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ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ФУНКЦИОНАЛЬНЫЙ АНАЛИЗ
INTEGRO-DIFFERENTIAL EQUATIONS AND FUNCTIONAL ANALYSIS
Серия «Математика»
2021. Т. 37. С. 17—30
Онлайн-доступ к журналу: http://mathizv.isu.ru
УДК 517.957, 517.958, 532.5.032
MSC 35N10, 76D05, 76D17
DOI https://doi.org/10.26516/1997-7670.2021.37.17
Exact Solutions to the Oberbeck-Boussinesq Equations for Shear Flows of a Viscous Binary Fluid with Allowance Made for the Soret Effect *
N. V. Burmasheva1, E. Yu. Prosviryakov1
1 Institute of Engineering Science of UB RAS, Ekaterinburg, Russian Federation
Abstract. The paper considers an exact solution to the equations of thermal diffusion of a viscous incompressible fluid in the Boussinesq approximation with neglect of the Dufour effect for a steady shear flow. It is shown that the reduced system of constitutive relations is nonlinear and overdetermined. A nontrivial exact solution of this system is sought in the Lin-Sidorov-Aristov class. The resulting family of exact solutions allows one to describe steady-state inhomogeneous shear flows. This class generalizes the classical Couette, Poiseuille, and Ostroumov-Birikh solutions. It is demonstrated that the system of ordinary differential equations reduced within this class retains the properties of nonlinearity and overdetermination. A theorem on solvability conditions for the overdetermined system is proved; it is reported that, when these conditions are met, the solution is unique. The overdetermined system is solvable owing to the algebraic identity relating the horizontal velocity gradients, which are linear functions of the vertical coordinate. The constructive proof of the computation of hydrodynamic fields consists in the successive integration of the polynomials, the polynomial degree being dependent on the values of the boundary parameters.
Keywords: viscous binary fluid, exact solution, Soret effect, shear flow, overdetermined system.
1. Introduction
The Boussinesq approximation is known to be used in studying con-vective flows of binary fluids [6; 14; 18]. The study of the properties of the solutions to the Oberbeck-Boussinesq equations describing thermal diffusion flows is complicated by the Soret and Dufour thermodynamic cross effects between the temperature field and the field of dissolved substance concentration [13; 22]. This is mathematically expressed in the coupled nature of the equations of heat transfer and concentration. A class of exact solutions of the Navier-Stokes equations for describing the flows of incompressible binary fluids with allowance made for both effects was proposed in [3] The idea that the Dufour parameter, due to its smallness, can be neglected in the description of fluid flows dominates in the current scientific literature [11; 12; 15; 16;23]. Exact solutions to the Oberbeck—Boussinesq equations are sought based on this physical assumption.
The finding of exact solutions to the thermal diffusion equations has started fairly recently from the study of unidirectional (stratified) flows in infinite plane and infinitely long cylindrical channels [4; 21]. Various modifications of the Ostroumov-Birikh family are used in the construction of exact solutions [5; 19]. In other words, the exact solutions whose velocity fields are described by the exact Couette solution are studied, and the horizontal (longitudinal) gradients of temperature, pressure, and concentration are taken into account.
For different applications, it is necessary to have a collection of exact solutions characterizing the distribution of hydrodynamic fields in shear flows (the vertical component of the velocity vector is zero) [7-11]. In this case, the Oberbeck—Boussinesq equation system is overdetermined. Five unknown functions (two velocity vector components, pressure, temperature, and concentration) are computed from six equations (the Oberbeck--Boussinesq system and the equations of incompressibility, heat conduction, and concentration). In the study of shear flows, the inhomogeneity of the fields of temperature, pressure, and concentration can induce an inhomo-geneous velocity field [1;3;17;20]. Note that, according to the Onsager principle, the inverse effect is true [1;3;17;20].
The solvability of the overdetermined systems describing inhomogeneous flows, which resulted from the reduction of the Navier-Stokes equations in various force fields, was studied in [9; 10]. An exact solution was constructed in the Lin-Sidorov-Aristov class [1;17;20]. It describes spatially inhomogeneous vortex flows. Investigations for inhomogeneous shear flows of viscous binary fluids have yet to be started. The study of fluid flows with a homogeneous two-dimensional velocity field depending only on the transverse coordinate (two-dimensional Couette flow), was started in [11]. The
* This work was supported by the Russian Science Foundation (project 19-19-00571).
presented mathematical model describes the convection of a binary fluid in view of the Soret effect in an exact statement. It was shown in [11] that, in a steady-state flow of a fluid with a dissolved substance, there was a more pronounced stratification of hydrodynamic fields as compared to Marangoni heat convection first discussed in [2]. Besides, the exact solutions reported in [8] and taking into account the spatial inhomogeneity of velocities also describe hydrodynamic fields with a greater number of stratification points than in a homogeneous velocity field. Thus, a fluid flow structure (first of all, in the velocity field) is affected by the spatial inhomogeneity of hydrodynamic fields, boundary conditions, and flow-inducing forces.
This paper, applying the technique proposed in [1; 11; 17; 20], studies the solvability of the Oberbeck—Boussinesq equation system for describing the spatially inhomogeneous thermal-diffusion shear flows of incompressible binary fluids in the Lin—Sidorov-Aristov class with allowance made for the Soret effect.
2. Problem Statement
Steady-state shear flows of binary viscous incompressible fluids are described by the following relationships [11]:
dVx dVx
Vx^A + Vy—
dx
dy
dP /d2Vx , d2Vx , d2Vx
dx + V \ dx2
+
dy2
+
dz2
Fx ^ + Vy dV<
dx
dy
dP
= + v
dy
id 2Vy d 2Vy d 2Vy
V dx2
+
= g (ß\T + ß2C),
dy2
dVx + dVy
+
dz2
dx
dy
0,
dT dx
dT
Vx— + Vy— = %
y dy
'd 2T d 2T d2T
+ — +
dC
dC
Vx——h Vy—~ = d
dx
dy
'd2C d 2 C
dx2
d2c
dy
dx2 dy2 dz2
^ + ad ^
dz2 d2T
),
d2T d2T
dx2 dy2 dz2
(2.1)
Here, V (x,y,z) = (Vx,Vy, 0) is fluid velocity; g is free fall acceleration; Pi, are the thermal and concentration expansion coefficients, respectively; T (x, y, z) is temperature deviation from the equilibrium state; C (x, y, z) is the deviation of the concentration of the light phase (admixture) in a binary fluid mixture from the equilibrium value; P (x, y, z) is deviation from hydrostatic pressure, normed to the average constant fluid density p; v is kinematic fluid viscosity; % is the temperature conductivity coefficient; d is the diffusion coefficient; a is the thermal diffusion coefficient (the Soret parameter) [22].
The quadratically nonlinear system (2.1) is overdetermined since four functions of the five equations (velocities Vx,Vy , pressure P, temperature T, and concentration С) need to be determined. A nontrivial solution to system (2.1) will be sought in the Lin-Sidorov-Aristov class, where the required unknowns are represented by complete linear forms in terms of coordinates [3; 9; 10; 17]:
Vx = U (z) + ui (z) x + U2 (z) y, Vy = V (z) + vi (z) x + V2 (z) y,
P = Po (z) + Pi (z) X + P2 (z) y, T = To (z) + Ti (z) X + T2 (z) y,
С = Co (z)+ Ci (z) x + C2 (z) y. (2.2)
Having substituted the exact solution (2.2) into system (2.1), applying the method of undetermined coefficients, we arrive at a system of ordinary differential equations, which, for the convenience of analysis, is here presented as follows:
ui + u2v1 = wu'l, u1u2 + u2v2 = wu'2,
u1 v1 + v1v2 = vv'l, u2 v1 + = vv'2, u1 + v2 = 0; (2.3)
U1T1 + V1T2 = xT'-l, U2 T1 + V2T2 = XT%; (2.4)
U1C1 + V1C2 = dC'l + adT'1, U2C1 + V2C2 = dC'2 + adT%; (2.5)
P1 = g (P1T1 + P2C1), P2 = 9 (P1T2 + P2C2); (2.6)
U1U + U2V = -P1 + uU", V1U + V2 V = -P2 + vV''; (2.7)
ОД + FT2 = XT0', UC1 + FC2 = dC'0 + adTO'; (2.8)
P0 = ^ (^1^0 + AO)). (2.9)
The prime in system (2.3)-(2.9) marks derivation with respect to the variable z. In the transition from system (2.1) to system (2.3)-(2.9) the property of overdetermination is inherited, i.e. we have sixteen equations for the determination of fifteen nonlinear coefficients of the linear forms in Eq. (2.2). Subsystem (2.3) alone has a "redundancy" of equations with respect to the number of unknowns. Thus, if it proves possible to find a nontrivial uultaneous solution, i.e. the one satisfying all the equations of the overdetermined subsystem (2.3), successive integration can yield a solution to any of the subsystems (2.4)-(2.9).
3. Analyzing the Solvability of the Equation System
The existence of a nontrivial exact solution of system (2.3)-(2.9) is ensured by the fulfilment of the following theorem.
Theorem 1. The overdetermined system (2.3) has a nontrivial exact solution in class (2.2), which is an exact solution of system (2.1), if and only if the spatial accelerations ui,u2,vi and v2 are described by the functions
ui = u cos § sin § = —v2, u2 = u cos2 v\ = —u sin2
where u is a function satisfying the equation u" = 0 and •& is some number.
Proof. Transform subsystem (2.3) by reducing the number of equations in it. To do this, use the relationship ui + v2 = 0 and express the spatial acceleration v2 via the component ui as v2 = —u\. Substitute this expression into all the differential equations of subsystem (2.3). This yields
uf + u2vi = w![, uf + u2vi = —w![, u'2 = 0, v'1 = 0. (3.1)
It follows from the last two equations in system (3.1) that the components v\,u2 are linear functions,
U2 = bfZ + b2, V\ = b3z + b4. (3.2)
Besides, from the first two equations of system (3.1) it follows that the spatial acceleration uf linearly depends on the variable z,
ui = b5z + be. (3.3)
Herewith, solutions (3.2),(3.3) must satisfy the algebraic consistency condition
u2 + u2vi = 0. (3.4)
The substitution of Eqs. (3.2) and (3.3) into the condition expressed by Eq. (3.4) results in the following system of conditions imposed on the coefficients bi,b2,b3,b4,b5 and b6 :
b25 + bib3 = 0, 2b5b6 + bib4 + b2b3 = 0, b2 + b2b4 = 0. (3.5)
The solutions of system (3.5) for the spatial accelerations represented by Eqs. (3.2), (3.3) are particular cases of the following representation:
ui = u cos § sin u2 = u cos2 vf = —u sin2 (3.6)
where § is some number and the function u satisfies the equation u" = 0, i.e. it is representable as u = ciz + c2 .
The substitution of the solution for the spatial accelerations represented by Eq. (3.6) into the consistency condition (3.4) gives the true identity
u2 + u2vi = (u cos $ sin $)2 + (u cos2 $) (—u sin2 $) = 0.
Consequently, the overdetermined system (2.3) has a simultaneous solution. Thus, the theorem has been constructively proved. □
4. Constructing an Exact Solution
The above exact solutions for the spatial accelerations ui,vi,andv2 enable us to find the unknown functions U, V, Ti, Ci and Pi (i = 0,1,2) defining the hydrodynamic fields.
Theorem 2. If the simultaneous solution of the overdetermined system (2.3) has the form of Eq. (3.6), the solution of Eqs. (2.3)-(2.9) does exist.
Proof. Transform Eqs. (2.4) and (2.5) enabling us to determine, with Eq. (3.6) taken into account, the longitudinal (horizontal) gradients Ti,T2 of the temperature field T and the gradients Ci,C2 of the concentration field С. The substitution gives
XT" = и cos fi sin §Ti - и sin2 §T2,
XT'2, = и cos2 VTx - и cos fi sin 1ЭТ2,
dC" + adT'-l = и cos fi sin fiCi - и sin2 VC2,
dC'2 + adT% = и cos2 fiCx - и cos fi sin VC2. (4.1)
Multiply the first equation in system (4.1) by cos •& and subtract the second equation multiplied by sin •& from it. After double integration, we obtain the relationship
cos - sin §T2 = 71Z + j2- (4.2)
Hereinafter, the coefficients of the form ji (i e N) are integration constants. It follows from the third and fourth equations in system (4.1) that
cos fiCi + a cos fiTi - sin §C2 - a sin §T2 = ^3z + j4. (4.3)
We express the gradients C2,T2 from the relations (4.2), (4.3). To do this, we represent system (4.2), (4.3) in the matrix form as
( sin fi 0 T2 \ = ( cos m - Hz - ъ A (44) ya sin fi sin fi J \C2 J у a cos °9Ti + cos fiCi - j3z - j4 J ' (
The structure of the solution of the matrix equation (4.4) depends on the value of the coefficient sin fi.
Let us first consider the particular case of fi being such that sin fi = 0. Then the left parts of Eq. (4.4) are identically equal to zero and system (4.4) becomes
Ti - jiz - J2 = 0, aTi + С- - - = 0. Hence, the solution is
Ti = jiz + J2, С- = -aTi + J3z + = (73 - "7") 2 + (74 - «72). (4.5)
The obtained solutions (4.5) are substituted into the second equation of system (4.1). In view of Eq. (3.6), when sin § = 0, Eqs. (2.6), (2.7) acquire the form
XT2 = uTi, C2 = aT2 + 2u°i, pi = 9 (PiTi + P2C1),
P2 = g (PiT2 + P2C2), vU" = Vu + Pi, vV" = P2. (4.6)
Expressions (4.5) and (3.6) are substituted into Eq. (4.6). Successive integration of the ordinary differential equations of this tenth-order system results in a polynomial solution where the gradient Pi of the pressure field is a second-order polynomial, the gradient P2 is a fifth-order one, the gradient T2 of the temperature field and the gradient C2 of the concentration field are fourth-order polynomials, and the velocities U and V are tenth- and seventh-order ones, respectively.
We now return to the general case of § being such that sin § = 0. This means that the determinant D = sin2 § of system (4.4) is nonzero; therefore, system (4.4) is uniquely solvable,
( T2 \ = 1 / — sin § 0 \/ — cos VTi + 7iz + ^2 A \C2 J sin2 § \ a sin •& — sin § J y —a cos rdTi — cos °9Ci + j3z + j4 J '
T2 = ctgtfTi + , C2 = ctgdCi + (a7i — 73) ' +(ar<2 — 74).
sin V sin V
(4.7)
Substituting Eq. (4.7) first into the first equation of system (4.1), we have
sin $
Ti =-(7^ + 72) u. (4.8)
X
With Eq. (4.8) taken into account, the transformation of the third equation of system (4.1) gives
sin $
C'i = [(—«71 (x + d)+ 7sx) 2 + (—072 (x + d) + 74X)j u. (4.9)
Integrating twice Eqs. (4.8) and (4.9), we arrive at the following solution for the longitudinal (horizontal) gradients T\,C\:
T\ = (ci^iZ4 + 2 (^72 + C27l) Z3 + 6C2^2Z2 + 75Z + ,
^ sin ■& r 4
= V2XA ^ 71 + + * + +2 [ci (—072 (x + d)+ 74X) + ^2 (—an (x + d) + 73X)] ¿3+
+6C2 (—aj2 (x + d) + J4x) + 17Z + 7^ . (4.10)
Substituting Eq. (4.10) into the linear relations (4.7), we find the form of the remaining gradients of the temperature and concentration fields. Note that, as distinct from the above-discussed case sin § = 0, all the components
T2, C\, C2 of the representation expressed by Eq. (2.2) are fourth-order polynomials. Then, from the obtained polynomials, the form of the exact solution for the gradients P\ and P2 of the pressure field is determined by single integration of Eq. (2.6).
Let us now turn to the integration of subsystem (2.7) enabling us to find the form of the homogeneous components U and V of the velocity field. The substitution of Eq. (2.7) for the spatial accelerations u\,u2,v\ and v2 into Eq. (3.6) gives the following system of equations:
Uu cos § sin § + Vu cos2 § = -P\ + vU",
-Uu sin2 § - Vu cos $ sin § = -P2 + vV". (4.11)
We multiply the first equation in system (4.11) by sin •& and the second one by cos •&; summing the resulting relations, we obtain the equation
U" = -ctg^" +sin m +c°s m. (4.12)
V sin V
By a double integration of Eq. (4.12) taking into account the expressions obtained for the gradients P\,P2, we arrive at a relation between the components U and V of the velocity field represented by Eq. (2.2). Note that the resulting dependence has a linear form and that the coefficients at the velocities U and V in it are constant. The obtained exact solution is then substituted into the second equation of system (4.11) thus yielding an equation for the velocity V. Solving this equation in view of the above-mentioned linear relationship between the components U and V, we can find an exact solution for the velocity U.
In what follows, we find solutions to Eq. (2.8), which gives the form of the background components of the temperature field T and the concentration field C. Similarly to solving system (4.2), (4.3), to find a solution to subsystem (2.8),we represent it as an inhomogeneous matrix algebraic equation with respect to the second derivatives of the background components C0 and T0 of the linear forms in Eq. (2.2),
( 0 X V C'0 ) = ( UTl + VT2 \ (413)
\d ad) \ T0 ) = ^ UC± + VC2) . ( 3)
Note that the determinant of the matrix of the coefficients in the left-hand side of system (4.13) is nonzero; therefore, Eq. (4.13) is solvable with respect to the derivatives of background temperature Tq and background concentration Cq ,
( ^ ) = ^ ( ft )( ^ + № ) • <4,4)
The right-hand side of Eq. (4.14) contains the known polynomial functions; consequently, Eq. (4.14) can be easily integrated.
A solution to subsystem (2.9) is found thereafter. The background pressure P0 results from the integration of the already known polynomial function. Hence, the pressure P0 itself is a polynomial. As before, the result of this integration can be easily found by the known inhomogeneity contained in the right-hand side of Eq. (2.9). These solutions are rather lengthy and omitted here. □
To summarize the foregoing, it is worth noting that the algorithm of solving the overdetermined system (2.3)-(2.9) is reducible to the integration of the polynomial functions. Such integration is known to be always feasible. Consequently, if a solution of the form of Eq. (3.6) is obtained, the solution of the other subsystems of the overdetermined system under study does exist.
5. Result Interpretation
The obtained exact solutions seem to be of great value for theoretical and applied research. Without constraining the generality of the reasoning, we illustrate the novelty of the solution class by studying the velocity field. To be specific, we will consider Marangoni convection in an infinite horizontal fluid layer, which, for a steady-state flow, is determined by the boundary conditions
Vf M dT dC Vf M dT dC
VVX (x,y,h) = —— ^ — , ,Vy (x,y,h) = — a,— — a2 — ,
T (x,y, 0) = Ay, T (x,y,h) = ft + Bx, C (x,y, 0) = 0, C (x,y,h) = Dx + Ey,
P (x,y,h)=0, Vx (x,y,h) = W cos a + Qy, Vy (x,y,h) = W sin a.
Here, is the dynamic viscosity of the fluid. The quantity h determines the thickness of the extensive horizontal fluid layer under study; A, B, D, E, W, a, ft, Q are the parameters specified at the fluid layer boundaries.
The deduction of these conditions is similar to the relations obtained in [2; 8]. The analysis of the velocity field shows that specific kinetic energy can have up to six zero values (Fig. 1).
This testifies to the presence of countercurrents and a cellular nature of the flow caused by the inhomogeneous velocity distribution, the interaction of the pressure, temperature, and concentration fields and their effect on fluid velocity. The obtained stratification of the velocity field is much more pronounced than that reported in [2; 8]. It was shown in [2; 8] that the number of stagnation points (zeros of kinetic energy) does not exceed four.
6. Conclusion
The paper has studied the overdetermined system of nonlinear partial equations that describes a steady-state shear flow of a binary viscous incompressible fluid. The study has shown the existence of a nontrivial exact solution of the reduced Navier-Stokes equations in the Boussinesq approximation (the Oberbeck-Boussinesq system), supplemented by the incompressibility, heat conduction, and concentration equations. The found polynomial exact solution satisfying the algebraic consistency condition belongs to the Lin—Sidorov—Aristov class. This solution describes convection and advection (horizontal convection) in fluid moving beyond the Coriolis field and in the rotating equatorial zone of the World Ocean in the approximation of the /-plane. The formulas presented in the paper describe the stratification of hydrodynamic fields from the reference values. Of special interest is the study of the velocity field, which illustrates the existence of several stagnation points and countercurrents in fluid flow, particularly in sea water.
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Natalya Burmasheva, Candidate of Sciences (Engineering Sciences), Senior Researcher, Institute of Engineering Science of UB RAS, 34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation, email: [email protected],
ORCID iD https://orcid.org/0000-0003-4711-1894
Evgeniy Prosviryakov, Doctor of Sciences (Physics and Mathematics), Chief Researcher, Institute of Engineering Science of UB RAS, 34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation, email: [email protected], ORCID iD https://orcid.org/0000-0002-2349-7801
Received 23.10.2020
Точные решения уравнений Обербека — Буссинеска для сдвиговых течений вязкой бинарной жидкости с учетом эффекта Соре
Н. В. Бурмашева1, Е. Ю. Просвиряков1
1 Институт машиноведения УрО РАН, Екатеринбург, Российская Федерация
Аннотация. Рассмотрено точное решение уравнений термодиффузии вязкой несжимаемой жидкости в приближении Буссинеска при пренебрежении эффектом Дюфора для установившегося сдвигового течения. Обозначено, что редуцированная система определяющих соотношений является нелинейной и переопределенной. Нетривиальное точное решение данной системы ищется в классе Линя - Сидорова - Аристова. Полученное семейство точных решений позволяет описывать установившиеся сдвиговые неоднородные течения. Данный класс позволяет обобщить классические решения Куэтта, Пуазейля и Остроумова - Бириха. Показано, что редуцированная в рамках этого класса система обыкновенных дифференциальных уравнений сохраняет свойства нелинейности и переопределенности. Доказана теорема об условиях разрешимости переопределенной системы, и показано, что при их выполнении решение единственно. Решение переопределенной системы возможно благодаря алгебраическому тождеству, связывающее горизонтальные градиенты скоростей, которые являются линейными функциями от вертикальной координаты.
Конструктивное доказательство вычисления гидродинамических полей заключается в последовательном интегрировании полиномов, причем степень полиномов зависит от значений краевых параметров.
Ключевые слова: вязкая бинарная жидкость, точное решение, эффект Соре, сдвиговое течение, переопределенная система.
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Наталья Владимировна Бурмашева, кандидат технических наук, старший научный сотрудник, сектор нелинейной вихревой гидродинамики, Институт машиноведения УрО РАН, Российская Федерация, 620049, г. Екатеринбург, ул. Комсомольская, 34; email: [email protected],
ORCID iD https://orcid.org/0000-0003-4711-1894
Евгений Юрьевич Просвиряков, доктор физико-математических наук, главный научный сотрудник, сектор нелинейной вихревой гидродинамики, Институт машиноведения УрО РАН, Российская Федерация, 620049, г. Екатеринбург, ул. Комсомольская, 34; email: [email protected], ORCID iD https://orcid.org/0000-0002-2349-7801
Поступила в 'редакцию 23.10.2020
