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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki
[J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 3, pp. 528-541 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1770
Mathematical Modeling, Numerical Methods and Software Complexes
MSC: 76F02, 76M45, 76F45, 76R05, 76U05
Convective layered flows of a vertically whirling viscous incompressible fluid. Temperature field investigation
© N. V. Burmasheva1'2, E. Yu. Prosviryakov1
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
The paper discusses a class of exact solutions of the Oberbeck-Boussinesq equations suitable for describing three-dimensional nonlinear layered flows of a vertically swirling viscous incompressible fluid. An inhomogeneous distribution of the velocity field (there is a dependence of the field components on the horizontal coordinates) generates a vertical swirl in the fluid without external rotation (excluding Coriolis acceleration). Setting the linearly distributed heat field and the field of shear stresses at the boundaries of the flow region is one of the reasons inducing convection in a viscous incompressible fluid. The main attention is paid to the study of the temperature field properties. The effect of vertical twist on the distribution of isolines of this field is studied. It is shown that the homogeneous component of the temperature field can be stratified into several zones relative to the reference value, and the number of such zones does not exceed nine. The inclusion of inhomoge-neous components of the temperature field can only decrease this number. It is also demonstrated that the class discussed in the paper allows one to generalize the previously obtained results on modeling convective flows of viscous incompressible fluids.
Research Article
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BurmashevaN. V.,Prosviryakov E. Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Temperature field investigation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 3, pp. 528-541. https://doi.org/10.14498/vsgtu1770. 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 Theoretical 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; e-mail: [email protected]
528
© Samara State Technical University
Keywords: exact solution, layered convection, shear stress, counterflow, stratification, system of Oberbeck-Boussinesq equations, vertical twist.
Received: 22nd January, 2020 / Revised: 23rd July, 2020 / Accepted: 24th August, 2020 / First online: 30th September, 2020
Introduction
One of the main factors causing convection in a viscous fluid is the uneven heating of this fluid. The reasons for the heterogeneity of the temperature field distribution are different, e.g. the presence of a heat source inside the volume occupied by the fluid, heating/cooling of the boundaries of this volume, etc.[1,2].
The energy equation (or the heat equation as one of its simplest versions) is known to depend, besides the physical characteristics of the liquid, on the value of the flow velocity V in view of mixing. Thus, it is necessary to take into account the mutual influence of hydrodynamic fields. Another important case illustrating this dependence is a model of viscous fluid flow based on the Boussinesq hypothesis [2]. This hypothesis suggests a linear relationship between fluid density p and temperature T. As a result, the specific gravity pg appearing in the Navier-Stokes vector equation is substituted by the term gftT, where ft is the volume expansion coefficient, and it is neglected in the inertia forces. In this case, the fluid is considered incompressible. Thus, the relationship between the flow velocity field determined by the velocity vector V and the temperature field T becomes mutual in the sense that both the equation of motion and the heat equation include the components of both fields: the temperature field and the velocity field.
In addition to the Navier-Stokes equation and the heat equation, the constitutive equations for constructing models of viscous fluid mechanics include the law of mass conservation [1-11]. In the case of incompressible fluids, this law is written in the divergent form of the incompressibility equation V-V = 0 [1,2]. The resulting system consists of five scalar equations with respect to five unknowns, namely the components Vx,Vy,VZ of the velocity vector V, pressure P, and temperature T. When considering a number of practically important flows belonging to the class of layered and shear (unidirectional and non-one-dimensional) flows, a problem arises related to the overdetermination of the Oberbeck-Boussinesq system since Vz = 0 for these flows [12-25].
One can resolve such an overdetermined system if, for example, one selects the projections of the velocity vector from a certain generalized class of exact solutions which allows one to satisfy the "unnecessary" equations [12-14,16-19,26,27]. The families of such classes differ, among other things, in that some of them can describe only flows of vertically unvortexed fluids, while others are suitable for modeling flows of fluids with nonzero vertical swirl [12-19, 28-37]. Moreover, taking into account the vertical twist is certain to complicate the structure of the solution to the boundary value problem under study.
The velocity field of convective flows of a vertically swirling fluid was studied in [13,18,19]. It was shown that the vertical vorticity component can exist when the fluid does not rotate. Isothermal flows of this kind were studied in [38,39]. When considering thermal factors, it is important not only to study their influence on the velocity field, but also to evaluate the contribution of the velocity field to the stratification of the temperature field.
This paper considers the exact solution of a boundary value problem describing the convective flow of a viscous fluid under the action of a given field of shear stresses. The effect of a constant vertical swirl on the temperature field is studied, as well as the features of the temperature field distribution depending on the given shear and normal stresses at the boundaries of a horizontal infinite fluid layer.
1. Problem statement. The exact solution of the Oberbeck—Boussinesq system
A system of equations of thermal shear convection in the Boussinesq approximation is considered. For shear flows (the component Vz of the velocity vector V is assumed to be identically equal to zero), this system takes the form [12-15,18,19]:
T. dVE dvx dP , AT. ^dVy dVy dp vx—— + Vy—- = -— + v AVX; + Vy—1 = -— + V AVy;
Ox Oy OX OX Oy Oy /-
dP _dT _dT dVr dV,,, (1)
= g/3T; + = XAT; ^ += 0.
OZ OX oy OX oy
Here, P is the deviation of the pressure from hydrostatic, divided by the constant mean density p of the fluid; T is the deviation from the average temperature; u, x are the coefficients of kinematic viscosity and thermal diffusivity of the fluid, respectively; A = + 2 + -j^ is the Laplace operator.
The system of equations (1) is overdetermined. It was shown in [13,18,19] that, if we consider the flow velocity field of the form
Vx = U (z)+ u(z)y, Vy = V (z), (2)
the incompressibility equation in system (1) is satisfied identically. In this case, the temperature field T and the pressure field P are be described by linear functions of the longitudinal (horizontal) coordinates as
P = PQ (z) + Pi (z) x + P2 (z) y; T = To (z) + (z) x + T2 (z) y. (3)
It was also shown in [13,18,19] that, substituting the families of generalized solutions (2), (3) we can reduce system (1) to the ordinary differential equations system of the following form:
u" = 0, Ti' = 0, Pi = gfiTi, XT2' = uTi, P2 = g$T2, vV" = P2, vU" = Vu + Pi, XT0 = UTi + VT2, P0 = g/3T0. ( )
Moreover, system (4) is integrated uniquely, and it can have a solution different from the trivial one. Here, the prime denotes derivation with respect to the vertical coordinate z. In what follows, we consider the case of constant vertical twist by setting u = Q = const.
2. Boundary value problem
We choose the conditions described in [12, 14,18] as the boundary conditions for the horizontal temperature gradients Ti, T2, the horizontal pressure gradients Pi, P2, the background temperature To, the background pressure P0, and the velocities U and V. We assume that the fluid flows in a horizontal infinite layer,
the lower surface of which = 0 is absolutely solid and selected by the reference level of temperature measurement. Without loss of generality, we assume that the reference temperature is zero,
T(x,y, 0) =0.
The velocity of the lower boundary = 0 is given as
Vx (0) = Qy, Vy (0) = 0.
At the undeformed (free) upper boundary z = h, constant atmospheric pressure is set and, by analogy with the temperature setting, it is counted from zero,
P(x, y, h) = 0.
We also assume that the field of shear stresses is set at the upper boundary as
dVx dU A dVv dV ,
^ = "to = ^ ^ = ^ = ^
Here, r] is the dynamic viscosity coefficient. Note that, due to the structure of the velocity field V, the resulting shear stress field is homogeneous, as in [12, 14, 18]. In addition, thermal sources are set at both boundaries of the fluid layer,
T(x, y, 0) = Ax + By, T(x, y,h) = + Cx + Dy.
In view of the class of generalized solutions (2), (3), the selected boundary conditions are written as follows:
U (0) = F (0) = 0, rjU'(h) = 6, vV'(h)= 6, Tq(0) = 0, Ti(0) = A, 22(0) = B,
(5)
To(h) = #, Ti (h) = C, T2 (h) = D,
Po(h) = Pi(h) = P2(h) = 0.
The exact polynomial solution of the boundary value problem (4), (5) for the velocity field components for the special case B = D = 0 was given and analyzed in [18]; therefore, we restrict ourselves to the exact solution for the temperature field and the pressure field, which has the form
Ti = A + (C -A)Z;
Pi = f ((C — A)Z2 + 2AZ - (C + A));
Qh2 (6)
22 = — ^ (1 — Z)Z ((C + 2A) + (C — A)Z);
P2 = (1 — Z )2 ((C + A) + 2(C + A)Z + (C — A)Z2);
24 X
T° = ^ + X2XZ) {—223760^^^X2h5X
x [5 A2(2 — Z)(1 + 2Z — Z2)(4 — 2Z + Z2)+ + 5AC (26 + 26Z — 30 Z2 + 5 Z3 + 5 Z4 — 2 Z5) +
+ C2 (82 + 82Z + 82 Z2 — 58Z3 + 5Z4 + 5Z5)] — — 3gftr?z/Q2h9 [2A2(2 — Z)(912 + 1368Z + 1596Z2 — 3680Z3+ + 1998Z4 + 294Z5 — 833Z6 + 336Z7 — 42 Z8) + + AC (7851 + 7851Z + 7851Z2 — 27569Z3 + 15397Z4+ + 921 Z5 — 3479Z6 — 14 Z7 + 756Z8 — 168 Z9) + + 2C2 (2088 + 2088Z + 2088Z2 — 4072 Z3 — 607Z4+
+ 1857Z5 — 343 Z6 — 343 Z7 + 42 Z8 + 42 Z9)] — — r?xQ2h9 [A2(2 — Z)(19456 + 29184Z — 9512Z2 — 7080Z3+ + 3838Z4 — 636Z5 — 398Z6 + 216 Z7 — 27 Z8) + + AC (82985 + 82985Z — 23935Z2 — 14035Z3 + 8141Z4 —
— 637Z5 — 637Z6 — 142 Z7 + 243 Z8 — 54 Z9)+ + C2(43268 + 43268Z + 43268Z2 — 10192Z3 — 10192Z4+
+ 4592 Z5 — 358Z6 — 358Z7 + 27 Z8 + 27 Z9)] — — 9979200frV 2ax2 [A(1 + Z — Z2) + C (1 + Z + Z2)] + + 332640h5^ 26xQ [A(3 + 3Z + 3Z2 — 7Z3 + 2Z4) +
+ C (3 + 3Z + 3Z2 — 2Z3 — 2Z4)] + + 332640h5^x2Q [A(14 + 14Z — 16Z2 — Z3 + 2Z4) +
+ C (13 + 13Z + 13Z2 — 2Z3 — 2Z4)]J. (7)
Here, Z = z/h e [0,1] is the dimensionless vertical coordinate.
The expression for the background pressure P° is not given here since it is cumbersome; however, it can be easily obtained by integrating the corresponding equation of system (4) due to the exact solution (7).
Note that the condition u = 0, which determines the degeneracy of the class (2) to the class
Vx = U (*), Vy = V (*),
considered in [12,14,16,17], is equivalent to the condition Q = 0; therefore, the effect of the parameter Q on the temperature field topology will be studied in more detail below.
3. Temperature field analysis
For further convenience, we introduce the functions T-p=°, T^=°, T(p=°, which are obtained from the exact solution (6), (7) when the vertical twist Q proves to be zero. In this case we have
T-p=° = A + (C — A)Z; T2n=° = 0;
T?=° = tfZ + 4h\fv;xZ) {-23760,f3r]h2x
x [5 A2(2 -Z)(1 + 2Z - Z2)(4 - 2Z + Z2)+ + 5 AC(26 + 26Z - 30 Z2 + 5 Z3 + 5 Z4 - 2Z5) + + C2(82 + 82Z + 82Z2 - 58Z3 + 5Z4 + 5Z5)] -
- 9979200v{i [A(1 + Z -Z2) + C(1 + Z + Z2)] }. (8)
3.1. Analysis of the properties of the longitudinal gradients T\,T2.
Note that, regardless of the magnitude of the twist Q, the component Ti = T-p=0 is a monotonic function, and it can take a single zero value in the layer [0,1] only if the inequality
Ti (0) ■ Ti (1) < 0 is valid. This inequality is equivalent to the condition
A C < 0.
Thus, when the longitudinal temperature gradients A and C take values of different signs, the thermal field T\x (and the field T^=°x, respectively) admits stratification at the point Z = A/(A - C).
We now consider the second longitudinal gradient (T2). Obviously, T2}=°y = 0; therefore, this thermal field does not admit stratifications. Let us now study the behavior of the field T2y determined from the expression (6) when Q = 0. It is easy to see that, in the degenerate case A = C, the component T2 takes values of the same sign; therefore, everywhere in the layer, the field T2y is determined by either heating or cooling of the fluid.
Let now the horizontal temperature gradients be different ( A = C); therefore, C -A = 0 and consequently, by virtue of (6), the longitudinal gradient T2 can be represented as
T2 = _Q!«^{i -z)z{Z + a),
6X
where a = (C + 2A)/(C - A). It is easy to verify that the function (1-Z)Z(Z+a) can have a single zero inside the layer [0,1] only when -1 < a < 0. Therefore, the thermal field T2y can change its sign no more than once inside the studied fluid layer.
3.2. Analysis of the properties of background temperature T0.
We now study the features of the behavior of the background temperature. We begin with the case that the vertical vorticity component is zero, Q = 0. In this case, the background temperature is determined by the expression (8). The field T°l=° results from the interaction of several individual thermal fields induced by heating the boundaries of the layer under study and setting the shear stress field at the upper boundary. If both gradients A and C are simultaneously zero, the background temperature, according to (8), is determined only by a homogeneous term (with respect to the horizontal coordinates),
T0n=° = §Z,
the temperature T being unaffected by the value of the components of the shear stress field.
Let us now consider the case that only one of the longitudinal temperature gradients is zero. Assume that, for definiteness, A = 0. In this case, the expression (8) can be represented as
T°
n=o
= z ■ I (Z ) = z
i - *4ft3(1 - Z) X
1008^%
;(2 - Z)(1 + 2Z - Z2)(4 - 2Z + Z2) + 84^(1 + Z - Z2U . (9
It is obvious from (9) that, if some point Z1 e (0,1) is the zero of the auxiliary function f (Z), the stratification of the thermal field T^=0 can occur at this point. Note that the polynomials
fi (Z) = (1 - Z)(2 - Z) (1 + 2Z - Z2) (4 - 2Z + Z2) ,
/2 (Z) = (1 - Z) (1 + Z - Z2) ,
included in the solution (9) are strictly monotonic inside the layer under study. Therefore, the background temperature T$=0 can have no more than two zero points in the layer (0,1) (Fig. 1).
Figure 1. Profile of the temperature T0 0 defined by the expression (9) when A = 0, C = 0
Assume now that the horizontal gradient is nonzero, C = 0. We write the solution (8) as
T
n=° = z ■ F(Z) = z
0 - ^(1 - Z)X 7\ux
X 15c2(2 - Z) (1 + 2Z - Z2) (4 - 2Z + Z2) +
+ 5c(26 + 26Z - 30Z2 + 5Z3 + 5Z4 - 2Z5) +
+ (82 + 82Z + 82Z2 - 58Z3 + 5Z4 + 5Z5) +
+ ^ [' <1 + z - z 2> + <1 + z+z 2
x
where c = A/C is a dimensionless parameter. Obviously, all the zero points of the function F (Z) will automatically be the zero points of the background temperature (8). The structure of the function F (Z), in addition to the above polynomials /1 and /2, includes the polynomials
/3 (Z) = (1 - Z) (26 + 26Z - 30Z2 + 5Z3 + 5Z4 - 2Z5) ,
/4 (Z) = (1 - Z) (82 + 82Z + 82 Z2 - 58Z3 + 5Z4 + 5Z5) , /5 (Z) = (1 - Z) (1 + Z + Z2) .
The functions /3, /4, f5 are also strictly monotonic on the interval [0,1]. There are only four coefficients in front of the polynomials fi (i = 1,6) in the solution (8). All these coefficients can be considered independent of each other due to the arbitrary choice of the values of the shear stress {1, the temperature gradient A, the temperature and the physical constants determining the viscous fluid under study. The analysis of the properties of the polynomial (8) has shown that, in view of these circumstances, the maximum number of zero points of the background temperature (8) does not exceed three (Fig. 2).
z
l.Oh
Figure 2. Profile of the temperature 70 0 defined by the expression (8)
Consequently, the thermal field T,p=0 can both heat and cool the fluid layer; the type of the thermal effect can change no more than three times with the distancing from the lower boundary Z = 0 in the direction of the upper boundary Z = 1.
Now, let the vertical twist Q be nonzero. In this case, according to (7), the terms reflecting the presence of nonzero vorticity in the fluid layer are added to the above-mentioned individual thermal fields of various nature.
It is easy to verify that the number of points at which the background temperature To (7) takes a zero value inside the fluid layer [0,1] does not exceed eleven since the exact solution (7) is an 11th degree polynomial. Moreover, the number of polynomials in the exact solution (7) increases sharply compared with the same number for the thermal field T,p=0. Their number increases to fifteen, all of them are strictly monotonic at Z e [0,1]. However, the number of independent coefficients in front of these polynomials increases to a lesser extent, i.e., only four coefficients are added, which are determined by two new independent parameters, namely the stress {2 and the actual vertical vorticity component Q. The study of the localization of the zeros of the polynomial (7) has shown that their number in the layer [0,1] does not exceed eight (Fig. 3).
Z
Thus, taking into account the constant spatial acceleration u (z) = Q = const leads to a significant increase in the number of zero points of the thermal field T0.
4. Superposition of thermal fields
The resulting temperature field is determined by the interaction of three thermal fields: T\x, T2y, and T0. As a result of their superposition, the number of zero points of the temperature T can change as the values of the longitudinal coordinates x and y change. As an illustrative example, we consider the case that the background temperature has four zero points (Fig. 4). This case corresponds to the value c = -2.04182 of the dimensionless parameter characterizing the ratio of the longitudinal temperature gradients A and C.
Since the ratio A/C proves to be negative, according to the above analysis, the thermal field T\x admits one stratification point (Fig. 5). Herewith, the parameter a = (C + 2A)/(C — A) = -1.01375 determining the presence of zero points of the longitudinal temperature gradient T2 does not fall in the interval (—1,0), and this means the absence of zero points of the gradient T2 (Fig. 5). For definiteness, when constructing the profiles of the temperature field components, the following values of the parameters were taken: C = 1, Qh2/(6%) = 1.
The resulting temperature field isolines T are shown in Figs. 6 and 7.
The change in the location of the isolines is considered as an example of the displacement of the zero isotherm in the characteristic isolines of the sections y = 0 (Fig. 8) and x = 0 (Fig. 9).
Figure 4. Profile of the temperature To with Figure 5. Profile of temperature gradients Ti four zero points (solid line) and T2 (dashed line)
Figures 8, and 9 clearly illustrate the change (decrease) in the number of stratification points of the final temperature field T, even for small values of the longitudinal coordinates x and y, compared to the number of zero points of the background temperature T0 marked by a bold line in Figs. 8 and 9.
Figure 6. Isolines of the temperature T in the section y = 0 z
1,0, o
U.Ö
/ 0.4 )
/' oV - ;
Figure 8. Zero isotherm in the section y = 0 for x = 0.9, x = 0.3, x = 0, x = -0.3 and x = -0.9 (from left to right)
Figure 7. Isolines of the temperature T in the section x = 0 z
Figure 9. Zero isotherm in the section x = 0 for y = -0.9, y = -0.3, y = 0, y = 0.3 and y = 0.9 (from left to right)
Competing interests. We declare that we have no conflicts of interest in the authorship and publication of this article.
Authors' contributions and responsibilities. We are fully responsible for submitting the final manuscript in print. Each of us has approved the final version of manuscript. Funding. The research has no funding from any party.
References
1. Landau L. D., Lifshits E. M. Course of Theoretical Physics, vol.6, Fluid Mechanics. New York, Pergamon Press, 1959, 539 pp.
2. Gershuni G. Z., Zhukhovitskii E. M. Convective Stability of Incompressible Fluids, Israel Program for Scientific Translations. Jerusalem, Keter Publishing House, 1976, 330 pp.
3. Kochin N. E., Kibel I. A., Roze N. V. Theoretical Hydromechanics. New York, Wiley Interscience, 1964, 577 pp.
4. Falkovich G. Fluid Mechanics: A Short Course for Physicists. Cambridge, Cambridge University Press, 2011, xii+180 pp. https://doi.org/10.1017/cbO9780511794353.
5. Navier M. Mémoire sur les lois du mouvement des fluides, Mém. Acad. Sci. Inst. France, 1827, vol. 6, pp. 389-440.
6. Stokes G. G. On the theories of internal friction of fluids in motion, and of the equilibrium and motion of elastic solids, In: Mathematical and Physical Papers, vol. 1. Cambridge, Cambridge University Press, 2009, pp. 75-129. https://doi.org/10.1017/cbo9780511702242. 005.
7. Drazin P. G. Introduction to hydrodynamic stability. Cambridge, Cambridge University Press, 2002, xviii+258 pp. https://doi.org/10.1017/CBO9780511809064.
8. Chandrasekhar S. Hydrodynamic and hydromagnetic stability. New York, Dover Publications, 1981, 708 pp.
9. Poisson M. Mémoire sur les équations générales de l'équilibre et du mouvement des corps solides élastiques et des fluides, Journ. de l'Ecole Polytechn., 1831, vol. 13, pp. 1-174.
10. Lin C. C. Note on a class of exact solutions in magneto-hydrodynamics, Arch. Rational Mech. Anal., 1858, vol. 1, pp. 391-395. https://doi.org/10.1007/BF00298016.
11. de Saint-Venant B. Note à joindre au Mémoire sur la dynamique des fluides, Comptes rendus, 1843, vol. 17, no. 22, pp. 1240-1244.
12. Burmasheva N. V., Prosviryakov E. Yu. A large-scale layered stationary convection of an incompressible viscous fluid under the action of shear stresses at the upper boundary. Velocity field investigation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2017, T. 21, №1, C. 180-196 (In Russian).
13. Burmasheva N. V., Prosviryakov E. Yu. Temperature field investigation in layered flows of a vertically swirling viscous incompressible fluid under two thermocapillar forces at a free boundary, Diagnostics, Resource and Mechanics of Materials and Structures, 2019, no. 1, pp. 6-42 (In Russian). https://doi.org/10.17804/2410-9908.2019.1.006-042.
14. Burmasheva N. V., Prosviryakov E. Yu. A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and presure field investigation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2017, vol. 21, no. 4, pp. 736-751 (In Russian). https://doi.org/10.14498/vsgtu1568.
15. Burmasheva N. V., Prosviryakov E. Yu. Investigation of a velocity field for the Marangoni shear convection of a vertically swirling viscous incompressible fluid, AIP Conf. Proc., 2018, vol. 2053, 040011. https://doi.org/10.1063A.5084449.
16. Burmasheva N.V., Prosviryakov E. Yu. Exact solution for the layered convection of a viscous incompressible fluid at specified temperature gradients and tangential forces on the free boundary, AIP Conf. Proc., 2017, vol. 1915, 040005. https://doi.org/10.1063/1.5017353.
17. Burmasheva N.V., Prosviryakov E. Y. Exact solutions for layered large-scale convection induced by tangential stresses specified on the free boundary of a fluid layer, IOP Conf. Ser.: Mater. Sci. Eng., 2017, vol.208, 012010. https://doi.org/10.1088/1757-899X/208/ 1/012010.
18. Burmasheva N. 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. https://doi.org/10.14498/vsgtu1670.
19. Burmasheva N. V., Prosviryakov E. Yu. Thermocapillary convection of a vertical swirling liquid, Theor. Found. Chem. Eng., 2020, vol. 54, no. 1, 230-239. https://doi.org/10.1134/ S0040579519060034.
20. Liapidevskii V. Y. A mixing layer in a homogeneous fluid, J. Appl. Mech. Tech. Phys., 2000, vol.41, no. 4, pp. 647-657. https://doi.org/10.1007/BF02466862.
21. Georgievsky D. V. Tensor-nonlinear shear flows: Material functions and the diffusionvortex solutions, Rus. J. Nonlin. Dyn., 2011, vol.7, no. 2, 451-463. https://doi.org/ 10.20537/nd1103004.
22. Taylor G. I. The transport of vorticity and heat through fluids in turbulent motion, Proc. Roy. Soc. London. Ser. A, 1932, vol.135, no. 828, pp. 685-705. https://doi.org/10.1098/ rspa.1932.0061.
23. Kuznetsova Yu. L., Skul'skiy O. I. Effect of different flows on the shear branding of a liquid with a non-monotonic flow curve., J. Appl. Mech. Tech. Phys., 2019, vol. 60, no. 1, pp. 22-30. https://doi.org/10.1134/S0021894419010048.
24. Rodi W. Hydrodynamic and Hydromagnetic Stability. Boca Raton, CRC Press, 1973, 124 pp.
25. Georgievsky D. V. Generalized Joseph Estimates of Stability of Plane Shear Flows with Scalar Nonlinearity, Bull. Russ. Acad. Sci. Phys., 2011, vol.75, no. 1, 149-152. https://doi.org/10.3103/S1062873810121044.
26. Knutova N. S., Shvarts K. G. A study of behavior and stability of an advective thermocap-illary flow in a weakly rotating liquid layer under microgravity, Fluid Dyn., 2015, vol. 50, no. 3, pp. 340-350. https://doi.org/10.1134/S0015462815030047.
27. Chikulaev D. G., Shvarts K. G. Effect of rotation on the stability of advective flow in a horizontal liquid layer with solid boundaries at small Prandtl numbers, Fluid Dyn., 2015, vol.50, no. 2, pp. 215—222. https://doi.org/10.1134/S0015462815020052.
28. Sidorov A. F. Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory, J. Appl. Mech. Tech. Phys., 1989, vol. 30, no. 2, pp. 197203. https://doi.org/10.1007/BF00852164.
29. Knyazev D. V. Two-dimensional flows of a viscous binary fluid between moving solid boundaries, J. Appl. Mech. Tech. Phys., 2011, vol.52, no. 2, pp. 212-217. https://doi.org/10.1134/ S0021894411020088.
30. Schwarz K. G. Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries, Fluid Dyn., 2014, vol.49, no. 4, pp. 438-442. https://doi.org/10.1134/ S0015462814040036.
31. Brutyan M. A., Kovalev V. E. Vortex flows of a micropolar fluid, TsAGI Sci. Journal, 2010, vol.41, no. 4, pp. 52-61 (In Russian). https://doi.org/10.1615/TsAGISciJ.v41.i4.60.
32. Aristov S. N., Prosviryakov E. Y. Stokes waves in vortical fluid, Rus. J. Nonlin. Dyn., 2014, vol.10, no. 3, pp. 309-318. https://doi.org/10.20537/nd1403005.
33. Nikulin V. V. Analytical model of motion of turbulent vortex rings in an incompressible fluid, J. Appl. Mech. Techn. Phys., 2014, vol.55, no. 4, pp. 558-564. https://doi.org/10.1134/ S0021894414040026.
34. Kovalev V. P., Sizykh G. B. Axisymmetric helical flows of an ideal fluid, Trudy MFTI, 2016, vol. 8, no. 3, pp. 171-179 (In Russian).
35. Brutyan M. A., Krapivskiy P. L. The exactsolution of the Navier-Stokes equations for the evolution of the vortex structure in a generalized shear-flow, Comput. Math. Math. Phys., 1992, vol. 32, no. 2, pp. 270-272.
36. Morozov K. I. Rotation of a droplet in a viscous fluid, J. Exp. Theor. Phys., 1997, vol. 85, no. 4, pp. 728-733. https://doi.org/10.1134/1.558360.
37. Greenspan H. P. The Theory of Rotating Fluids. Cambridge, Cambridge University Press, 1968, xii+328 pp.
38. Aristov S. N., Prosviryakov E. Y. Large-scale flows of viscous incompressible vortical fluid, Russ. Aeronaut., 2015, vol.58, no. 4, pp. 413-418. https://doi.org/10.3103/ S1068799815040091.
39. Privalova V. V., Prosviryakov E. Yu., Simonov M. A. Nonlinear gradient flow of a vertical vortex fluid in a thin layer, Rus. J. Nonlin. Dyn., 2019, vol. 15, no. 3, pp. 271-283. https:// doi.org/10.20537/nd190306.
Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. 2020. Т. 24, № 3. С. 528-541 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1770
УДК 532.51, 517.958:531.3-324
Конвективные слоистые течения вертикально завихренной вязкой несжимаемой жидкости. Исследование температурного поля
© Н. В. Бурмашева1'2, Е. Ю. Просвиряков1
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
Приведен класс точных решений уравнений Обербека—Буссинеска, подходящих для описания трехмерных нелинейных слоистых течений вертикально завихренной вязкой несжимаемой жидкости. Неоднородное распределение поля скорости (имеет место зависимость компонент поля от горизонтальных координат) генерирует вертикальную закрутку в жидкости без внешнего вращения (без учета Кориолисова ускорения). Задание на границах области течения линейно распределенных теплового поля и поля касательных напряжений является одной из причин, индуцирующих конвекцию в вязкой несжимаемой жидкости. Основное внимание уделено исследованию свойств температурного поля. Изучено влияние вертикальной закрутки на распределение изолиний этого поля. Показано, что однородная составляющая температурного поля может стратифицироваться на несколько зон относительно отсчетного значения, причем число таких зон не превосходит девяти. Учет неоднородных составляющих поля температуры может приводить только к уменьшению этого числа. Также показано, что представленный в статье класс позволяет обобщить ранее полученные результаты по моделированию конвективных течений вязких несжимаемых жидкостей.
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
Burmasheva N. V., Prosviryakov E. Yu. Convective layered flows of a vertically whirling viscous incompressible fluid. Temperature field investigation, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 3, pp. 528-541. https://doi.org/10.14498/vsgtu1770.
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 теоретической 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; e-mail: [email protected]
540
© Самарский государственный технический университет
Ключевые слова: точное решение, слоистая конвекция, касательное напряжение, противотечение, стратификация, система уравнений Обер-бека—Буссинеска, вертикальная закрутка.
Получение: 22 января 2020 г. / Исправление: 23 июля 2020 г. / Принятие: 24 августа 2020 г. / Публикация онлайн: 30 сентября 2020 г.
Конкурирующие интересы. Мы заявляем, что у нас нет конфликта интересов в отношении авторства и публикации этой статьи.
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