Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation Текст научной статьи по специальности «Физика»

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N. V. Burmasheva, E. Yu. Prosviryakov, Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.J, 2019, Volume 23, Number 2, 341-360

DOI: https://doi.org/10.14498/vsgtu1670

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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki

[J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2019, vol. 23, no. 2, pp. 341-360 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1670

Mathematical Modeling, Numerical Methods and Software Complexes

MSC: 76F02, 76M45, 76F45, 76R05, 76U05

Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation

N. V. Burmasheva1'2, E. Yu. Prosviryakov1'2

1 Institute of Engineering Science, Urals Branch, Russian Academy of Sciences,

34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation.

2 Ural Federal University named after the First President of Russia B. N. Yeltsin,

19, Mira st., Ekaterinburg, 620002, Russian Federation.

Abstract

This article discusses the solvability of an overdetermined system of heat convection equations in the Boussinesq approximation. The Oberbeck-Boussinesq system of equations, supplemented by an incompressibility equation, is overdetermined. The number of equations exceeds the number of unknown functions, since non-uniform layered flows of a viscous incompressible fluid are studied (one of the components of the velocity vector is identically zero). The solvability of the non-linear system of Oberbeck-Boussinesq equations is investigated. The solvability of the overdetermined system of non-linear Oberbeck-Boussinesq equations in partial derivatives is studied by constructing several particular exact solutions. A new class of exact solutions for describing three-dimensional non-linear layered flows of a vertical swirling viscous incompressible fluid is presented. The vertical component of vorticity in a non-rotating fluid is generated by a non-uniform velocity field at the lower boundary of an infinite horizontal fluid layer. Convection in a viscous incompressible fluid is induced by linear heat sources. The main attention is paid to the study of the properties of the flow velocity field. The dependence of the structure of this field on the magnitude of vertical twist

Research Article

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BurmashevaN. V.,Prosviryakov E. Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2019, vol. 23, no. 2, pp. 341-360. doi: 10.14498/vsgtu1670. Authors' Details:

Natalya V. Burmasheva https://orcid.org/0000-0003-4711-1894 Cand. Tech. Sci.; Researcher; Sect. of Nonlinear Vortex Hydrodynamics1; Associate Professor; Dept. of Applied Mathematics and Mechanics2; e-mail: [email protected] Evgeniy Yu. Prosviryakov © https://orcid.org/0000-0002-2349-7801

Dr. Phys. & Math. Sci.; Head of Sector; Sect. of Nonlinear Vortex Hydrodynamics1; Professor; Dept. of Theoretical Mechanics2; e-mail: [email protected]

is investigated. It is shown that, with nonzero vertical twist, one of the components of the velocity vector allows stratification into five zones through the thickness of the layer under study (four stagnant points). The analysis of the velocity field has shown that the kinetic energy of the fluid can twice take the zero value through the layer thickness.

Keywords: exact solution, layered convection, tangential stress, stagnation point, counterflow, stratification, Oberbeck-Boussinesq equation system, vertical twist.

Received: 16th January, 2019 / Revised: 27th March, 2019 / Accepted: 29th April, 2019 / First online: 2nd May, 2019

Introduction. Mathematical models that describe viscous fluid flow are generally based on the Navier-Stokes equations [1-5]. Assumptions regarding specific mass forces involved in these equations make it possible to distinguish regularities that are imperceptible when these equations are considered in general terms. One of the most well-known and widely used assumptions is the linear temperature dependence of fluid density: p = p0 (1 — fiT), where p0 is the average density, ft is the coefficient of volumetric expansion. After substituting the expression relating density and temperature into the Navier-Stokes equation, we obtain the equation of the motion of a viscous fluid in the Boussinesq approximation (the Oberbeck-Boussinesq system) [6-10]. In addition to the velocity vector components, the equations of the Oberbeck-Boussinesq system include scalar pressure and temperature fields. The system of equations is not closed. To close the Navier-Stokes equations and the continuity equation, use the energy equation (heat equation) [6, 11].

The difficulty of finding the exact solutions of the system of differential Oberbeck-Boussinesq equations (partial differential equations) stems from its nonlin-earity due to the presence of a convective derivative in the equations describing pulse transfer and in the heat equation. The properties of the solution are influenced by the boundary conditions, the physical parameters of the fluid and the environmental characteristics [12-14].

A number of interesting flows arising in technical problems and technological processes, for example, a submerged jet [15-17], trail behind the body [18-20], or the flow of fluid or gas from a hole [2,21] belong to the class of so-called shear flows [22-26]. Shear flows have the property that one of the three velocity components is assumed to be zero. In this case, the closed system of equations describing the motion of a fluid becomes overdetermined (the number of equations exceeds the number of unknown functions).

One of the approaches that allow one to solve overdetermined systems arising in mathematical physics when considering shear flows is the construction of generalized classes of exact solutions [27-29], the substitution of which into the system of equations under consideration leads to the identical satisfaction of some of the equations from the Oberbeck-Boussinesq system and reduces the initially nonlinear system of partial differential equations to a simpler system.

These solution families differ, among other things, in the way the vorticity calculated for the selected class behaves. A family of exact solutions for a vectorial velocity field generating no vertical twist was discussed in [30-34]. Taking into

account vertical twist [35-43] changes the form of the particular solution of the boundary value problem and complicates its structure.

This article discusses the effect of constant vertical twist on the topology of the velocity field of the flow in a boundary value problem describing the flow of a fluid in an infinite horizontal layer, induced by tangential stresses specified on the free surface. A comparison is made with the case when the vertical twist is set equal to zero in the selected velocity class.

1. Problem statement. An exact solution to the Oberbeck—Boussi-nesq system. The following system of equations of thermal shear convection in the Boussinesq approximation is considered [30, 31, 35, 36, 44, 45]:

dVx dVx dP - + Vy x dy

dV,, „dV„ dP

Vx—^ + Vy—- = -— + v AVX, dx dy dx

V,-^ + Vy—^ = + vAVy, d x d d

d P

n = (1)

dT dT

V'âx + V' dH = xAT'

dVT dV,

+

dx dy

Here, P is pressure deviation from hydrostatic, divided by constant average fluid density p; T is deviation from the average temperature; v and % are the coefficients of kinematic viscosity and thermal diffusivity of the fluid, respectively; △ = ^ + -^¡2 + j^2! is the Laplace operator.

The system of equations (1) is overdetermined, since it consists of five equations for the determination of four unknown functions Vx, Vy, P, and T. For the solvability of system (1), it is necessary to make sure that the equations involved in it are compatible and to construct exact solutions that are non-trivial. By choosing a class of generalized solutions of a special type, one can achieve identical satisfaction of "extra" equations. It was shown in [30, 31, 46-48] that, for the velocity field

Vx = U (z), Vy = V (z) (2)

the incompressibility equation holds identically. The choice of the class (2) allows system (1) to be reduced to the form

jd2u _ dp dP - ^

d z2 dx' dz2 dy,dz ' (3)

id2T d2T d2T\ ^^dT dT ( )

\ dx2 dy2 d z2 J dx dy'

In the system of equations (3), the number of equations coincides with the number of unknowns.

Ekman was the first to suggest considering solutions in the form (2) for the description of large-scale flows of rotating fluids [49]. An exact solution of the

form (2) generalizes the unidirectional Couette flow [50, 51] and the Birich-Ost-roumov flow [52, 53] in the inertial reference system. Note that the exact solution (2) is not the only family, the substitution of which into the incompressibility equation leads to an identity. Velocities of a more general form [54-56],

Vx = Vx(y,z), Vy = Vy (x,z), (4)

also possess the property under study and allow one to reduce the number of equations in system (1). Substituting class (4) into system (1) also leads to the identical satisfaction of the incompressibility equation:

vdVx _ dP /d2VX d2Vx \ y dy dx V dy2 dz2 /' vdVy = _ dP + v / + s2Vy A x dx dy V dx2 dz2 /' dP

IT = 9ßT, dz

ydT ydT_ f o^T ^T \

x dx y dy v dx2 dy2 dz2)

The procedure of constructing such classes is considered in detail in [8, 30, 31, 34-36, 45, 57, 58]. For the velocity field (4), it is possible, using a number of transformations, to construct exact solutions to the three-dimensional Ober-beck-Boussinesq system (1). In contrast to class (4), for which all the vorticity components Q = rot V,

o =_ ^ o _^ o = ^ (5)

^x — M , oy — M , oz — M M (5)

dz dz dx dy

are non-zero in the general case, the vertical component O^ of vorticity (5) is always zero for the velocity field (2),

_ dV dU 0

dz dz

In other words, the family of velocities (4) can describe vertical spin in a fluid, which occurs without setting rotation at the boundaries of the region in question. The class of exact solutions (4) for the Oberbeck-Boussinesq equation system allows large-scale flows in the equatorial zone of the World Ocean to be studied with the use of the traditional approximation for the angular velocity vector (one Coriolis parameter is used) [38, 44, 55, 56, 59, 60].

We set the task to analyze how the consideration of vertical twist affects the behavior of the flow. For convenience and clarity, we choose a family of exact solutions [35, 36, 44, 61]

Vx — U (*)+ u(z)y, Vy — V (z), (6)

which is a special case of class (4). The vorticity vector components calculated for it take the form

dV dU

O — -—— O — — O — —v

°x — , oy — , oz — a.

dz dz

Class (6) differs from class (2) in that the additional term u(z)y in the expression for the velocity Vx is taken into account. Note that, when u = 0, class (6) degenerates into family (2).

The substitution of class (6) into the system of Oberbeck-Boussinesq equations brings the system (1) to the form

'd2U d2u \ dP

vfd <J d"u \ dP ^ V V dz2 dz2 V dx U ,

^ d2V = dP dP = dz2 dy,dz '

dT 9T_ f d^T d^T d^T)

U dx dy v dx2 dy2 dz2 J'

It follows from the second equation of system (7) that the horizontal pressure gradient dP/dy depends only on the transverse (vertical) coordinate z; therefore, the pressure P can be represented as

P = Pi (x, z) + P2(z)y'

Substituting the partial derivative dP/dx = dP\/dx into the first equation of system (7), we obtain that dP\/dx depends only on z, i.e. the pressure P proves to be linear in the coordinate x. Finally, we arrive at the form

P = Po (z) + Pi (z) x + P2 (z) y. (8)

At the end, we substitute (8) into the third equation of system (7),

dPo dPi dP2 _

d d d

and we find that the temperature T is a linear function of the horizontal coordinates, i.e.

T = To (z) + Ti (z) x + T2 (z) y. (9)

In view of the chosen structure of the temperature and pressure fields (8), (9), the equations of system (1) take the form

fd2U d2u \ „ d2V „

"brr + TP2 y )=Pi +uV, V —2 = P2, d 2 d 2 d 2

dP- + d^x + d^v = gP(T0 + Tix + T2V), (10)

UTi + VT2+uTiy = %( % + x +

The equations of system (10) are equalities of the form

ak (z) + bk (z)x + ck (z)y = 0. (11)

Applying the method of undetermined coefficients, we equate to zero the coefficients at the independent variables x, and the free terms in the polynomial

expressions (11). Inasmuch as all the required functions depend only on z, we denote the derivatives with respect to the coordinate 2 by a prime. As a result, we obtain the following system of equations to determine the unknown components of the hydrodynamic fields (the equations in the system are written in the order of integration):

u" — 0, T[' — 0, P[ — gfiTx, yT% — uT1, P.. — gpT.,

(12)

vV" — P2, uU" — Vu + P1, XT0 — UTl + VT2, P0 — gfjT0.

Note that only the first two equations in system (12) are isolated. After integrating the differential equations in order to determine the functions u and Tl, the exact solutions for which are the linear functions

u — clz + c2, Tl — c3z + c4,

we arrive at a solution for the remaining functions involved in system (1). Hereinafter, we discuss the case of constant vertical twist, setting u — O — const.

2. Boundary value problem. As boundary conditions for the horizontal temperature gradients Tl and T2, the horizontal pressure gradients Pl and P2, the background temperature To, the background pressure Po and the velocities U and V, we select the conditions described in [30, 31]. The absolutely solid bottom surface 2 — 0 of the infinite horizontal layer under study is the reference level of temperature measurement. Without loss of generality, we assume the reference temperature to be zero,

T(x,y, 0) —0. The velocity of the lower boundary 2 — 0 is set as

V, (0) — Oy, Vy (0) — 0.

On the upper undeformed (free) boundary 2 — h, a constant atmospheric pressure is set and, by analogy with temperature setting, it is measured from zero,

P (x,y,h) — 0.

We also assume that a homogeneous field of tangential stresses is specified on the upper boundary as

dVx dU A dVy dV A

Here, is the dynamic viscosity coefficient. Note that, due to the structure of the velocity field V, the resulting tangential stress field, as well as in [30, 31], is homogeneous. In addition, on both boundaries of the fluid layer, heat sources are specified as

T(x, y, 0) — Ax + By, T(x, y,h) — $ + Cx + Dy.

Taking into account the class of generalized solutions (6), (8), and (9), we write the selected boundary conditions as follows:

U (0) — V (0) — 0,

nU'(h)— il, nV'(h)— 6, To(0) — 0, Tl(0)— A, T2 (0) — B, (13)

To(h) — V, Tl (h)— C, T2 (h)— D, Po(h) — Pl(h) — P2 (h) — 0.

In what follows, we will study the velocity field in detail; therefore, the exact polynomial solution of the boundary problem (12), (13) is not completely given. The expressions for the functions T0 and P0 are cumbersome; however, they can be easily obtained by integrating the corresponding equations of system (12). The exact solution of the boundary value problem (12), (13) has the form

C — A nB

u — O; Tl — A + —f^z; Pi — ((C - A)z2 + 2Ahz - (C + A)h2) ;

T2 — B + D - Bz -~^(h - z)z(2Ah + Ch - Az + Cz); h 3! hx

Q0

P2 = -^(h - z)(Bh + Dh - Bz + Dz) +

+ (h - z)2(Ah2 + Ch2 + 2Ahz + 2Chz - Az2 + Cz2); 4! hXK

v = ^ + Ifc. [B(4h3 - 6h2z + 4hz2 - z3) + D(8h3 - 6h2z + z3)] -

Qz rtfiA u5 iri4 , miL^iJ

,, ■ [A(14h5 - 15h4z + 10hV - 6hz4 + z5) + 6! hux

+ C(16h5 - 15h4z + 5h2z3 - /)]; (14)

U = \A(4h3 - 6h2z + 4hz2 - z3) + C(8h3 - 6h2z + z3)! -6! hu

1,5 or»i,3 „2 i 1 r 1,2 „3 ck ~4

■ [5(24h5 - 20h3z2 + 15hV - 6hz4 + z5)+ + D(66h5 - 40h3z2 + 15hV - /)] +

+ 9l3^2z0 |A(528h7 - 392h5z2 + 210hV - 56hV + 24h/ - 3/) + 3 ■ 8! hu2x

+ C(648h7 - 448h5z2 + 210hV - 28h2z5 + 3/)] -

(3h {2 - ) + —.

3! r/v -q

Note that, in view of the exact solution (14), the condition u = 0 determining the degeneracy of class (6) into class (2), is equivalent to the condition Q = 0. Therefore, the effect of the parameter Q on the topology of the velocity field will be further studied in detail.

3. Velocity field analysis. We set the horizontal temperature gradients B = D = 0 in the boundary conditions (13) and the exact solution (14). The flow induced by this heating of the boundaries is a generalization of the unidirectional Birich convective flow [52].

As we pass to the dimensionless coordinate Z = z/h G [0,1], the expressions for the velocities U and V assume the form

V = ahz - gf^z\A(14 - 15Z + 10Z3 - 6Z4 + Z5) + rj 6\vx

+ C(16 - 15Z + 5Z3 -Z5)], (15)

U = 9-^-Z\A(4 - 6Z + 4Z2 - Z3) + C(8 - 6Z + Z3)1 + 6\u

+ ¡¡Pa ^ Z[A(528 - 392Z2 + 210Z3 - 56Z5 + 24Z6 - 3Z7) +

gf3 Q2h7 3^8~\v2x

+ C(648 - 448Z2 + 210Z3 - 28Z5 + 3Z7)] -

Q^ ,

, Z(3 - Z2) + ^Z. (16) 3\r}v V ' 7] '

The velocity field (15), (16) describes the convective flow of a viscous incompressible fluid, which cannot be reduced to unidirectional flow at Q = 0. Thus, the boundary value problem (12), (13) is essentially non-one-dimensional.

Denote by U0 and V0 the velocities (15), (16) calculated in the absence of vertical twist (Q = 0),

V0 = ^Z,

U0 = gPh^Z\A(4 - 6Z + 4Z2 - Z3) + C(8 - 6Z + Z3)1 + ^Z. (17) 6\u f]

Let us now study how the inclusion of the terms containing spatial acceleration Q in Eqs. (15), (16) changes the structure of the flow velocities in comparison with the velocity field V0, U0 when different values of the temperature gradients A and C are specified. We start with the simplest case, namely, the case of a uniform heat source (A = B = C = D = 0).

When a uniform heat source T\ = T2 = 0, T0 = -&Z is set, the velocities U and V are determined by linear functions as

U = U0 = ^Z, V = V0 = ^Z.

Thus, in the direction of both longitudinal axes, the flow is reduced to a combination of unidirectional flows of the Couette type [50], which correspond to a constant field of tangential stresses

dU_r]dU _ dV _r]dV

}~dz = hdZ = ^ Tyz = = hdZ

xz

Moreover, the direction of the vortex O remains unchanged everywhere inside the layer. Hereinafter, it is assumed that the flow is convective, i.e. that A2 + C2 = 0.

We start by analyzing the velocity V (15), since the structure of Eq. (15) is simpler than Eq. (16) determining the velocity U. According to Eq. (15), the velocity V is determined by the superposition of the flow caused by setting the tangential stresses at the upper boundary and two convective flows induced by setting the heat sources. Note that the contribution of each of these flows not only is determined by the values of the parameters A, C, £2, Q, but also depends on the thickness of the layer h. By choosing h, one can make the linear term ^Z prevail over the non-linear terms in the velocity expression (15). Let us consider two limiting cases, A = 0 and C = 0, which allow us to reduce the number of streams contributing to the resultant flow.

Assume that A = 0, then the expression (15) for the velocity V becomes

Vi = ^Z - ^^z(16 - 15Z + 5Z3 - Z5) = V 6! vx

_C9PQh5Z \ *7 5 , 6! met*

Z5 - 5Z3 + 15Z - 16 +

6! vx L CgfrQh4}'

wherefrom it follows that the velocity V can have stagnant points only if the polynomial equation

Z5 - 5Z3 + 15Z + ai = 0,

with some

~ . 6!^x6

ai = -16 +

Cgf rQh4'

has roots within the interval (0,1). The analysis of the solvability of the equation shows that such a root in the (0,1) interval is unique and that it exists only when the control parameters of the problem satisfy the condition

1 < ^X 6 < 144 < Cgf r]Qh4 < 45'

The dependence of the position of the stagnant point of the velocity V on the value of the parameter ai is shown in Fig. 1 (curve 1).

Note that in the case under consideration, for A = 0, there is such a value Zi of the vertical coordinate Z that the tangential stress

* = If = ^[6Z5 - 20Z3 +30Z + .] (18)

vanishes. In this case, the stress changes its type (from tensile to compressive). Such Zi exists only for ai £ [-16,0], i.e. when

vxS.2 1

0 <-———j < —.

Cgf r]Qh4 45

The dependences of the coordinate Z (the zeros of the polynomial (18)) on the parameter ai are shown in Fig. 1 (curve 2).

ai, 02, a3, 04

Figure 1. Dependencies of the position of the stagnant points of the velocities V1, V2, U0, U2 and the critical point of the corresponding stresses TyZ, ry^, tton the parameters a1, a2, a3, a4: curve 1 — the set of points satisfying the condition V1 = 0; curve 2 — the set of points satisfying the condition t^ = 0; curve 3 — the set of points satisfying the condition V2 = 0; curve 4 — the set of points satisfying the condition TyJz = 0; curve 5 — the set of points satisfying the condition U0 = 0; curve 6 — the set of points satisfying the condition t01 = 0; curve 7 — the set of points satisfying the condition U^ = 0; curve 8 — the set of points satisfying the

condition = 0

Consider another limiting case, assuming that C = 0 in (15). Then the velocity V2 = V |C=Q takes the form

V2 = &hZ - nh Z(14 - 15Z + 10Z3 - 6Z4 + Z5) = V 6! vx

A9f^h5r7 {„5 6! "X&

Z

Z5 - 6Z4 + 10Z3 - 15Z + 14 -

6!ux L Agf3i]tth4\

Obviously, the velocity V2 can have stagnant points only if the equation Z5 - 6Z4 + 10Z3 - 15Z + a2 = 0 has solutions inside the layer (0, 1) for some

6! "X&

a0 = \4 — -

2 Agfr/Qh4'

The tangential stress

= V_ dV2 = - MrW4 5 - 30Z4 + 40Z3 - 30Z + 0,2] h aZ 6! vx

corresponding to the velocity V2 can also change its type. The corresponding dependencies are shown in Fig. 1 (curves 3 and 4). Similarly, one can obtain estimates for the control parameters of the boundary value problem, at which the velocity V2 and the tangential stress t?z vanish. We have the following inequality

for the velocity V2:

1 180

<

VXÎ 2

<

7

360 '

For the tangential stress t"2z ,

Agßr] Qh4 it is written as follows:

0 <

VXÎ 2

<

7

Agß r/Qh4 360

Note that, even in these limiting cases, the velocity V can have one stagnant point.

Next, we assume in (15) that A = 0, C = 0,

V =

Cgßüh5 6! vx

Z [Z5 - 5Z3 + 15Z + ai - a(Z5 - 6Z4 + 10Z3 - 15Z + 14)].

Here, a = A/C is a dimensionless parameter. The velocity V can have two stagnant points (Fig. 2).

Thus, in the chosen case of heating of the boundaries ( B = D = 0), the velocity V (15), when the vertical twist (Q = 0) is taken into account, can have up to two zero points.

We now study in a similar way the velocity U (16) of fluid flow along the Ox axis. The expression for the velocity U, in contrast to the velocity V (15), is determined by the superposition of six streams: two streams caused by setting the tangential stresses and £2 and four streams induced by the temperature gradients A and C. Besides, some of these streams are caused by the presence of a vertical twist in the fluid, characterized by the Q parameter. In the same way as

1.0

0.8

— 0.6

Q 0.4

-0.15 -0.1 -0.05

0.05

0.15

Figure 2. The profile of the velocity V for ai = -15.2, a = -1.1

in the analysis of the velocity V (15), we start with the limiting cases, when one of the longitudinal temperature gradients appears to be zero. The vanishing of each temperature gradient reduces the number of flows forming the velocity (16) by two.

Note that, when there is no vertical twist (when Q = 0 and the velocity U is determined by the expression (17)), the type of velocities

U°° = U°1 ,Q = ^f^Z(8 - 6Z + Z3) + №Z = 1 ia=° 6!u r]

= C9fh3 { Z 3 6Z + 8 + \

= ~6~u~Z\ - 6Z + 8 + C^Khl),

U°° = U°L ° = M^Z(4 - 6Z + 4Z2 -Z3) + ^—Z =

2 ]G=° 6\v V ' 7]

= AA^Z(-Z3 + 4Z2 - 6Z + 4+^)

is similar to the form of the velocities V1 and V2 in the sense that any of them is determined by the interaction of two streams, linear and nonlinear. By analogy, it can be shown that the velocities U(° and U°, as well as the corresponding tangential stresses = t®z| a=°, and t{°2 = t®z|c=°, can vanish inside the layer. The location of the stagnation points depends on the combination of the parameters

c^ and (Fig. 1, curves 5 to 8).

Let us now study the effect of the vertical twist Q on the behavior of the velocity U (16) in the limiting cases A = 0 and C = 0. If A = 0, then the velocity U1 = U |a=° takes the form

U1 = Z

CfL ^ - 6Z + Z 3) - ^ (3 -Z 2) + &h + 6!u ?Arqv r]

+ Cgf Q2h? (648 - 448Z2 + 210Z3 - 28Z5 + ?Z7) ? ■ 8! u2x

Cgf3Q2h7 r ~ . , o o , , ? ■ 8! v2xti

Z

?Z' - 28Z5 + 210Z3 - 448Z2 + 648 + „ , 1 +

? ■ 8! v2x L CgfQ2 r]h6

+ 18uX(8 6Z +Z3) 4 ■ 7! vx^2 (? 72)

+ W(8 - 6Z + Z) - CgffQVh4 (? -Z). by virtue of (16).

The polynomials ? Z7 - 28 Z5 + 210Z3 - 448Z2 + 648, 8 - 6Z + Z3, and ? - Z2 involved in the expression of the velocity U1 are strictly monotonic on the domain of definition [0,1], and each coefficient in front of these polynomials contains at least one independent control parameter. The profile of the velocity U\ with three stagnant points is shown in Fig. 3.

When C = 0, it follows from (16) that the velocity U = U|c=° can be transformed in the same way

Z

Figure 3. The profiles of the velocities U1 and U2 when they have three stagnant points

u = z

Aafjh3 (4 - 6Z + 4Z2 - Z3) - ^ (3 - z2) + ^+

6!z/ 3! r¡v r¡

l2h7 v2X Agf№2 h7

+ J'J, 0 (528 - 392Z2 + 210Z3 - 56Z5 + 24Z6 - 3Z7) 3 ■ 8! v2X

Z

-3 Z7 + 24 Z6 - 56 Z5 + 210 Z3 - 392 Z2 + 528+

3 ■ 8! u2X

+ ^(4 - 6Z + 4Z2 - Z3) - í^(3 - Z2) + 3 ■ 8!

Q2h4 v y ^Agf]Qh4 ' Ag/3Q2h7^ J'

whence it follows that U2 also can have up to three stagnant points (Fig. 3).

Thus, in the considered limiting cases ( A = 0 and C = 0) with Q = 0, the velocity U can have no more than one critical point, and when the vertical twist (Q = 0) is taken into account, their number can reach three.

In the case A = 0, C = 0, the structure of the velocity U0 (17) is similar to the structure of the expression (15) for the velocity V; namely, the expression of velocity U0, as well as the velocity V, is determined by the superposition of three flows: two non-linear flows, caused by temperature factors, and one linear flow induced by tangential stresses specified on the upper boundary 2 = h. Consequently, the velocity U0 (17) can have up to two stagnant points.

Let us now analyze how the velocity U determined by the expression (16) is affected by the contribution of the terms caused by the presence of a vertical twist. We write (16) as follows:

U = |"a(4 - + 2 - z3) + C(8 - 6Z + Z3)1 +

+ Q h z I"A(528 - 392Z2 + 210Z3 - 56Z5 + 24Z6 - 3Z7) + 3 ■ 8\ u2x

+ C(648 - 448Z2 + 210Z3 - 28Z5 + 3Z7)1 - —¡^^—Z(3 - Z2) + ^Z =

3\

= z

g/h3 720v

{Ag i(Z )+Cg 2(Z )) +

g/3 Q2h7 120960v2x

{Ag 3 (Z ) + Cg ¿Z )) -

n 6h3 ( z 6h 1

-9 5(Z) +- .

6

(19)

All the polynomials gi{Z) in (19) are strictly monotonic. The study of the spectral properties of the polynomial (19) shows that the velocity (19) can have no more than four stagnant points. The coefficient in front of g5 and the free term are independent due to the arbitrariness of the choice of the control parameters {i and (2; it can be seen from (19) that the coefficients in front of the polynomials 9i, 92, g3, and g4 are related,

gf3Q2h7 gph3 Q2h4

3 ■ 8! u2x 168z/x'

i.e. we have only three independent parameters for all these four coefficients, namely, A, C, Q, and this imposes additional restrictions on the behavior of the function U if we consider the properties of the flow of a particular fluid in a horizontal layer of a fixed thickness h.

Thus, in the case AC = 0, the number of critical points of the velocity U, as in the cases A = 0 and C = 0, increases by two when the vertical twist characterized by the parameter Q is taken into account. The qualitatively different profiles of the velocity U are shown in Fig. 4.

______ 1.0

* **

is

i J % /

* * 0.6 \ / " ■/

ly''

» A4

3 ** *

* ___ ■ ■ I* '1

-0.6

-0.4

-0.2

U

Figure 4. The profiles of the velocity U with different numbers of stagnant points: curve 1 two stagnant points; curve 2 — three stagnant points; curve 3 — four stagnant points

Conclusion. This article provides a new exact solution for the Oberbeck-Boussinesq equations describing the shear convection of a vertically swirling fluid. The resulting exact solution allows you to resolve this overdetermined system. Fluid motion is induced by specifying heat sources at both boundaries of an infinite horizontal layer and taking into account external friction at the free boundary (specifying tangential stresses). It has been demonstrated that no more than two stagnant points can exist in a fluid, although one of the components of the velocity vector can vanish up to four times through the layer thickness.

Competing interests. We declare that we have no conflicts of interest in the authorship

or publication of this contribution.

Authors' contributions and responsibilities. We are fully responsible for submitting

the final manuscript in print. Each of us has approved the final version of the manuscript.

Funding. This work was supported by the Foundation for Assistance to Small Innovative Enterprises in Science and Technology (the UMNIK program, agreement 12281GU/2017).

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ISSN: 2310-7081 (online), 1991-8615 (print)

d https://doi.org/10.14498/vsgtu1670

УДК 532.51, 517.958:531.3-324

Конвективные слоистые течения вертикально завихренной вязкой несжимаемой жидкости. Исследование поля скоростей

Н. В. Бурмашева1'2, Е. Ю. Просвиряков1'2

1 Институт машиноведения УрО РАН,

Россия, 620049, Екатеринбург, ул. Комсомольская, 34.

2 Уральский федеральный университет

им. первого Президента России Б. Н. Ельцина,

Россия, 620002, Екатеринбург, ул. Мира, 19.

Аннотация

Обсуждается разрешимость переопределенной системы уравнений тепловой конвекции в приближении Буссинеска. Система уравнений Обербека-Буссинеска, дополненная уравнением несжимаемости, является переопределенной. Количество уравнений превосходит количество неизвестных функций, поскольку изучаются неоднородные слоистые потоки вязкой несжимаемой жидкости (одна из компонент вектора скорости тождественно равна нулю). Проведено исследование разрешимости нелинейной системы уравнений Обербека-Буссинеска. Исследование разрешимости переопределенной системы нелинейных уравнений в частных производных Обербека-Буссинеска осуществлялось при помощи построения нескольких частных точных решений. Приведен новый класс точных решений для описания трехмерных нелинейных слоистых течений вертикальной завихренной вязкой несжимаемой жидкости. Вертикальная компонента завихренности в невращающейся жидкости генерируется неоднородным полем скоростей на нижней границе бесконечного горизонтального слоя жидкости. Конвекция в вязкой несжимаемой жидкости индуцируется линейными источниками тепла. Основное внимание уделено исследованию свойств поля скоростей течения. Исследована зависимость структуры этого поля от величины вертикальной закрутки. Показано, что одна из компонент вектора скорости при ненулевой вертикальной закрутке допускает расслоение на пять зон по толщине рассматриваемого слоя (четыре застойные точки). Анализ поля скоростей

Научная статья

3 ©® Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru) Образец для цитирования

BurmashevaN. V.,Prosviryakov E. Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2019, vol. 23, no. 2, pp. 341-360. doi: 10.14498/vsgtu1670.

Сведения об авторах

Наталья Владимировна Бурмашева А https://orcid.org/0000-0003-4711-1894 кандидат технических наук; научный сотрудник; сектор нелинейной вихревой гидродинамики1; доцент; кафедра прикладной математики и механики2; e-mail: [email protected]

Евгений Юрьевич Просвиряков © https://orcid.org/0000-0002-2349-7801 доктор физико-математических наук; заведующий сектором; сектор нелинейной вихревой гидродинамики1 ; профессор; кафедра теоретической механики2 ; e-mail: [email protected]

показал, что кинетическая энергия жидкости может дважды принимать нулевой значение по толщине слоя.

Ключевые слова: точное решение, слоистая конвекция, касательное напряжение, застойная точка, противотечение, стратификация, система уравнений Обербека-Буссинеска, вертикальная закрутка.

Получение: 16 января 2019 г. / Исправление: 27 марта 2019 г. / Принятие: 29 апреля 2019 г. / Публикация онлайн: 2 мая 2019 г.

Конкурирующие интересы. Мы заявляем, что у нас нет конфликта интересов в отношении авторства и публикации этой статьи.

Авторская ответственность. Мы несем полную ответственность за предоставление окончательной рукописи в печать. Каждый из нас одобрил окончательную версию рукописи.

Финансирование. Работа выполнена при поддержке фонда содействия развитию малых форм предприятий в научно-технической сфере (программа УМНИК, договор 12281ГУ/2017).