Computational experiment for a class of Mathematical models of magnetohydrodynamics Текст научной статьи по специальности «Математика»

КРАТКИЕ СООБЩЕНИЯ

MSC 35G61 DOI: 10.14529/mmpl70110

COMPUTATIONAL EXPERIMENT FOR A CLASS OF MATHEMATICAL MODELS OF MAGNETOHYDRODYNAMICS

A.O. Kondyukov, Novgorod State University, Velikiy Novgorod, Russian Federation, [email protected],

T.G. Sukacheva, Novgorod State University, Velikiy Novgorod, Russian Federation, [email protected],

S.I. Kadchenko, Nosov Magnitogorsk State Technical University, Magnitogorsk, Russian Federation, [email protected],

L.S. Ryazanova, Nosov Magnitogorsk State Technical University, Magnitogorsk, Russian Federation, [email protected]

The first initial-boundary value problem for the system modelling the motion of the incompressible viscoelastic Kelvin - Voigt fluid in the magnetic field of the Earth is investigated considering that the fluid is under external influence. The problem is studied under the assumption that the fluid is under different external influences depending not only on the coordinates of the point in space but on time too. In the framework of the theory of semi-linear Sobolev type equations the theorem of existence and uniqueness of the solution of the stated problem is proved.The solution itself is a quasi-stationary semi-trajectory. The description of the problem's extended phase space is obtained.The results of the computainal experiment are presented.

Keywords: magnetohydrodynamics; Sobolev type equations; extended phase space; incompressible viscoelastic fluid; explicit one-step formulas of Runge - Kutta.

Introduction

Consider the first initial-boundary value problem for the system modeling the motion of the incompressible viscoelastic Kelvin - Voigt fluid in the magnetic field of the Earth

(1 - = vAv - (v • V)v - -Vp - 2Q x v + — (V x b) x b + f1,

P Pf i1)

Vv = 0, Vb = 0, bt = SAb + Vx (v x b) + f2,

v(x, 0) = v0(x), b(x, 0) = b0(x), x e D, (2)

v(x, t) = 0, b(x, t) = 0, (x,t) e dD x R+. (3)

vb

p is the pressure, vector-functions f1 and f2 correspond to external influences on the fluid and magnetic field respectively; П = ^ V x v is angular velocity, V is operator of Hamilton,

S is the magnetic viscosity, fi is the magnetic permeability, p is the density, D С Rn is the cylinder's domain with the boundary dD of the class C(X.

This work continues investigations of magnetohydrodynamics models which were started by the authors in [2,3]. Its distinctive feature is that there are vector-functions fk = (fi, fk,■■■, fn) in the right part of equations (1).

The first initial-boundary value problem (1) - (3) is reduced to an abstract Cauchy problem for a semi-linear non-autonomous Sobolev type equation. On the base of the solvability theory of the indicated problem the theorem of existence and uniqueness of the solution of the stated problem is proved. The solution itself is a quasi-stationary semi-trajectory. The description of the problem's extended phase space is obtained. The results of the computational experiment are represented. The results of part 1 are taken from [4], the results of part 2 are obtained on the base of the results [2]. During the computational experiment the methods of solution of the initial-boundary problem (1) - (3) are used. They were described in [3]. Also we used the software package [5].

1. Semi-Linear Non-Autonomous Sobolev Type Equations

Consider the Cauchy problem for the semi-linear non-stationary Sobolev-type equation

u(0) = u0, Lu = Mu + F (u) + f (t). (4)

Here the operator L E L(U; F), i.e. it is linear and continuous, ker L = {0}; the operator M : dom M — F is linear and closed and it is densely defined in U, i.e. M E Cl(U; F); U and F are Banach spaces. Let UM be the lineal dom M equipped with the norm of the graph ||| • ||| = \\M • ^ + || • ||M■ We assume that F E C^(UM; F), and the function f EC +; F )■

Consider (4) under condition that the operator M is strongly (L,p)-sectorial [4]. It is well known that in this case a solution of this problem exists not for all u0 E UM, and even if it exists it may be non-unique. So we introduce two definitions: the extended phase space and quasi-stationary semi-trajectory.

It is well known that if the operator M is strong ly (L,p)-sectorial, t hen U = U0 ®U1, F = F0 ®F1, where

U0 = {p EU : UV = 0 3t E R+}, F0 = {^ EF : F= 0 3t E R+}

are the kernels, and

U1 = {u EU : lim U'u = u}, F1 = {f EF : lim Ftf = f}

are the images of the analytic solving semigroups

r

Ft = hjL*

(M(r C S@a(M) is a contour such that arg¡i — ±6 when

r

Ill — of the linear homogeneous equation Lu = Mu^ Let Lk (Mk) be a restriction of the operator L (M) on Uk (Uk Pi domM), k = 0, L Then Lk : Uk — Fk, Mk : Uk P dom M — Fk, k = 0, 1; and restrictions M0 and L1 of ^te operators M and L on the spaces U0 P dom M and U1 respectively are linear continuous operators and they have bounded inverse operators.

So we reduce (4) to

Ru° = u0 + G(u) + g{t), u0(0) = u0 и1 = Su1 + H (u) + hit), u1(0) = u0

1_ C„,1 I ил,л I hf+\ „.Ifn\ — „°1 (5)

where uk e Uk, k = 0,1, u = u0 + u1, operators R = M0 1L0, S = L- 1M1, G = M-1 (I-Q)F, H = L-lQF, g = M-1(I-Q)f, h = L^Qf.EereQ e L(F)(= L(F; F)) is the corresponding projector.

We study such quasi-stationary semi-trajectories of (4), for which Rii° = 0. Assume that the operator R is bi-splitting , i.e. its kernel ker R and image im R are complemented in the space U. Denote U00 = ker R, and U01 = U0 QU00 is a complement of the subspace U00. Then the first equation of (5) is reduced to the form Ru01 = u00 + u01 + G(u) + g(t), where u = u00 + u01 + ul.

Theorem 1. Let the operator M be strongly (L,p)-sectorial, and the operator R be bisplitting. Let there exist a quasi-stationary semi-trajectory u = u(t) of equation (4). Then it satisfies the following relations

0 = u00 + u01 + G(u) + g(t), u01 = const. (6)

It is known that if the operator M is strongly (L,p)-sectorial then the op erator S is sectorial. So it generates an analytic semigroup on U\ Denote it as {Uf : t > 0} because the operator U1 in fact is a restriction of the operator U* on U1. The fact that U = U0 ®U1 shows that there exists a projector P e L(U) corresponding to this splitting. It can be shown that P e L(UM). Then the space UM splits into the direct sum Um — UM Ф UM so that the embedding UM С Uk, k = 0,1 is dense and continuous. Symbol A'v denotes the Frechet derivative at v e V of the operator A, defined от some Banach space V.

Theorem 2. Let the operator M be strongly (L,p)-sectorial, the operator R be the bisplitting one, the operator F e C^(UM; F), the vector-function f e C; F). Let the following conditions be satisfied:

(i) 0 = u01 + (I - Pr)(G(u00 + u01 + u1) + g(t)) in the neighborhood OUQ С Um of the u0

(ii) the projector PR e L(UM), and the operator I+PRG'u0 : UM UM is the topological linear isomorphism (UM = UM П U00);

(iii) I \\UH\l{u ui )dt< жУт e R+.

0M

Then there exists a unique solution of (4), which is a quasi-stationary semi-trajectory of equation (4).

Now let Uk and Fk be Banach spaces, operators Ak e L(Uk, Fk), and operators Bk : dom Bk — F be linear and closed with domains dom Bk dense in Uk, k = 1, 2. Construct spaces U = U1 x U2, F = F1 x F2 and operators L = A1 ® A2, M = B1 ® B2. By construction operator L e L(U; F), and operator M : dom M — F is linear, closed and densely defined, in U dom M = dom B1 x dom B2.

Theorem 3. Let the operators Bk be strongly (Ak,pk)-sectoriaI, k = 1, 2; and p1 > p2. Then the operator M is strong ly (L,p1)-sectorial.

2. The Existence of the Unique Solution

Following [2] consider problem (2), (3) for system (1) transformed to the form

(1 - xA)vt = vAv - (v • V)v - 1 Vp - 2Q x v + -(Vx b) x b + f1,

p pi V(V • v) = 0, V(V b) = 0, bt = SAb + Vx (v x b) + f 2.

U F L

M (L, 1)

sectorialness of the operator M under the assumption x-1 E a(A),A = V2En (En is identity n x n matrix) is established using the Theorem 3. The form of the nonlinear F

column include the term f \ and the last dement includes f2. Belonging of F to the class C^(Um; F) is proved by calculation of the Frechet derivatives of this operator. Therefore, all assumptions of Theorem 1 and Theorem 2 are satisfied. Consequently, the following result holds.

Theorem 4. If x-1 E ^(A) U a(Aa), then for all u0 E MM and so me T E R+ there exists a unique solution u = (ua, 0,up,ub) of (1) - (3) which is a quasi-stationary trajectory. Moreover, u(t) E MM for a 111 E (0,T).

Here Aa is a restriction of the operator A on the corresponding subspace of solenoidal vectors. ua, 0, up, ub are the elements of the corresponding subspaces of Banach space U [2]; M

MM = {u EUM : un = 0,bn = 0,up = n(vAa - (ua • V)ua-2Q x ua + (V x ba) x ba + f1 )}■ 3. Computational Experiment

Introduce a cylindrical coordinate system (r, p, z) with center O on one side of the surface of the cylinder D and combine Oz axis with the cylinder axis. In the future, we will assume that a flow of fluid is axially symmetric.

Figures 1 — 3 show the charts of surfaces of fluid flow velocity components at time t* = 2 s, obtained for the following values of the problem parameters: x = 2, 7 m/s2, v = 0,00328 m2/s, i = 1 p = 1000 kg/m3, S = 0,1, r0 = 0,1m, z0 = 0,2 m. Vector-functions' components f \ f2 had the form f1 = 0, 001r(r0 - r) sin(nt) kg • m/s2, f?p = fz =0 f2 = 0, 001r cos(nt) T, f2 = fZ = 0 Vector functions v0(r, z^d b0(r, z) in the initial conditions (2) are given in the form v0 = u0rir, b0 = br0ir, where u0 = 0, 25 1/s, br0 = 0, 00005 T. ^te initial ^^^^^tons for vector-functions ^ A are given in the form: ip(r, z, 0) = 0, 25u0r(2zir - riz), A(r, z, 0) = -br0ziv. The boundary and initial conditions for vector-functions ^ and A are given in the form: ^(r,z,t) = 0, A(r,z,t) = 0, (x,r,t) E dD x R+.

Conducted computational experiments show a computational stability and computational efficiency of the developed algorithm for numerical solution of initial-boundary problem (1) - (3).

Fig. 1. Chart of surface radial component of fluid flow velocity vr = vr (r, z, t*)

Fig. 2. Chart of surface transversal component of fluid flow velocity vv = Vp(r,z,t*)

Fig. 3. Chart of surface axial component of fluid flow velocity vz = vz(r,z,t*)

References

1. Hide R. On Planetary Atmospheres and Interiors. Mathematical Problems in the Geophisical Sciences, American Mathematical Society, 1971.

2. Sukacheva T.G., Kondyukov A.O. Phase Space of a Model of Magnetohydrodynamics. Differential Equations, 2015, vol. 51, no. 4, pp. 502-509. DOI: 10.1134/S0012266115040072

3. Kadchenko S.I., Kondyukov A.O. Numerical Study of a Flow of Viscoelastic Fluid of Kelvin - Voigt Having Zero Order in a Magnetic Field. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 2, pp. 40-47. DOI: 10.14529/jceml602005

4. Sukacheva T.G., Kondyukov A.O. On a Class of Sobolev Type Equations. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 4, pp. 5-21. DOI: 10.14529/mmpl40401

5. Kondyukov A.O., Kadchenko S.I., Kakushkin S.N. Numerical Modelling of the Motion of the Viscoelastic Conductive Fluid in the Magnetic Field. The copyright holder: Federal state budgatary educational institution of higher education "Yaroslav-the-Wise Novgorod State University" (RU), №2016619268, registered 17.08.2016, the registry of computer programs.

Received 24 December, 2016

УДК 517.9 DOI: 10.14529/mmpl70110

ВЫЧИСЛИТЕЛЬНЫЙ ЭКСПЕРИМЕНТ ДЛЯ ОДНОГО КЛАССА МАТЕМАТИЧЕСКИХ МОДЕЛЕЙ МАГНИТОГИДРОДИНАМИКИ

А. О. Кондюков, Т.Г. Сукачева, С.И. Кадченко, Л.С. Рязанова

Исследуется первая начально-краевая задача для системы уравнений, моделирующей движение несжимаемой вязкоупругой жидкости Кельвина - Фойгта в магнитном поле Земли с учетом внешнего воздействию на жидкости. Задача изучается в предположении, что жидкость находится под влиянием различных внешних воздействий, зависящих не только от координаты точки в пространстве, но и от времени. В рамках теории полулинейных неавтономных уравнений соболевского типа доказана теорема о существовании и единственности решения, которое является квазистационарной полутраекторией, а также дано описание расширенного фазового пространства. Приведены результаты вычислительного эксперимента

Ключевые слова: магнитогидродинамика; уравнения соболевского типа; расширенное фазовое пространство; несжимаемая вязкоупругая жидкость; явные одноша-говые формулы Рунге - Кутты.

Литература

1. Hide, R. On Planetary Atmospheres and Interiors / R. Hide // Mathematical Problems in the Geophysical Sciences. - American Mathematical Society, 1971.

2. Сукачева, Т.Г. Фазовое пространство одной модели магнитогидродинамики / Т.Г. Сукачева, А.О. Кондюков// Дифференциальные уравнения. - 2015. - Т. 51, № 4. - С. 495-501.

3. Kadchenko, S.I. Numerical Study of a Flow of Viscoelastic Fluid of Kelvin - Voigt Having Zero Order in a Magnetic Field / S.I. Kadchenko, A.O. Kondyukov // Journal of Computational and Engineering Mathematics. - 2016. - V. 3, № 2. - P. 40-47.

4. Sukacheva T.G. On a Class of Sobolev Type Equations / T.G. Sukacheva, A.O. Kondyukov // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. -2014. - Т. 7, № 4. - С. 5-21.

5. Численное моделирование течения вязкоупругой электропроводной жидкости в магнитном поле / Кондюков А.О., Кадченко С.И., Какушкин С.Н.; правообладатель: ФГБОУ ВО «Новгородсикй государственный университет имени Ярослава Мудрого. -№2016619268, зарегистр. 17.08.2016, реестр программ для ЭВМ.

Алексей Олегович Кондюков, аспирант, кафедра алгебры и геометрии, Новгородский государственный университет имени Ярослава Мудрого (г. Великий Новгород, Российская Федерация), [email protected].

Тамара Геннадьевна Сукачева, доктор физико-математических наук, профессор, кафедра алгебры и геометрии, Новгородский государственный университет имени Ярослава Мудрого (г. Великий Новгород, Российская Федерация), [email protected].

Сергей Иванович Кадченко, доктор физико-математических наук, профессор, кафедра прикладной математики и информатики, Магнитогорский государственный технический университет им. Г.И. Носова (г. Магнитогорск, Российская Федерация), [email protected].

Любовь Сергеевна Рязанова, кандидат педагогических наук, доцент, кафедра прикладной математики и информатики, Магнитогорский государственный технический университет им. Г.И. Носова (г. Магнитогорск, Российская Федерация), [email protected].

Поступила в редакцию 24 декабря 2016 г.