment of momentum equation. They contain additional viscosity coeffi-
cients characterizing microstructural properties of the liquid. The main peculiarity
of the governing equations for the micropolar fluids is that they contain a char-
acteristic flow scale. So, the solution does not possess a similarity property and
depends on the microscale of the flow. Nevertheless, in the limiting case the sys-
tem of governing equations reduces to the classical Navier-Stokes equations, which
describe the flow of Newtonian liquids.

Consideration of liquids with microstructure is especially relevant to the filtration
problems, when the most intensive interaction of flow with the surrounding medium
takes place. The close position of boundaries makes ineffective the Darcy law for
the filtration modeling in the framework of the cell model. Darcy law requires some
additional assumptions on the mechanism of the flow interaction with solid surfaces.
At the same time, Newtonian liquid flows in porous media are successfully described
by the Brinkman equation [1]. The extension of this equation to the micropolar
fluids was made in [12]. The cell model technique was applied to the micropolar



EJDE-2021/74 ESTIMATES OF CHARACTERISTICS OF A MICROPOLAR FLOW 3

fluid flows in fibrous and globular membranes in [17]-[19]. The cells of cylindrical
and spherical shapes were considered and the cell shape certainty allows to obtain
a solution in an explicit analytical form. Here we extend the study to a cell of
arbitrary shape with axial symmetry. This property of the cell geometry plays a
key role in the formulation of the field equations in the form allowing for their
integration. The solution to our problem is treated in the weak sense. We derive
a priori estimates for the behavior of flow. The obtained inequalities allow making
easy estimates of linear and angular velocities at micro scale in filtration flows of
micropolar fluids. Moreover, with the help of the derived estimates we have shown
the boundedness of weakly defined velocity fields. The exact analytical formulae
solving our problem for the case of spherical particle are given in [19].

2. Statement of the problem

Assume that a porous cell Ω = Ω1 ∪ Ω2 ∈ R3 is bounded and has an arbitrary
shape with axial symmetry (see Figure 1). Its boundary is assumed to be smooth
and consists of the parts ∂Ω1 = Γ1 ∪ Γ2, ∂Ω2 = Γ2 ∪ Γ3. The direction of the

uniform incoming flow of velocity Ũ coincides with the symmetry axis. Here and
further we use symbol ˜ for the dimensioned flow characteristics.

Figure 1. Cell domain

The flow domain Ω is split into two regions Ω1 and Ω2 in order to represent
separately porous and liquid parts of the cell. Micropolar fluid occupies domain
Ω2, while domain Ω1 represents porous material with pore space fully saturated
with the fluid. The filtration flow in domain Ω1 is governed by the system of
equations which differs from the system governing the free flow in domain Ω2. Now
we proceed to the discussion of these systems.

Micropolar media principally differ from Newtonian liquids by the construction
of their elements, which can rotate in addition to the translation motion. The
angular velocity of microrotation ω̃ contributes to the deformation rate tensor γ̂,
which becomes a non-symmetric and equals to γ̂ = (∇ṽ)T − ε̂ · ω̃, where ∇ṽ is
the gradient of linear velocity ṽ, ε̂ is the Levi-Civita tensor, superscript T implies
transposed values. Apart from the volumetric, shear and rotational deformations
described by tensor γ̂, an element of a micropolar medium may twist and bend due
to the dependence of ω̃ on the spatial coordinate. The rate of these deformations

are characterized by the curvature-twist rate tensor χ̂ = (∇ω̃)
T

, which is also non-
symmetric due to the definition of the angular velocity gradient ∇ω̃. Hence, the



4 YU. O. KOROLEVA, D. YU. KHANUKAEVA EJDE-2021/74

stress tensor t̂ and couple stress tensor m̂ of a micropolar fluid are both non sym-
metric. Meanwhile, strictly speaking micropolar media cannot be included in the
list of non-Newtonian liquids. The reason is that the theory of simple micro fluids
uses linear relation of stresses and couple stresses with the deformation rate ten-
sors, as accepted in classical hydrodynamics. Besides, in the limiting case the state
equations should transfer to the classical Navier-Stokes law, which linearly relates
the symmetric stress tensor with the symmetric deformation rate tensor, dynamic
viscosity µ being the coefficient of proportionality. We use the state equations of
simple micro fluids in the form traditional to non symmetric theory of elasticity
[25], namely

t̂ = (−p̃+ λ tr γ̂)Ĝ+ 2µγ̂(S) + 2κγ̂(A),

m̂ = α(tr χ̂)Ĝ+ 2δχ̂(S) + 2ςχ̂(A),

where p̃ is the hydrostatic pressure, λ is the second classical viscosity coefficient,
Ĝ is a metric tensor, κ is the rotational viscosity coefficient, α, δ, ς are the angular
viscosity coefficients, superscripts (S) and (A) denote symmetric and skew sym-
metric parts of tensors correspondingly. The advantage of this form of the state
equations is that in the limiting case of κ = 0 one obtains the classical expression
for the stress tensor with coefficient µ coinciding with the dynamic viscosity of the
Newtonian liquid.

In the further notations subscripts 1 or 2 are applied to show that the flow is
considered in the corresponding subdomain Ω1,Ω2. For example, ṽi denotes the
dimensioned linear velocity of flow passing through Ωi : ṽi = ṽ|Ωi

. The notations
ω̃i, p̃i are analogous.

Thus, the field equations in the domain Ω2 are the following [7]. The mass
conservation law gives the continuity equation for an incompressible liquid

div ṽ2 = 0. (2.1)

Combined with the definition of tensor γ̂ equation (2.1) yields tr γ̂ = 0 that excludes
coefficient λ from the problem statement. The momentum conservation law with
substituted stress tensor reads

∇p̃2 = (µ+ κ)∆ṽ2 + 2κ curl ω̃2. (2.2)

The moment of the momentum balance with substituted couple stress tensor gives
the differential form of the moment of momentum equation

(α+ δ − ζ)∇ div ω̃2 + (δ + ζ)∆ω̃2 + 2κ curl ṽ2 − 4κω̃2 = 0. (2.3)

Governing equations for the stationary filtration of the micropolar fluid in porous
media were derived in [12] with the use of a standard averaging technique. So, for
region Ω1 they look as follows:

div ṽ1 = 0, (2.4)

∇p̃1 =
µ+ κ

ε
∆ṽ1 +

2κ

ε
curl ω̃1 −

µ+ κ

k
ṽ1, (2.5)

(α+ δ − ζ)∇ div ω̃1 + (δ + ζ)∆ω̃1 + 2κ curl ṽ1 − 4κω̃1 = 0, (2.6)

where k is the permeability and ε is the porosity of the porous medium (given
constants). It is worth mentioning, that the values of linear and angular velocities
ṽ1, ω̃1 and pressure p̃1 in equations (2.4)-(2.5) are averaged. Hence, they can be
directly compared with the corresponding values measured in experiments.



EJDE-2021/74 ESTIMATES OF CHARACTERISTICS OF A MICROPOLAR FLOW 5

Owing to the axial symmetry of the problem, all the unknown functions are
independent of the angle in the plane perpendicular to the axis of revolution. The
linear velocity lies in the plane of any cross-section containing the axis of revolution.
That is for the section, shown in Figure 1 it lies in the plane of the figure. The
angular velocity is always perpendicular to the linear velocity. This fact leads to
the divergence-free property of the spin field

div ω̃1 = 0, div ω̃2 = 0. (2.7)

Thus, the first member in equation (2.3) and (2.6) vanishes. Hence, we can rewrite
(2.3) and (2.6) as

(δ + ζ)∆ω̃2 + 2κ curl ṽ2 − 4κω̃2 = 0, (2.8)

(δ + ζ)∆ω̃1 + 2κ curl ṽ1 − 4κω̃1 = 0. (2.9)

We set the no-slip and no-spin boundary conditions on Γ1:

ṽ1|Γ1 = 0, ω̃1|Γ1 = 0. (2.10)

We suppose the continuity of both linear and angular velocity vectors on Γ2:

ṽ1

∣∣
Γ2

= ṽ2

∣∣
Γ2
, ω̃1

∣∣
Γ2

= ω̃2

∣∣
Γ2
. (2.11)

Finally, we set conditions at the boundary Γ3, which is responsible for the cell
interactions. A rather wide but finite variety of types of conditions at this bound-
ary is considered in the literature. Since the outer cell surface is hypothetical, the
conditions at this boundary should represent some physical processes. Happel’s
condition [14] assumes the outer cell boundary is a rigid envelope providing the
absence of energy exchange between the cell and the environment. Kuwabara [20],
on the opposite, used an analog of a free surface model expressed as the absence of
vorticity at the cell outer surface. Kvashnin [21] generalized the condition of the
velocity profile symmetry which takes place at the points where the cells come into
contact with each other and applied it to the whole boundary. Cunningham’s con-
dition [5] considers each cell as if placed in a uniform free stream. The conservation
of the flow uniformity condition, that is the free stream velocity is set at the outer
cell boundary. For a cell of an arbitrary shape this condition seems to be the most
appropriate:

ṽ2|Γ3
= Ũ. (2.12)

One more condition deals with the angular velocity or couple stress. Either
the former or the latter can be set to be equal to zero. There is no any physical
evidences in favor of any of them. So, we can use the no-spin condition

ω̃2

∣∣
Γ3

= 0. (2.13)

Passing to non-dimensional variables

x =
x̃

dΩ2

, v =
ṽ

|Ũ|
, ω =

ω̃dΩ2

|Ũ|
, p =

p̃dΩ2

µ|Ũ|
,



6 YU. O. KOROLEVA, D. YU. KHANUKAEVA EJDE-2021/74

where dΩ2
is the diameter of Ω2, we rewrite the original system in the non-dimensional

form
div v1 = 0,

ε
( 1

N2
− 1
)
∇p1 =

1

N2
∆v1 + 2 curlω1 −

εσ2

N2
v1,

L2∆ω1 +
1

2

N2

1−N2
curl v1 −

N2

1−N2
ω1 = 0.

(2.14)

div v2 = 0,( 1

N2
− 1
)
∇p2 =

1

N2
∆v2 + 2 curlω2,

L2∆ω2 +
1

2

N2

1−N2
curl v2 −

N2

1−N2
ω2 = 0.

(2.15)

Here we use three non-dimensional parameters N,L and σ:

N2 =
κ

µ+ κ
, L2 =

δ + ζ

4µd2
Ω2

, σ =
dΩ2√
k
. (2.16)

Coupling parameter N2 - demonstrates the fraction of rotational viscosity in the
sum of rotational and translational viscosities, L2 represents the relation between
the micro scale of the problem and its macro scale, the parameter σ represents the
ratio of the macro scale of the cell to the micro scale of the porous layer. Observe
that 0 < N < 1 due to its definition. The boundary conditions in non-dimensional
form are as follows:

v1|Γ1
= 0, ω1|Γ1

= 0, (2.17)

v1|Γ2 = v2|Γ2 , ω1|Γ2 = ω2|Γ2 , (2.18)

v2|Γ3 =
Ũ

|Ũ|
= U, (2.19)

ω2|Γ3
= 0. (2.20)

3. Weak solution

We suppose that the incoming velocity U was a smooth given vector-function.
All unknown vector-functions shall be considered in the weak sense, i.e. as general
functions from Sobolev space H1 satisfying a corresponding integral identity. The
notation f ∈ H1(Ω;R3) means that each component of the vector-function f =
(f1, f2, f3) belongs to the same space: fi ∈ H1(Ω). The notation H1(Ω,Γ;R3) is
reserved for all functions from the space H1(Ω;R3) having a zero trace on the set
Γ ⊂ ∂Ω. The space H−1 is the dual one to H1. Let us denote further the outer
normal vector to the boundary Γ by nΓ = (n1

Γ, n
2
Γ, n

3
Γ). In particular, n∂Ωi

means
the outer normal vector to the domain Ωi. Define the spaces

A1 = {u ∈ H1(Ω;R3) : div u = 0},
A2 = {u ∈ H1(Ω;R3) : div u = 0, u = U on Γ3}.

Now we can formulate the definition of the weak solution.

Definition 3.1. The triplet (v1,ω1, p1) ∈ (H1(Ω1,Γ1;R3), H1(Ω1,Γ1;R3), H−1(Ω1))
is the weak solution to equations (2.14) and boundary conditions (2.17)-(2.18) if



EJDE-2021/74 ESTIMATES OF CHARACTERISTICS OF A MICROPOLAR FLOW 7

the following integral equalities hold

ε
( 1

N2
− 1
)∫

Γ2

up1 · nΓ2
dS = − 1

N2

∫
Ω1

∇v1∇u dx+
1

N2

∫
Γ2

u
∂v1

∂nΓ2

dS

+ 2

∫
Ω1

curlω1u dx−
εσ2

N2

∫
Ω1

uv1 dx,

(3.1)

for any test-function u = (u1, u2, u3) ∈ H1(Ω1,Γ1;R3) ∩A1;

− L2

∫
Ω1

∇ϕ∇ω1 dx+ L2

∫
Γ2

ϕ
∂ω1

∂nΓ2

dS

+
1

2

N2

1−N2

∫
Ω1

ϕ curl v1 dx−
N2

1−N2

∫
Ω1

ϕω1 dx = 0

(3.2)

for any test-function ϕ ∈ H1(Ω1,Γ1;R3) ∩A1;∫
Ω1

v1∇q1 dx =

∫
Γ2

v2q1 · nΓ2
dS, for all q1 ∈ L2(Ω1); (3.3)∫

Ω1

ω1∇q2 dx =

∫
Γ2

ω2q2 · nΓ2
dS, for all q2 ∈ L2(Ω1). (3.4)

Definition 3.2. The triplet (v2,ω2, p2) ∈ (H1(Ω2;R3), H1(Ω2,Γ3;R3), H−1(Ω2))
is the weak solution to equations (2.15) and (2.7)-(2.20) if the following integral
identities hold:

−
( 1

N2
− 1
)∫

Γ2

gp2 · nΓ2
dS +

( 1

N2
− 1
)∫

Γ3

Up2 · nΓ3
dS

= − 1

N2

∫
Ω2

∇v2∇g dx− 1

N2

∫
Γ2

g
∂v2

∂nΓ2

dS +
1

N2

∫
Γ3

U
∂v2

∂nΓ3

dS

+ 2

∫
Ω2

curlω2g dx,

(3.5)

for any test-function g = (g1, g2, g3) ∈ H1(Ω2;R3) ∩A2;

− L2

∫
Ω2

∇ψ∇ω2 dx+ L2

∫
Γ2

ψ
∂ω2

∂nΓ2

dS

+
1

2

N2

1−N2

∫
Ω2

ψ curl v2 dx−
N2

1−N2

∫
Ω2

ψω2 dx = 0

(3.6)

for any test-function ψ ∈ H1(Ω2,Γ3;R3) ∩A1;∫
Ω2

v2∇q3 dx = −
∫

Γ2

v1q3 · nΓ2 dS +

∫
Γ3

Uq3 · nΓ3 dS, for all q3 ∈ L2(Ω2),

(3.7)∫
Ω2

ω2∇q4 dx =

∫
Γ2

ωq4 · nΓ2 dS, for all q4 ∈ L2(Ω2) (3.8)

4. A priori estimates for the weak solution

In this section we derive a priori estimates for the velocity functions v1,v2 and
ω1,ω2 solving the original problem in the weak sense. We recall the definition of
L2 and H1 norms in the Sobolev space H1(Ω) which will be used in our analysis:

‖v‖2L2(Ω) =

∫
Ω

v2 dx,



8 YU. O. KOROLEVA, D. YU. KHANUKAEVA EJDE-2021/74

‖v‖2H1(Ω) =

∫
Ω

v2 dx+

∫
Ω

|∇v|2 dx.

We will estimate the unknown velocities with respect to L2 and H1 norms.
The pressure p1, p2 are obviously defined up to the constant. First we state an
a priori estimate of Friedrichs-type valid for functions vanishing on a part of a
boundary. The technique of the proof in case when functions vanishes not on the
whole boundary, was demonstrated in [2, 3].

Lemma 4.1 (Friedrichs inequality). Let Ω be a bounded domain with boundary
∂Ω satisfying a local Lipschitz condition. Consider f ∈ H1(Ω) such that f = 0 on
Γ ⊂ ∂Ω, where meas(Γ) 6= 0. Then there exists a constant C > 0 which depends
on diameter of the domain Ω such that∫

Ω

f2 dx ≤ C
∫

Ω

|∇f |2 dx. (4.1)

Moreover, following to the proof of [23, Lemma 2.2.2] it is possible to show that
the sharp constant C ≤ d2

Ω even for the case when Γ 6= ∂Ω, see also [22, p. 22].
Boundary conditions (2.17) and (2.20) imply the validity of (4.1) for v1,ω1 in the
domain Ω1 as well as for ω2 in the domain Ω2:∫

Ω1

v2
1 dx ≤ d2

Ω1

∫
Ω1

|∇v1|2 dx, (4.2)∫
Ω1

ω2
1 dx ≤ d2

Ω1

∫
Ω1

|∇ω1|2 dx, (4.3)∫
Ω2

ω2
2 dx ≤ d2

Ω2

∫
Ω2

|∇ω2‖2 dx, (4.4)

where dΩi
is the diameter of domain Ωi, i = 1, 2. A similar inequality for v2 is

slightly different due to the non-zero trace v2|∂Ω2
on the boundary (see [11, 27]):∫

Ω2

v2
2 dx ≤ d2

Ω2

(∫
Ω2

|∇v2|2 dx+
1

|∂Ω2|

∫
∂Ω2

|v2|2 dS
)
. (4.5)

Here |∂Ω2| is measure of the boundary ∂Ω2. We use this fact to derive estimates
for the weak solution in the next theorem. Observe that dΩ2

= 1 since we have
passed to non-dimensional values.

Theorem 4.2. The solution to (3.1)–(3.4) and (3.5)–(3.8) satisfies the following
estimates:

‖v1‖2L2
< (2ε)−1σ−2 + ε−1σ−2 max{2N2, 4L2(1−N2)}(‖ω1‖2H1 + ‖ω2‖2H1),

(4.6)

‖v2‖2L2

<
(1 +

d2
Ω1

2

1−
d2

Ω1

2

)(1

2
+ max{2N2, 4L2(1−N2)}(‖ω1‖2H1 + ‖ω2‖2H1)

)
+ 1.

(4.7)

(
1−

d2
Ω1

2

)
‖v1‖2H1 + ‖∇v2‖2L2

<
1

2
+ max{2N2, 4L2(1−N2)}(‖ω1‖2H1 + ‖ω2‖2H1),

(4.8)



EJDE-2021/74 ESTIMATES OF CHARACTERISTICS OF A MICROPOLAR FLOW 9

‖ω1‖2H1 + ‖ω2‖2H1 ≤ max{ N2

4L2(1−N2)
, 1}

(
‖ curl v1‖2L2

+ ‖ curl v2‖2L2

)
(4.9)

‖ω1‖2H1 + ‖ω2‖2H1 ≤ max{ N2

2L2(1−N2)
, 2}

(
‖v1‖2H1 + ‖v2‖2H1

)
(4.10)

‖ω1‖2H1
≤ 1

8(1− d2
Ω1

)2

N4

L4(1−N2)2
‖ curl v1‖2L2

, (4.11)

‖ω2‖2H1
≤ max{1

3
,

1

3

N2

(1−N2)L2
}‖ curl v2‖2L2

, (4.12)

‖v1‖2H1 + ‖v2‖2H1 <
1

C
, (4.13)

where

C =


2− d2

Ω1
+ 16L2N2 − 16L2, if N2

L2(1−N2) < 2,

2− d2
Ω1
− 8N2, if 2 ≤ N2

L2(1−N2) < 4,

2− d2
Ω1
− 2N4

L2(1−N2) , if N2

L2(1−N2) ≥ 4.

Proof. We substitute u = v1 ≡ (v1
1 , v

2
1 , v

3
1) in (3.1), ϕ = ω1 in (3.2) and take into

the account q1 = p1 in (3.3). One obtains

ε
( 1

N2
− 1
) 3∑
i=1

∫
Γ2

vi1p1n
i
Γ2
dS +

εσ2

N2

∫
Ω1

|v1|2 dx+
1

N2

∫
Ω1

|∇v1|2 dx

=
1

N2

∫
Γ2

v1
∂v1

∂nΓ2

dS + 2

∫
Ω1

v1 curlω1 dx,

(4.14)

L2

∫
Ω1

|∇ω1|2 dx+
N2

1−N2

∫
Ω1

|ω1|2 dx

=
1

2

N2

1−N2

∫
Ω1

ω1 curl v1 dx+
L2

2

∫
Γ2

ω1
∂ω1

∂nΓ2

dS.

(4.15)

Observing that

v1
∂v1

∂nΓ2

=
1

2

∂(v1)2

∂nΓ2

,

we rewrite (4.14) and (4.15) equivalently as

ε
( 1

N2
− 1
) 3∑
i=1

∫
Γ2

vi1p1n
i
Γ2
dS +

εσ2

N2

∫
Ω1

|v1|2 dx+
1

N2

∫
Ω1

|∇v1|2 dx

=
1

2N2
|v1|2Γ2

+ 2

∫
Ω1

v1 curlω1 dx,

(4.16)

L2

∫
Ω1

|∇ω1|2 dx+
N2

1−N2

∫
Ω1

|ω1|2 dx

=
1

2

N2

1−N2

∫
Ω1

ω1 curl v1 dx+
L2

2
|ω1|2Γ2

.

(4.17)

Multiplying (4.16) by 1/2 ,and (4.17) by −2(1 −N2)/N2, after the summation
one obtains



10 YU. O. KOROLEVA, D. YU. KHANUKAEVA EJDE-2021/74

ε

2

( 1

N2
− 1
) 3∑
i=1

∫
Γ2

vi1p1n
i
Γ2
dS +

εσ2

2N2

∫
Ω1

|v1|2 dx+
1

2N2

∫
Ω1

|∇v1|2 dx

+
L2

N2
(1−N2)|ω1|2Γ2

=
1

2N2
|v1|2Γ2

+

∫
Ω1

div[v1 × ω1] dx

+

∫
Ω1

|ω1|2 dx+
2L2

N2
(1−N2)

∫
Ω1

|∇ω1|2 dx.

(4.18)

Here we used the formula

div[v1 × ω1] = v1 curlω1 − ω1 curl v1.

Substituting g = v2 in (3.5), ψ = ω2 in (3.6) and using (3.7) with q3 = p2 one gets
the following integral identity after integration by parts,( 1

N2
− 1
) 3∑
i=1

∫
∂Ω2

vi2p2n
i
Γ2
dS +

1

N2

∫
Ω2

|∇v2|2 dx

=
1

N2

∫
∂Ω2

v2
∂v2

∂nΩ2

dS + 2

∫
Ω2

v2 curlω2 dx,

(4.19)

L2

∫
Ω2

|∇ω2|2 dx+
N2

1−N2

∫
Ω2

|ω2|2 dx

=
1

2

N2

1−N2

∫
Ω2

ω2 curl v2 dx+ L2

∫
∂Ω2

ω2
∂ω2

∂nΩ2

dS.

(4.20)

Analogously, observing that

v2
∂v2

∂nΩ2

=
1

2

∂(v2)2

∂nΩ2

,

we can simplify (4.19) to the form

( 1

N2
− 1
) 3∑
i=1

∫
∂Ω2

vi2p2n
i
Γ2
dS +

1

2N2
|v2|2Γ2

+
1

N2

∫
Ω2

|∇v2|2 dx

=
1

2N2
|v2|2Γ3

+ 2

∫
Ω2

v2 curlω2 dx.

(4.21)

From the boundary conditions (2.18)-(2.19) we have

−
( 1

N2
− 1
) 3∑
i=1

∫
Γ2

vi2p2n
i
Γ2
dS +

( 1

N2
− 1
)∫

Γ3

3∑
i=1

Uip2n
i
Γ3
dS

+
1

2N2
|v2|2Γ2

+
1

N2

∫
Ω2

|∇v2|2 dx

=
1

2N2
+ 2

∫
Ω2

v2 curlω2 dx.

(4.22)



EJDE-2021/74 ESTIMATES OF CHARACTERISTICS OF A MICROPOLAR FLOW 11

In a similar way, (4.20) is equivalent to

L2

∫
Ω2

|∇ω2|2 dx+
N2

1−N2

∫
Ω2

|ω2|2 dx+
L2

2
|ω2|2Γ2

=
1

2

N2

1−N2

∫
Ω2

ω2 curl v2 dx.

(4.23)

Multiplying (4.22) by 1/2, and (4.23) by −2(1−N2)
N2 , after its summation one gets

− 1

2

( 1

N2
− 1
) 3∑
i=1

∫
Γ2

vi2p2n
i
Γ2
dS +

1

2

( 1

N2
− 1
)∫

Γ3

3∑
i=1

Uip2n
i
Γ3
dS

+
1

4N2
|v2|2Γ2

+
1

2N2

∫
Ω2

|∇v2|2 dx

=
1

4N2
+

∫
Ω2

div[v2 × ω2] dx+
2L2

N2
(1−N2)

∫
Ω2

|∇ω2|2 dx

+

∫
Ω2

|ω2|2 dx+
L2

N2
(1−N2)|ω2|2Γ2

.

(4.24)

Integrating by parts
∫

Ωi
div[vi×ωi] dx for i = 1, 2. Applying boundary conditions,

we obtain ∫
Ω1

div[v1 × ω1] dx =

∫
Γ2

[v1 × ω1] · nΓ2
dS,∫

Ω2

div[v2 × ω2] dx = −
∫

Γ2

[v2 × ω2] · nΓ2
dS.

Taking this into account, we sum (4.18), (4.24) and use the continuity of linear and
angular velocities on the boundary Γ2:(

− 1

2
+
ε

2

)( 1

N2
− 1
) 3∑
i=1

∫
Γ2

vi2p2n
i
Γ2
dS +

1

2

( 1

N2
− 1
)∫

Γ3

3∑
i=1

Uip2n
i
Γ3
dS

+
1

2N2

∫
Ω2

|∇v2|2 dx+
εσ2

2N2

∫
Ω1

|v1|2 dx+
1

2N2

∫
Ω1

|∇v1|2 dx

=
1

4N2
|v2|2Γ2

+

∫
Γ2

[v1 × ω1] · nΓ2
dS +

∫
Ω1

|ω1|2 dx

+
2L2

N2
(1−N2)

∫
Ω2

|∇ω2|2 dx+
1

4N2
−
∫

Γ2

[v2 × ω2] · nΓ2
dS

+
2L2

N2
(1−N2)

∫
Ω2

|∇ω2|2 dx+

∫
Ω2

|ω2|2 dx.

This equality implies

εσ2‖v1‖2L2
+ ‖∇v1‖2L2

+ ‖∇v2‖2L2

<
1

2
|v2|2Γ2

+
1

2
+ max{2N2, 4L2(1−N2)}(‖ω1‖2H1 + ‖ω2‖2H1).

(4.25)

The trace |v2|2Γ2
on the boundary can be estimated as follows

|v2|2Γ2
= |v1|2Γ2

≤ d2
Ω1
‖∇v1‖2L2

;



12 YU. O. KOROLEVA, D. YU. KHANUKAEVA EJDE-2021/74

therefore,

εσ2‖v1‖2L2
+ (1−

d2
Ω1

2
)‖∇v1‖2L2

+ ‖∇v2‖2L2

<
1

2
+ max{2N2, 4L2(1−N2)}(‖ω1‖2H1 + ‖ω2‖2H1),

(4.26)

min
{
εσ2,

(
1−

d2
Ω1

2

)}
‖v1‖2H1 + ‖∇v2‖2L2

<
1

2
+ max{2N2, 4L2(1−N2)}(‖ω1‖2H1 + ‖ω2‖2H1).

(4.27)

In practical applications the parameter εσ2 > 1, while d2
Ω1
< 1, therefore one arrives

at (
1−

d2
Ω1

2

)
‖v1‖2H1 + ‖∇v2‖2L2

<
1

2
+ max{2N2, 4L2(1−N2)}(‖ω1‖2H1 + ‖ω2‖2H1).

(4.28)

An estimate for v2 in L2 follows from the Friedrichs inequality (4.5) and (4.26).
Namely,

‖v2‖2L2
≤ ‖∇v2‖2L2

+ |v1|2Γ2
+ 1

≤
(1 +

d2
Ω1

2

1−
d2

Ω1

2

)(1

2
+ max{2N2, 4L2(1−N2)}(‖ω1‖2H1 + ‖ω2‖2H1)

)
+ 1.

Hence, inequalities (4.6), (4.7) and (4.8) are proved.
Let us derive estimates for norms of angular velocities. An analysis of identity

(4.17) leads to the restriction

L2

∫
Ω1

|∇ω1|2 dx ≤
1

2

N2

1−N2

∫
Ω1

ω1 curl v1 dx+
L2

2
|ω1|2Γ2

.

Applying Hölder’s inequality, Friedrichs inequality (4.3) and the estimate

|ω1|2Γ2
≤ d2

Ω1

∫
Ω1

|∇ω1|2 dx,

one deduces that

L2

∫
Ω1

|∇ω1|2 dx

≤ 1

4α2
1

N2

1−N2

∫
Ω1

|ω1|2 dx+
α2

1

4

N2

1−N2

∫
Ω1

(curl v1)2 dx+
L2d2

Ω1

2

∫
Ω1

|∇ω1|2 dx

≤ d2
Ω1

( 1

4α2
1

N2

1−N2
+
L2

2

)∫
Ω1

|∇ω1|2 dx+
α2

1

4

N2

1−N2

∫
Ω1

(curl v1)2 dx.

Finally, we obtain∫
Ω1

|∇ω1|2 dx ≤
(
L2 − d2

Ω1

( 1

4α2
1

N2

1−N2
+
L2

2

))−1α2
1

4

N2

1−N2

∫
Ω1

(curl v1)2 dx,

where an arbitrary constant α1 is chosen such that the coefficient in the right-hand
side of the derived inequality was positive. To prove the estimate (4.11) for H1



EJDE-2021/74 ESTIMATES OF CHARACTERISTICS OF A MICROPOLAR FLOW 13

norm one needs to apply the Friedrichs inequality (4.3) to express the bound for
L2 norm of function ω1 via L2 norm of its gradient,

‖ω1‖2H1 ≤ max{1, dΩ1
}
(
L2−d2

Ω1

( 1

4α2
1

N2

1−N2
+
L2

2

))−1α2
1

4

N2

1−N2

∫
Ω1

(curl v1)2 dx.

We shall choose the constant α1 such that 1
4α2

1

N2

1−N2 = L2

2 . Observing that dΩ1
< 1

from the geometry of our problem, we simplify the obtained inequality to the form

‖ω1‖2H1 ≤
1

8(1− d2
Ω1

)2

N4

L4(1−N2)2

∫
Ω1

(curl v1)2 dx

and prove (4.11).
Observe that left-hand side of (4.23) is non-negative, therefore its right-hand side

is also nonnegative and each term in (4.23) can be estimated by N2

1−N2

∫
Ω2
ω2 curl v2 dx.

In particular,

N2

1−N2

∫
Ω2

|ω2|2 dx ≤
1

2

N2

1−N2

∫
Ω2

ω2 curl v2 dx

if and only if∫
Ω2

|ω2|2 dx ≤
1

2

∫
Ω2

ω2 curl v2 dx ≤
1

4

∫
Ω2

|ω2|2 dx+
1

4

∫
Ω2

| curl v2|2 dx.

Thus, ∫
Ω2

|ω2|2 dx ≤
1

3

∫
Ω2

| curl v2|2 dx. (4.29)

Similarly, applying (4.29) and Hölder’s inequality we can estimate

L2

∫
Ω2

|∇ω2|2 dx ≤
1

2

N2

1−N2

∫
Ω2

ω2 curl v2 dx

≤ 1

4

N2

1−N2

∫
Ω2

|ω2|2 dx+
1

4

N2

1−N2

∫
Ω2

| curl v2|2 dx

≤ 1

3

N2

1−N2

∫
Ω2

| curl v2|2 dx,

one obtains ∫
Ω2

|∇ω2|2 dx ≤
1

3

N2

(1−N2)L2

∫
Ω2

| curl v2|2 dx. (4.30)

Applying again the Friedrichs inequality (4.4), we derive the estimate∫
Ω2

|ω2|2 dx ≤
1

3

N2

(1−N2)L2

∫
Ω2

| curl v2|2 dx, (4.31)

thus we can improve the constant in the bound (4.29):∫
Ω2

|ω2|2 dx ≤ C
∫

Ω2

| curl v2|2 dx, (4.32)

where C = max{ 1
3 ,

1
3

N2

(1−N2)L2 }. Estimates (4.30) and (4.32) directly imply

‖ω2‖2H1 ≤ max{1

3
,

1

3

N2

(1−N2)L2
}‖ curl v2‖2L2

.



14 YU. O. KOROLEVA, D. YU. KHANUKAEVA EJDE-2021/74

Summing integral identities (4.23), (4.20), using the boundary conditions (2.18),
we obtain

L2
(∫

Ω1

|∇ω1|2 dx+

∫
Ω2

|∇ω2|2 dx
)

+
N2

1−N2

(∫
Ω1

|ω1|2 dx+

∫
Ω2

|ω2|2 dx
)

=
N2

2(1−N2)

(∫
Ω1

ω1 curl v1 dx+

∫
Ω2

ω2 curl v2 dx
)
.

This equality and the inequality
∫
fg dx ≤ 1

2α2

∫
f2 dx + α2

2

∫
g2 dx with nonzero

free coefficient α imply

2L2(1−N2)

N2

(∫
Ω1

|∇ω1|2 dx+

∫
Ω2

|∇ω2|2 dx
)

+
(

1− 1

2α2

)(∫
Ω1

|ω1|2 dx+

∫
Ω2

|ω2|2 dx
)

≤ α2

2

(∫
Ω1

(curl v1)2 dx+

∫
Ω2

(curl v2)2 dx
)
,

(4.33)

what can be reduced to the estimate

‖ω1‖2H1 + ‖ω2‖2H1

≤ α2

2
max{ N2

2L2(1−N2)
,

2α2

2α2 − 1
}
(
‖ curl v1‖2L2

+ ‖ curl v2‖2L2

)
.

(4.34)

Assuming α = 1, (4.34) gives

‖ω1‖2H1 + ‖ω2‖2H1 ≤ max{ N2

4L2(1−N2)
, 1}

(
‖ curl v1‖2L2

+ ‖ curl v2‖2L2

)
. (4.35)

Applying the definition of curl operator, one obtains the inequality∫
Ωi

(curl vi)
2 dx ≤ 2

∫
Ωi

|∇vi|2 dx ≤ 2‖vi‖2H1 , i = 1, 2.

Thus, (4.35) becomes

‖ω1‖2H1 + ‖ω2‖2H1 ≤ max{ N2

2L2(1−N2)
, 2}

(
‖v1‖2H1 + ‖v2‖2H1

)
, (4.36)

and we have obtained (4.10). Finally, combining (4.8) and (4.36), one obtains
(4.13). The boundedness of norms for angular velocity fields is the consequence of
inequalities (4.10) and (4.13). �

The estimates obtained here show that the weakly defined velocity fields v1, v2,
ω1, ω2 are bounded.

Acknowledgments. This research was supported by the Russian Science Foun-
dation, grant No. 20-19-00670.

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16 YU. O. KOROLEVA, D. YU. KHANUKAEVA EJDE-2021/74

Yulia O. Koroleva

Department of High Mathematics, Russian Gubkin State University of Oil and Gas,

Moscow & HSE University, Moscow, Russia
Email address: koroleva.y@gubkin.ru

Daria Yu. Khanukaeva
Department of High Mathematics, Russian Gubkin State University of Oil and Gas,

Moscow, Russia

Email address: khanuk@yandex.ru


	1. Introduction
	2. Statement of the problem
	3. Weak solution
	4. A priori estimates for the weak solution
	Acknowledgments

	References