Georgian Mathematical Journal 1(1994), No. 3, 251-266

TWO-DIMENSIONAL PROBLEMS OF STATIONARY FLOW OF A NONCOMPRESSIBLE VISCOUS FLUID IN

THE CASE OF OZEEN’S LINEARIZATION

T. BUCHUKURI AND R. CHICHINADZE

Abstract. Two-dimensional boundary value problems of flow of a viscous micropolar fluid are investigated in the case of linearization by Ozeen’s method.

1. Basic Equations. A system of equations of motion of a noncompressible micropolar fluid was obtained in 1964 by Condiff and Dahler [1] and, independently, in 1966 by Eringen. It is a generalization of the classical system of Navier–Stokes for micropolar fluids. In real life we observe such properties in fluids containing polymer particles as admixtures. When fluids of this kind flow along the body, surface friction is 30 to 35% less than in the case of flow of fluids without polymer admixtures [2]. It is impossible to predict such effects by the classical theory of Navier–Stokes, but a fairly good explanation can be found within the framework of the theory of micropolar fluids.

We consider a two-dimensional model of stationary flow of a micropolar fluid. A system of the basic equations then has the form

\begin{matrix}div ṽ = 0, \\ (μ + α)Δṽ_{1} + 2α\frac{∂ω̃}{∂x_{2}} − \frac{∂p}{∂x_{1}} + ρF_{1} = ρ ∑_{k=1}^{2} ṽ_{k} \frac{∂ṽ_{1}}{∂x_{k}} , \\ (μ + α)Δṽ_{2} − 2α\frac{∂ω̃}{∂x_{1}} − \frac{∂p}{∂x_{2}} + ρF_{2} = ρ ∑_{k=1}^{2} ṽ_{k} \frac{∂ṽ_{2}}{∂x_{k}} , \\ γΔω̃ − 4αω̃ + 2α(\frac{∂ṽ_{2}}{∂x_{1}} − \frac{∂ṽ_{1}}{∂x_{2}} ) + ρF_{3} = I ∑_{k=1}^{2} ṽ_{k} \frac{∂ω̃}{∂x_{k}} , \end{matrix}  (1)

1991 Mathematics Subject Classification. 76A05, 35Q20.

where ṽ = (ṽ_{1}, ṽ_{2}) is the velocity vector, F̃ = (F_{1}, F_{2}) is the mass force, p is the pressure, ρ is the density; μ, α, γ, I are positive constants. In the two-dimensional case the microrotation and mass moment vectors have one component each and are denoted by ω̃ and F_{3}, respectively.

Since the system (1) is nonlinear, we come across certain difficulties during its investigation. On the other hand, to solve many problems of applied nature it is sufficient to consider a linearized variant of the system (1). The equations of Navier-Stokes can be linearized by two well-known methods: that of Stokes and that of Ozeen. When using Stokes’ method of linearization, nonlinear terms are totally discarded. This method yields satisfactory results for small ṽ and ω̃ (note that in this case nonlinear terms are small values of higher order). However, if the fluid flow velocity ṽ is not a small value, this model leads to an essential error. In particular, the effects predicted by this method when a fluid flows along a solid body do not agree with experimental data.

A lesser error is obtained in the case of linearization by Ozeen’s method consisting in the following: it is assumed that fluid flow differs but little from flow along the x_{1}-axis with the constant velocity v_{0}. Then we set

ṽ_{k} = v_{0}δ_{k1} + v_{k}, k = 1, 2, ω̃ = ω,

where v_{k}, k = 1, 2, ω are small values; δ_{kj} is the Kronecker symbol.

On substituting these values in (1), we obtain an Ozeen-linearized system of equations of stationary flow of a micropolar fluid in the two-dimensional case:

\begin{matrix}div v = 0, \\ (μ + α)Δv_{1} + 2α\frac{∂ω}{∂x_{2}} − \frac{∂p}{∂x_{1}} + ρF_{1} = η_{1} \frac{∂v_{1}}{∂x_{1}} , \\ (μ + α)Δv_{2} − 2α\frac{∂ω}{∂x_{1}} − \frac{∂p}{∂x_{2}} + ρF_{2} = η_{1} \frac{∂v_{2}}{∂x_{1}} , \\ γΔω − 4αω + 2α(\frac{∂v_{2}}{∂x_{1}} + \frac{∂v_{1}}{∂x_{2}} ) + ρF_{3} = η_{2} \frac{∂ω}{∂x_{1}} . \end{matrix}  (2)

Here η_{1} = ρv_{0}, η_{2} = Iv_{0}.

The system (2) can be rewritten in the matrix form if we introduce the notation

L(∂_{x}) = ∥ \begin{matrix}(μ + α)Δ & 0 & 2α\frac{∂}{∂x_{2}} \\ 0 & (μ + α)Δ & − 2α \frac{∂}{∂x_{1}} \\ −2α\frac{∂}{∂x_{2}} & 2α\frac{∂}{∂x_{1}} & γΔ − 4α \end{matrix} ∥ ,

G(∂_{x}) = ∥ \frac{∂}{∂x_{1}} ; \frac{∂}{∂x_{2}} ; 0 ∥ , η = ∥ \begin{matrix}η_{1} & 0 & 0 \\ 0 & η_{1} & 0 \\ 0 & 0 & η_{2} \end{matrix} ∥ ,

(v_{1}, v_{2}, ω) ≡ u = (u_{1}, u_{2}, u_{3}) = ∥ \begin{matrix}u_{1} \\ u_{2} \\ u_{3} \end{matrix} ∥ , F = (F_{1}, F_{2}, F_{3}) = ∥ \begin{matrix}F_{1} \\ F_{2} \\ F_{3} \end{matrix} ∥ .

Now the system (2) takes the form

\begin{matrix}\frac{∂u_{1}}{∂x_{1}} + \frac{∂u_{2}}{∂x_{2}} = 0, \\ L(∂_{x})u − G(∂_{x})p + ρF = η \frac{∂u}{∂x_{1}} . \end{matrix}  (3)

Alongside with the system (3), we will also consider its conjugate

\begin{matrix}\frac{∂u_{1}}{∂x_{1}} + \frac{∂u_{2}}{∂x_{2}} = 0, \\ L(∂_{x})u − G(∂_{x})p + ρF = −η \frac{∂u}{∂x_{1}} , \end{matrix}  (4)

which is obtained from (3) if we replace η_{1} and η_{2} by −η_{1} and −η_{2}, respectively.

Let us formulate the boundary value problems for (2). Denote by D^{+} a finite domain in the Euclidean two-dimensional space ℝ^{2} bounded with a piecewise-smooth contour S. Let D^{−} ≡ ℝ^{2}\(D^{+} ∪ S). Denote by n(y) = (n_{1}(y), n_{2}(y)) the unit normal at a point y ∈ S, external with respect to the domain D^{+}.

The pair (u, p), where u = (v_{1}, v_{2}, ω), will be called regular in D^{+} if u ∈ ℂ^{2}(D^{+}) ∩ ℂ^{1}(D̄^{+}), p ∈ ℂ^{1}(D^{+}) ∩ ℂ(D̄^{+}).

The pair (u, p) will be called regular in D^{−} if u ∈ ℂ^{2}(D^{−}) ∩ ℂ^{1}(D̄^{−}), p ∈ ℂ^{1}(D^{−}) ∩ ℂ(D̄^{−}) and the conditions

u(x) = O(|x|^{− \frac{1}{2}} ), p(x) = o(1)  (5)

are fulfilled in the neighbourhood of |x| = ∞.

The boundary value problems for the system (3) are formulate as follows:

Problem (I)^{±}. In the domain D^{±} find a regular solution of the system (3) by the boundary condition

lim_{D^{±}∋x→y∈S} u(x) = f(y).  (6)

Problem (II)^{±}. In the domain D^{±} find a regular solution of the system (3) by the boundary condition

lim_{D^{±}∋x→y∈S} [P (∂_{x}, n(y))u(x)− \frac{1}{2}n_{1}(y)ηu(x)−N(y)p(x)]=f(y), (7)

where f = (f_{1}, f_{2}, f_{3}) is a given vector on S,

P (∂_{x}, n(y)) = ∥P_{ij}(∂_{x}, n(y))∥_{3×3},  (8)

P_{ij}(∂_{x}, n(y))=(μ+α)n_{j}(y)\frac{∂}{∂x_{i}} +(μ+α)δ_{ij} ∑_{k=1}^{2} n_{k}(y)\frac{∂}{∂x_{k}} , i, j =1, 2;

P_{i3}(∂_{x}, n(y)) = 2α(i − 1)n_{1}(y) + 2α(i − 2)n_{2}(y), i = 1, 2; P_{3j}(∂_{x}, n(y)) = 0, j = 1, 2;

P_{33}(∂_{x}, n(y)) = γ ∑_{k=1}^{2} n_{k}(y)\frac{∂}{∂x_{k}};

N_{i}(y)=(N_{1}(y), N_{2}(y), N_{3}(y)), N_{i}(y)=n_{i}(y), i=1, 2; N_{3}(y)=0.

The boundary value problems (Ĩ)^{±} and (ĨI)^{±} for the conjugate system (4) are formulated similarly. In that case the boundary condition of Problem (I)^{±} coincides with (6), while the boundary condition of Problem (II)^{±} is obtained from (7) if we replace η_{1} and η_{2} by −η_{1} and −η_{2}, respectively.

Remark. Our previous thorough treatment of the boundary value problems of stationary flow of a micropolar fluid under Ozeen’s linearization in the three-dimensional case is given in [3]. Since investigations of the twodimensional problems are mostly the same, we will dwell on only the part differing from the three-dimensional case.

2. On Fundamental solutions. The fundamental solutions of the system (1) are found from the relations

\begin{matrix}\frac{∂ v^{(m)}_{1}}{∂x_{1}} + \frac{∂ v^{(m)}_{2}}{∂x_{2}} = 0, \\ (μ + α)Δ v^{(m)}_{1} −η_{1} \frac{∂ v^{(m)}_{1}}{∂x_{1}} + 2α\frac{∂ ω^{(m)}}{∂x_{2}} − \frac{∂ p^{(m)}}{∂x_{1}}+ a^{(m)}_{1} δ(x) = 0, \\ (μ + α)Δ v^{(m)}_{2} −η_{1} \frac{∂ v^{(m)}_{2}}{∂x_{1}} − 2α\frac{∂ ω^{(m)}}{∂x_{1}} − \frac{∂ p^{(m)}}{∂x_{2}}+ a^{(m)}_{2} δ(x) = 0, \\ γΔ ω^{(m)} −4α ω^{(m)} −η_{2} \frac{∂ ω^{(m)}}{∂x_{1}} − 2α\frac{∂ v^{(m)}_{1}}{∂x_{2}} + 2α\frac{∂ v^{(m)}_{2}}{∂x_{1}} + b^{(m)} δ(x) = 0, \end{matrix}  (9)

where δ(x) is the Dirac distribution,

a^{(m)}_{k}= 2δ_{km}, b^{(m)}= 0, k, m = 1, 2; a^{(3)}_{k}= 0, k = 1, 2, b^{(3)}= 2.

Assume that the fundamental solutions ( v^{(m)}_{1}, v^{(m)}_{2}, ω^{(m)}, p^{(m)}), m = 1, 2, 3, satisfy the conditions

lim_{|x|→∞} ( v^{(m)}_{1} (x), v^{(m)}_{2} (x), ω^{(m)} (x), p^{(m)} (x)) = 0.  (10)

Then v^{(m)}_{1}, v^{(m)}_{2}, ω^{(m)} and p^{(m)} are gradually increasing distributions in ℝ^{2}, and, on subjecting the system (9) to the Fourier transformation, we obtain

\begin{matrix}ξ_{1} ˆ v^{(m)}_{1} + ξ_{2} ˆ v^{(m)}_{2} = 0, \\ iξ_{1} ˆ p^{(m)} − (μ + α)|ξ|^{2} ˆ v^{(m)}_{1} + iη_{1}ξ_{1} ˆ v^{(m)}_{1} − 2iαξ_{2} ˆ ω^{(m)}+ a^{(m)}_{1}= 0, \\ iξ_{2} ˆ p^{(m)} − (μ + α)|ξ|^{2} ˆ v^{(m)}_{2} + iη_{1}ξ_{1} ˆ v^{(m)}_{2} + 2iαξ_{1} ˆ ω^{(m)}+ a^{(m)}_{2}= 0, \\ − γ|ξ|^{2} ˆ ω^{(m)} − 4α ˆω^{(m)} + iη_{2}ξ_{1} ˆ ω^{(m)} + 2iαξ_{2} ˆ v^{(m)}_{1} − 2iαξ_{1} ˆ v^{(m)}_{2}+ b^{(m)}= 0. \end{matrix} (11)

Here |ξ| ≡ (ξ_{1}^{2} + ξ_{2}^{2})^{1/2}, f̂ denotes the Fourier transform of the gradually increasing distribution f:

f̂(ξ) = ∫_{ℝ^{2}} e^{iξ·x}f(x) dx, ξ = (ξ_{1}, ξ_{2}) ∈ ℝ^{2}, |ξ| ≡ (ξ_{1}^{2} + ξ_{2}^{2})^{1/2}.

Solving the system (11) with respect to ˆ v^{(m)}_{1}, ˆ v^{(m)}_{2} , ˆ ω^{(m)}, ˆ p^{(m)}, we have

ˆp^{(m)} = \frac{i}{|ξ|^{2}} ξ· a^{(m)},
ˆv^{(m)}_{1} = \frac{γ|ξ|^{2} − iη_{2}ξ_{1} + 4α}{Φ(ξ)} a^{(m)}_{1} −\frac{2iαξ_{2}}{Φ(ξ)} b^{(m)} −

− \frac{γ|ξ|^{2} − iη_{2}ξ_{1} + 4α}{|ξ|^{2}Φ(ξ)}(ξ· a^{(m)} )ξ_{1},

ˆv^{(m)}_{2} = \frac{γ|ξ|^{2} − iη_{2}ξ_{1} + 4α}{Φ(ξ)} a^{(m)}_{2} +\frac{2iαξ_{1}}{Φ(ξ)} b^{(m)} −

− \frac{γ|ξ|^{2} − iη_{2}ξ_{1} + 4α}{|ξ|^{2}Φ(ξ)}(ξ· a^{(m)} )ξ_{2},

ˆω^{(m)} = \frac{(μ + α)|ξ|^{2} − iη_{1}ξ_{1}}{Φ(ξ)} b^{(m)} +\frac{2iα}{Φ(ξ)}( a^{(m)}_{1} ξ_{2}− a^{(m)}_{2} ξ_{1}), m = 1, 2, 3,

(12) where

Φ(ξ) = γ(μ + α)|ξ|^{4} − i[η_{1}γ + η_{2}(μ + α)]ξ_{1}|ξ|^{2} − η_{1}η_{2}ξ_{1}^{2} + 4μα|ξ|^{2} − 4iαη_{1}ξ_{1}.

Denote by v^{(m)}_{1}, v^{(m)}_{2}, ω^{(m)}, p^{(m)}, m = 1, 2, 3, the inverse Fourier transforms of

ˆv^{(m)}_{1}, ˆ v^{(m)}_{2}, ˆ ω^{(m)}, ˆ p^{(m)}, respectively, and by Γ and Q the matrices

Γ = ∥Γ_{ik}∥_{3×3}, Γ_{ik} = v^{(k)}_{i}, i = 1, 2, k = 1, 2, 3; Γ_{3k} = ω^{(k)}; (13)

Q = ∥Q_{k}∥_{3×1}, Q_{k} = p^{(k)}, k = 1, 2; Q_{3} = 0.

Γ and Q will be called the matrices of fundamental solutions of the system (1). Their Fourier transform is

Q̂(ξ) = (\frac{2iξ_{1}}{|ξ|^{2}} , \frac{2iξ_{2}}{|ξ|^{2}} , 0),  (14)

Γ̂(ξ)=∥ \begin{matrix}2A(ξ)(1−\frac{ξ^{2}_{1}}{|ξ|^{2}} ) & −2A(ξ) \frac{ξ_{1}ξ_{2}}{|ξ|^{2}} & −2B(ξ)ξ_{2} \\ −2A(ξ) \frac{ξ_{1}ξ_{2}}{|ξ|^{2}} & 2A(ξ)(1−\frac{ξ^{2}_{2}}{|ξ|^{2}} ) & 2B(ξ)ξ_{1} \\ 2B(ξ)ξ_{2} & −2B(ξ)ξ_{1} & \frac{2}{Φ(ξ)}((μ+α)|ξ|^{2}−iη_{1}ξ_{1}) \end{matrix} ∥ , (15)

where

A(ξ) = \frac{γ|ξ|^{2} − iη_{2}ξ_{1} + 4α}{Φ(ξ)}, B(ξ) = \frac{2iα}{Φ(ξ)}.

Calculating the inverse Fourier transform of (14), we get

Q(x) = (\frac{1}{π} \frac{x_{1}}{|x|^{2}} , \frac{1}{π} \frac{x_{2}}{|x|^{2}} , 0).  (16)

Though the inverse transform of Γ̂ is not expressed in terms of elementary functions, we can nevertheless obtain asymptotic representations of the fundamental matrix Γ in the neighbourhood of the points |x| = 0 and |x| = ∞, which is convenient in investigating the boundary value problems.

Represent Γ̂ in the form

Γ̂(ξ) = Γ̂^{(0)}(ξ) + Γ̂^{(1)}(ξ),

where

Γ̂^{(0)}(ξ) = ∥ \begin{matrix}\frac{2ξ_{1}^{2}}{(μ+α)|ξ|^{4}} & −\frac{2ξ_{1}ξ_{2}}{(μ+α)|ξ|^{4}} & 0 \\ −\frac{2ξ_{1}ξ_{2}}{(μ+α)|ξ|^{4}} & \frac{2ξ_{2}^{2}}{(μ+α)|ξ|^{4}} & 0 \\ 0 & 0 & \frac{2}{γ|ξ|^{2}} \end{matrix} ∥  (17)

and the elements of Γ̂^{(1)} are written as

Γ̂_{ij}^{(1)}(ξ) = ∑_{3≤2k−|α|≤4} a_{k,α}^{ij} \frac{|ξ|^{α}}{|ξ|^{2k}} + φ_{ij}(ξ), i, j = 1, 2, 3.  (18)

Here a_{k,α}^{ij} are some constants, φ_{ij} are functions admitting, in the neighbourhood of |x| = ∞, the estimate φ_{ij}(ξ) = O(|ξ|^{−5}).

Taking (18) into account, it can be proved [3] that, Γ_{ij}^{(1)} satisfies, in the neighbourhood of |x| = 0, the conditions

\begin{matrix}Γ_{ij}^{(1)}(x) = O(1), i, j = 1, 2, 3; \\ ∂^{α}Γ_{ij}^{(1)}(x) = O(|x|^{1−|α|} ln |x|), i, j = 1, 2, 3. \end{matrix}  (19)

Performing the inverse Fourier transform of (17), we obtain

Γ^{(0)}(x) = ∥\begin{matrix}−\frac{1}{4π(μ+α)}( ln |x| + \frac{x_{1}^{2}}{|x|^{2}} ) − \frac{x_{1}x_{2}}{4π(μ+α)|x|^{2}} & 0 \\ −\frac{x_{1}x_{2}}{4π(μ+α)|x|^{2}} − \frac{1}{4π(μ+α)}( ln |x| + \frac{x_{2}^{2}}{|x|^{2}} ) & 0 \\ 0  0 & −\frac{1}{πγ} ln |x| \end{matrix}∥ . (20)

Thus we have

Theorem 1. The fundamental matrix Γ is represented as

Γ = Γ^{(0)} + Γ^{(1)},  (21)

where Γ^{(0)} is defined from (20), Γ^{(1)} ∈ ℂ^{∞}(ℝ\{0}) and satisfies, in the neighbourhood of the point |x| = 0, the conditions (19).

To obtain the necessary representation of Γ in the neighbourhood of |x| = ∞ we represent Γ̂ as the sum

Γ̂(ξ) = Γ̂^{(∞)}(ξ) + Γ̂^{(2)}(ξ),  (22)

where

Γ̂^{(∞)}(ξ) = \frac{2}{(μ|ξ|^{2} − iη_{1}ξ_{1})} ∥ \begin{matrix}\frac{ξ_{2}^{2}}{|ξ|^{2}} & −\frac{ξ_{1}ξ_{2}}{|ξ|^{2}} & −\frac{iξ_{2}}{2} \\ −\frac{ξ_{1}ξ_{2}}{|ξ|^{2}} & \frac{ξ_{1}^{2}}{|ξ|^{2}} & \frac{iξ_{1}}{2} \\ \frac{iξ_{2}}{2} & −\frac{iξ_{1}}{2} & \frac{iη_{1}ξ_{1}}{4μ} \end{matrix} ∥ .  (23)

Then, taking (15) into account, it can be proved that the components of Γ̂^{(2)} satisfy the estimates

\begin{matrix}|∂^{α}Γ̂_{ij}^{(2)}(ξ)| ≤ \frac{c}{(|ξ|^{2} + |ξ_{1}|)^{|α|}} , |ξ| ≤ 1, |α| ≤ 2, i, j = 1, 2, \\ |∂^{α}Γ̂_{3j}^{(2)}(ξ)| ≤ \frac{c|ξ|}{(|ξ|^{2} + |ξ_{1}|)^{|α|}} , |ξ| ≤ 1, |α| ≤ 2, j = 1, 2, 3, \\ |∂^{α}Γ̂_{i3}^{(2)}(ξ)| ≤ \frac{c|ξ|}{(|ξ|^{2} + |ξ_{1}|)^{|α|}} , |ξ| ≤ 1, |α| ≤ 2, i = 1, 2, 3, \end{matrix} (24)

from which we obtain the estimates of Γ^{(2)} in the neighbouhood of the point |x| = ∞ [3]:

\begin{matrix}∂^{α}Γ̂_{ij}^{(2)}(x) ≤ o(|x|^{−1}), |α| ≥ 0, i, j = 1, 2, 3, \\ ∂^{α}Γ̂_{3j}^{(2)}(x) ≤ o(|x|^{−2}), |α| ≥ 1, j = 1, 2, 3, \\ ∂^{α}Γ̂_{i3}^{(2)}(x) ≤ o(|x|^{−2}), |α| ≥ 1, i = 1, 2, 3. \end{matrix}  (25)

Let us now calculate the inverse Fourier transform of Γ̂^{(∞)}. We obtain

Γ_{11}^{(∞)}(x) = \frac{1}{2πμ}(K_{0}(m|x|) − \frac{x_{1}}{|x|}K′_{0}(m|x|))e^{mx_{1}} − \frac{1}{2πmμ} \frac{x_{1}}{|x|^{2}} ,

Γ_{12}^{(∞)}(x) = Γ_{21}^{(∞)}(x) = −\frac{1}{2πμ} \frac{x_{2}}{|x|}K′_{0}(m|x|)e^{mx_{1}} − \frac{1}{2πmμ} \frac{x_{2}}{|x|^{2}} ,

Γ_{22}^{(∞)}(x) = \frac{1}{2πμ}(K_{0}(m|x|) + \frac{x_{1}}{|x|}K′_{0}(m|x|))e^{mx_{1}} + \frac{1}{2πmμ} \frac{x_{1}}{|x|^{2}} ,

Γ_{13}^{(∞)}(x) = −Γ_{31}^{(∞)}(x) = \frac{m}{2πμ} \frac{x_{2}}{|x|}K′_{0}(m|x|)e^{mx_{1}},

Γ_{23}^{(∞)}(x) = −Γ_{32}^{(∞)}(x) = −\frac{m}{2πμ}(K_{0}(m|x|) + \frac{x_{1}}{|x|}K′_{0}(m|x|))e^{mx_{1}},

Γ_{33}^{(∞)}(x) = −\frac{η_{1}m}{4πμ^{2}} (K_{0}(m|x|) + \frac{x_{1}}{|x|}K′_{0}(m|x|))e^{mx_{1}},

(26)

where K_{0} is the MacDonald function of zero order:

K_{0}(t) = ∫_{0}^{∞} e^{−t ch η} dη, m = \frac{η_{1}}{2μ}.

The equalities (22)–(26) imply the validity of

Theorem 2. The fundamental matrix Γ is represented in the form

Γ = Γ^{(∞)} + Γ^{(2)},  (27)

where the components Γ^{(∞)} are written as (26) and Γ^{(2)} admits the estimates (25) in the neighbourhood of the point |x| = ∞.

It is not difficult to obtain the following asymptotic representation in the neighbourhood of the infinity of the MacDonald function and its derivative:

K_{0}(t) = \begin{matrix}√ & \overline{π} \\ √ & \overline{2t} \end{matrix}e^{−t}(1 − \frac{1}{8t}) + O(\frac{e^{−t}}{t^{2}\sqrt{t}}),

K′_{0}(t) = −\begin{matrix}√ & \overline{π} \\ √ & \overline{2t} \end{matrix}e^{−t}(1 + \frac{3}{8t}) + O(\frac{e^{−t}}{t^{2}\sqrt{t}}),

K′′_{0}(t) = \begin{matrix}√ & \overline{π} \\ √ & \overline{2t} \end{matrix}e^{−t}(1 + \frac{7}{8t}) + O(\frac{e^{−t}}{t^{2}\sqrt{t}}).

Hence we obtain the asymptotic representations of the components of the matrix Γ^{(∞)} in the neighbourhood of the point |x| = ∞:

Γ_{11}^{(∞)}(x) = \frac{(|x| + x_{1})e^{−m(|x|−x_{1})}}{2\sqrt{2πm}|x|^{3/2}} − \frac{1}{2πmμ} \frac{x_{1}}{|x|^{2}} + O(|x|^{−3/2}),

Γ_{12}^{(∞)}(x) = Γ_{21}^{(∞)}(x) = \frac{x_{2}e^{−m(|x|−x_{1})}}{2\sqrt{2πm}μ|x|^{3/2}} − \frac{1}{2πmμ} \frac{x_{2}}{|x|^{2}} + O(|x|^{−3/2}),

Γ_{13}^{(∞)}(x) = −Γ_{31}^{(∞)}(x) = −\frac{\sqrt{m}}{2\sqrt{2π}} \frac{x_{2}e^{−m(|x|−x_{1})}}{|x|^{3/2}} + O(|x|^{−3/2}),

Γ_{22}^{(∞)}(x) = \frac{(|x| − x_{1})e^{−m(|x|−x_{1})}}{2\sqrt{2πm}|x|^{3/2}} + \frac{1}{2πmμ} \frac{x_{1}}{|x|^{2}} + O(|x|^{−3/2}),

Γ_{23}^{(∞)}(x) = −Γ_{32}^{(∞)}(x) = −\frac{\sqrt{m}(|x| − x_{1})e^{−m(|x|−x_{1})}}{2\sqrt{2π}μ|x|^{3/2}} + O(|x|^{−3/2}),

Γ_{33}^{(∞)}(x) = −\frac{η_{1}\sqrt{m}(|x| − x_{1})e^{−m(|x|−x_{1})}}{4\sqrt{2π}μ^{2}|x|^{3/2}} + O(|x|^{−3/2}).

(28)

In particular, (28) implies

|Γ_{kj}^{(∞)}(x)| ≤ c_{1} \frac{e^{−m(|x|−x_{1})}}{|x|^{1/2}} + c_{2}|x|^{−3/2}, k, j = 1, 2, 3.  (29)

In a similar manner we can obtain the estimates for the derivatives of Γ_{kj}^{(∞)} as well:

|∂^{α}Γ_{kj}^{(∞)}(x)| ≤ c_{1} \frac{e^{−m(|x|−x_{1})}}{|x|^{1/2}} + c_{2}|x|^{−2}, |α| ≥ 1, k, j = 1, 2, 3. (30)

Remark. As follows from (25), (29), in the case of Ozeen’s linearization the fundamental matrix has order O(|x|^{−1/2}) at infinity, but, as shown in [4], the fundamental matrix of the system obtained in the case of Stokes’ linearization has order O(ln |x|) at infinity, i.e. it is unbounded. Therefore the properties of solutions of the external boundary value problems are different in the two cases.

We also note that no such difference between the fundamental solutions is observed for Stokes’ and Ozeen’s linearizations. In that case both fundamental solutions have order O(|x|^{−1}) at infinity [3], [5].

3. Regular Solution Representation Formulas. The Uniqueness Theorems. Let D^{+} be a finite domain in ℝ^{2} bounded by the piecewisesmooth contour S; (u, p) and (u′, p′) be regular pairs satisfying the conditions

\frac{∂u_{1}}{∂x_{1}} + \frac{∂u_{2}}{∂x_{2}} = \frac{∂u′_{1}}{∂x_{1}} + \frac{∂u′_{2}}{∂x_{2}} = 0.

If, besides, L(∂_{x})u − η \frac{∂u}{∂x_{1}} − G(∂_{x})p ∈ L_{1}(D^{+}), L(∂_{x})u′ − η \frac{∂u′}{∂x_{1}} − G(∂_{x})p′ ∈ L_{1}(D^{+}), then the following identities of the Green formula type are fulfilled:

∫_{D^{+}} [u(L(∂_{x})u − η \frac{∂u}{∂x_{1}} − G(∂_{x})p) + E(u, u)]dx =

= ∫_{S} [u(P (∂_{y}, n)u − \frac{1}{2}n_{1}ηu − Np)]^{+}d_{y}S,  (31)

∫_{D^{+}} [u′(L(∂_{x})u − η \frac{∂u}{∂x_{1}} − G(∂_{x})p) −

−u(L(∂_{x})u′ + η \frac{∂u′}{∂x_{1}} − G(∂_{x})p)]dx =

= ∫_{S} [u′(P′(∂_{y}, n)u − \frac{1}{2}n_{1}ηu − Np) −

−u(P (∂_{y}, n)u′ + \frac{1}{2}n_{1}ηu′ − Np′)]^{+}d_{y}S,  (32)

where

E(u, u′) = (μ + α) ∑_{i,j=1}^{2} v_{ij}v′_{ji} + (μ − α) ∑_{i,j=1}^{2} v_{ji}v′_{ij} + γ ∑_{i=1}^{2} ω_{i}ω′_{i},

v_{ij} = \frac{∂v_{j}}{∂x_{i}} + (j − i)ω, i, j = 1, 2; ω_{i} = \frac{∂ω}{∂x_{i}} , i = 1, 2.

E(u, u′) is an analogue of the energetic form. In particular,

E(u, u′) = (μ + α)(v_{12}^{2} + v_{21}^{2}) + 2(μ − α)v_{12}v_{21} +
+ 2μ(v_{11}^{2} + v_{22}^{2}) + γ(ω_{1}^{2} + ω_{2}^{2}).  (33)

One can easily verify that for the form (33) to be positive definite with respect to v_{ij}, ω_{i}, it is necessary and sufficient that the conditions

μ > 0, α > 0, γ > 0  (34)

be fulfilled.

If we use the area potentials

U(F, x) = \frac{1}{2} ∫_{D^{±}} ρΓ(x − y)F (y) dy,

q(F, x) = \frac{1}{2} ∫_{D^{±}} ρQ(x − y)F (y) dy,

where F = (F_{1}, F_{2}, F_{3}) ∈ ℂ^{0,h}(D^{±}), 0 < h ≤ 1, and is the finite vector in the case of D^{−}, then the boundary value problems for the systems (3) and (4) can be reduced to the corresponding problems for the homogeneous systems [3]

\frac{∂u_{1}}{∂x_{1}} + \frac{∂u_{2}}{∂x_{2}} = 0, L(∂_{x})u − G(∂_{x})p − η \frac{∂u}{∂x_{1}} = 0  (35)

and

\frac{∂u_{1}}{∂x_{1}} + \frac{∂u_{2}}{∂x_{2}} = 0, L(∂_{x})u − G(∂_{x})p + η \frac{∂u}{∂x_{1}} = 0.  (36)

Therefore in the sequel we will consider the boundary value problems only for the homogeneous systems (35) and (36).

Like in the three-dimensional case [3], from (32) we can obtain the following formula for representation of solutions of the system (3) in the domain D^{+}:

u(x) = \frac{1}{2} ∫_{S} Γ(x − y)[P (∂_{y}, n)u(y) − \frac{1}{2}n_{1}ηu(y) − N(y)p(y)]^{+}d_{y}S −

−\frac{1}{2} ∫_{S} [P (∂_{y}, n)Γ′(x − y) + \frac{1}{2}n_{1}ηΓ′(x − y) −

−N(y) ∗ Q(x − y)]′u^{+}(y)d_{y}S,  (37)

p(x) = \frac{1}{2} ∫_{S} Q(x − y)[P (∂_{y}, n)u(y) − \frac{1}{2}n_{1}ηu(y) − N(y)p(y)]^{+}d_{y}S −

−\frac{1}{2} ∫_{S} [P (∂_{y}, n)Q(x − y) + \frac{1}{2}n_{1}ηQ(x − y) −

−η_{1}Q(x − y)N(y)]u^{+}(y)d_{y}S.  (38)

Here the prime above the matrix denotes the operation of transposition and

N(y) ∗ Q(z) = ∥ \begin{matrix}n_{1}(y)Q_{1}(z) & n_{1}(y)Q_{2}(z) \\ n_{2}(y)Q_{1}(z) & n_{2}(y)Q_{2}(z) \end{matrix} ∥ .

Taking the properties (29) and (30) of the matrix of fundamental solutions into account and repeating the reasoning from [3], one can prove that the formulas (37), (38) are also valid for regular solutions in the domain D^{−}.

Now let us prove that the equality

∫_{D^{−}} E(u, u) dx = − ∫_{S} u^{−}[P (∂_{y}, n)u − \frac{1}{2}n_{1}ηu − Np]^{−}d_{y}S (39)

is fulfilled for the regular solution (u, p) of the homogeneous system (35) in D^{−}.

To this effect we have to use the formula (31) in the domain D^{−}∩B(0, R), where B(0, R) is the circle of radius R centred at x = 0 and containing the domain D^{+}. Recalling that the pair (u, p) is the solution of the system (35), we obtain

∫_{D^{−}∩B(0,R)} E(u, u) dx = − ∫_{S} u^{−}[P (∂_{y}, n)u − \frac{1}{2}n_{1}ηu − Np]^{−}d_{y}S + I(R),

where

I(R) = ∫_{∂B(0,R)} u[P (∂_{y}, n)u − \frac{1}{2}n_{1}ηu − Np] d_{y}S.  (40)

Obviously, to prove (39) it is enough to prove

lim_{R→∞} I(R) = 0.  (41)

From (16), (25), (29) and the formula of representation of regular solutions in the domain D^{−} it follows that in the neighbourhood of infinity

|∂^{α}u(x)| = c_{1} \frac{e^{−m(|x|−x_{1})}}{|x|^{1/2}} + O(|x|^{−1}), |α| ≥ 0,

|p(x)| = O(|x|^{−1}).

Taking these estimates into account in (40) and passing to the limit, we obtain (41). The equality (39) is proved.

Let us turn to proving the uniqueness theorems. In particular, we will prove

Theorem 3. Each solution of the homogeneous problem (I)_{0}^{+} has the form

u = 0, p = p_{0},  (42)

where p_{0} is an arbitrary constant. The homogeneous Problems (II)_{0}^{+}, (I)_{0}^{−} and (II)_{0}^{−} can have only the trivial solutions u = 0, p = 0.

Proof. Let (u, p) be a solution of anyone of the considered problems. Then by virtue of (31) and (39)

∫_{D^{±}} E(u, u) dx = 0.

Therefore E(u, u) = 0, x ∈ D^{+} (x ∈ D^{−}). Hence on account of (34)

\frac{∂v_{j}}{∂x_{i}} + (j − i)ω = 0, \frac{∂ω}{∂x_{i}} = 0, i, j = 1, 2.

The general solution of the resulting system has the form

v_{1} = ax_{2} + b_{1}, v_{2} = −ax_{1} + b_{2}, ω = a.  (43)

Since for the external domain the regular solution must vanish at infinity, the external homogeneous boundary value problems have trivial solutions. In the case of Problem (I)_{0}^{+}, (43) also implies that v_{1} = v_{2} = ω = 0; substituting these value in the equation (35) we obtain

\frac{∂p}{∂x_{1}} = \frac{∂p}{∂x_{2}} = 0,

whence it follows that p = const.

Next, let us consider Problem (II)_{0}^{+}. By virtue of (43) and the homogeneous boundary condition we have a = 0, i.e. v_{i} = b_{i}. Then (35) yields p = p_{0}. Considering again the boundary condition, we obtain

η_{1}n_{1}b_{k} + 2p_{0}n_{k} = 0, k = 1, 2.

Since the boundary of D^{+} is not the straight line, we conclude that b_{1} = b_{2} = p_{0} = 0.

4. Reduction of the Boundary Value Problems to Integral Equations. Like in the three-dimensional case, we introduce the simple-layer potentials

V (φ)(x) = ∫_{S} Γ(x − y)φ(y) d_{y}S, a(φ)(x) = ∫_{S} Q(x − y)φ(y) d_{y}S

and the double-layer potentials

W (φ)(x) = ∫_{S} [P (∂_{y}, n)Γ′(x − y) + \frac{1}{2}n_{1}ηΓ′(x − y) −
− N(y) ∗ Q(x − y)]′φ(y)d_{y}S,
b(φ)(x) = ∫_{S} [P (∂_{y}, n)Q(x − y) + \frac{1}{2}n_{1}ηQ(x − y) −
− η_{1}Q_{1}(x − y)N(y)]φ(y)d_{y}S.

Denote by Ṽ(φ), ã(φ), W̃(φ), b̃(φ) the potential obtained from V (φ), a(φ), W (φ), b(φ) after replacing η_{1} and η_{2} by −η_{1} and −η_{2}, respectively (note that ã(φ) coincides with a(φ)).

We will seek for the solution of Problem (I)^{±} [(Ĩ)^{±}] in the form of doublelayer potentials

u(x) = W (φ)(x), p(x) = b(φ)(x)
[ u(x) = W̃(φ)(x), p(x) = b̃(φ)(x) ]

and the solution of Problem (II)^{±} [(ĨI)^{±}] in the form of single-layer potentials

u(x) = V (ψ)(x), p(x) = a(ψ)(x)
[ u(x) = Ṽ(φ)(x), p(x) = a(φ)(x) ].

Then, as was done in [3], we obtain, for the densities φ, ψ, the singular integral equations

∓φ(z) + ∫_{S} [P (∂_{y}, n)Γ′(z − y) + \frac{1}{2}n_{1}ηΓ′(z − y) −
− N(y) ∗ Q(y − z)]′φ(y)d_{y}S = f(z),  (I)^{±}
∓φ(z) + ∫_{S} [P (∂_{y}, n)Γ(z − y) − \frac{1}{2}n_{1}ηΓ(z − y) −
− N(y) ∗ Q(y − z)]′φ(y)d_{y}S = f(z),  (Ĩ)^{±}
±ψ(z) + ∫_{S} [P (∂_{y}, n)Γ(z − y) − \frac{1}{2}n_{1}ηΓ(z − y) −
− N(z) ∗ Q(y − z)]ψ(y)d_{y}S = f(z),  (II)^{±}
±ψ(z) + ∫_{S} [P (∂_{y}, n)Γ′(z − y) + \frac{1}{2}n_{1}ηΓ′(z − y) −
− N(z) ∗ Q(y − z)]ψ(y)d_{y}S = f(z).  (ĨI)^{±}

The investigation of these equations leads to the following result [3, 6]:

Theorem 4. If S ∈ L_{k+1}(h′), f ∈ ℂ^{k,h}(S), 0 < h < h′ ≤ 1, k = 1, 2, …, then the Fredholm theorems hold for the pairs of equations (I)^{+} and (ĨI)^{−}, (I)^{−} and (ĨI)^{+}, (II)^{+} and (Ĩ)^{−}, (II)^{−} and (Ĩ)^{+} in the space ℂ^{h,k}(S).

5. Existence Theorems for the Boundary Value Problems. In this paragraph we present the existence theorems for the boundary value problems. Their proofs are left out because they repeat the ones given in [3].

Theorem 5. If S ∈ L_{2}(h′), f ∈ ℂ^{1,h}(S), 0 < h < h′ ≤ 1, and f satisfies the condition

∫_{S} Nf dS = 0,  (44)

then Problem (I)^{+} has a regular solution. Moreover, if the condition

∫_{S} p dS = 0

is fulfilled, then this solution is unique.

We observe that the condition (44) is not only sufficient, but also necessary for the existence of a regular solution of Problem (I)^{+}.

Theorem 6. If S ∈ L_{2}(h′), f ∈ ℂ^{1,h}(S), 0 < h < h′ ≤ 1, then Problem (I)^{−} has a unique regular solution.

Theorem 7. If S ∈ L_{2}(h′), f ∈ ℂ^{0,h}(S), 0 < h < h′ ≤ 1, then Problem (II)^{+} has a unique regular solution.

References

1. D.W. Condiff and J.S. Dahler, Fluid mechanical aspects of antisymmetric stress. Physics of Fluids. 7(1964), No.6, 842-854.

2. A.C. Eringen, Theory of micropolar fluids. J. Math. Mech. 16(1966), No. 1, 1-18.

3. T.V. Buchukuri and R.K. Chichinadze, Boundary value problems of stationary flow of a viscous noncompressible micropolar fluid in the case of linearization by Ozeen’s method. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze, 96(1991), 29-60.

4. R.K. Chichinadze, Two-dimensional problems of stationary flow of a viscous noncompressible micropolar fluid. (Russian) Soobshch. Akad. Nauk Gruzin. SSR, 121(1986), No. 3, 485-488.

5. R.K. Chichinadze, Boundary value problems for stationary flow of a viscous noncompressible micropolar fluid. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze, 75(1984), 111-132.

6. V.D. Kupradze, T.G. Gegelia, M.O. Basheleishvili and T.V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. (Translation from the Russian) North-Holland Publishing Company, 1979; Russian original: ”Nauka”, Moscow, 1976.

(Received 18.03.1993)

Authors’ address: A.Razmadze Mathematical Institute Georgian Academy of Sciences 1, Z.Rukhadze St., 380093 Tbilisi Republic of Georgia