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0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 2, pp. 167-186. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230201
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 76F02, 76F45, 76M45, 76R05, 76U05
An Inhomogeneous Steady-State Convection of a Vertical Vortex Fluid
S. A. Berestova, E. Yu. Prosviryakov
An exact solution of the Oberbeck-Boussinesq equations for the description of the steady-state Benard - Rayleigh convection in an infinitely extensive horizontal layer is presented. This exact solution describes the large-scale motion of a vertical vortex flow outside the field of the Coriolis force. The large-scale fluid flow is considered in the approximation of a thin layer with nondeformable (flat) boundaries. This assumption allows us to describe the large-scale fluid motion as shear motion. Two velocity vector components, called horizontal components, are taken into account. Consequently, the third component of the velocity vector (the vertical velocity) is zero. The shear flow of the vertical vortex flow is described by linear forms from the horizontal coordinates for velocity, temperature and pressure fields. The topology of the steady flow of a viscous incompressible fluid is defined by coefficients of linear forms which have a dependence on the vertical (transverse) coordinate. The functions unknown in advance are exactly defined from the system of ordinary differential equations of order fifteen. The coefficients of the forms are polynomials. The spectral properties of the polynomials in the domain of definition of the solution are investigated. The analysis of distribution of the zeroes of hydrodynamical fields has allowed a definition of the stratification of the physical fields. The paper presents a detailed study of the existence of steady reverse flows in the convective fluid flow of Benard - Rayleigh - Couette type.
Keywords: exact solution, shear flow, inhomogeneous flow, convection, Oberbeck-Boussinesq system, class of Lin-Sidorov-Aristov solutions, vertical swirl of fluid, reverse flow, stratification
Received May 15, 2022 Accepted December 16, 2022
Svetlana A. Berestova s.a.berestova@urfu.ru
Ural Federal University
ul. Mira 19, Ekaterinburg, 620002 Russia
Evgenii Yu. Prosviryakov evgen_pros@mail.ru
Ural Federal University
ul. Mira 19, Ekaterinburg, 620002 Russia
Institute of Engineering Science of Ural Branch of the Russian Academy of Sciences ul. Komsomolskaya 34, Ekaterinburg, 620049 Russia
Introduction
The study of various natural, technological, biological and other processes is based on the application of the model of a dropping fluid [1-3]. The first equations for describing fluid motion were proposed by Euler [4]. These equations described the flows of the so-called ideal fluid. Further experimental and theoretical investigations of flows illustrated an inaccurate description by means of the Euler equations. The next step in the study of the flows of incompressible media was the derivation of equations which would take into account internal friction forces. In other words, mechanics began to consider continual media which are characterized by viscosity due to internal friction distributed according to the Newton law [1-3]. To describe the flows of such viscous fluids, use is made of the Navier-Stokes vector equation supplemented with a scalar continuity equation.
In the world's scientific literature, the year 1822 is regarded as the year of birth of the Navier -Stokes equations, although the derivation of these equations in modern form was performed in the 1820s-1850s. At present, it may be stated that the greatest contribution was made by Navier [5, 6], Poisson [7], Saint-Venant [8], and Stokes [9]. The Navier-Stokes equations turned out to be an extraordinarily difficult mathematical object of study. The main difficulty involved in solving these equations stems from the quadratic nonlinearity of the equations due to a convective derivative and a strong dependence on the initial and boundary conditions [10, 11]. Nowadays the Navier-Stokes equations find the widest application in problems of mathematics, mechanics, physics and various applied sciences.
In the middle of 2000, the Clay Mathematical Institute formulated seven so-called Millennium Prize Problems. In this list, number (section) six was assigned to the Navier-Stokes equation [12, 13]. The problem was formulated by the US mathematician C.Fefferman under the title "Navier-Stokes, Existence and Regularity" [12, 13]. The goal is to prove the existence and uniqueness of the solution for the Cauchy problem and for boundary conditions periodic in spatial variables (coordinates). Soon after publication of the Millennium Prize Problems, O. A. Ladyzhenskaya [14] set forth her view on finding a solution and a proof of their existence and uniqueness. She rightly pointed out that the focus of [12, 13] was on the so-called physically meaningful solutions (infinitely smooth functions), whereas she suggested seeking generalized solutions to use integral identities and estimates [14]. Despite the fact that O. A. Ladyzhenskaya offered a plan in [14] for solving the posed problem in order to automatically satisfy C.Feffer-man's formulation, multiple attempts at solving the Sixth Millennium Prize Problem have failed so far.
Over the last few decades, many interesting papers dealing with the mathematical properties of the Navier-Stokes equations have been published. Below we give references to papers that are, in our opinion, the most significant in that they allow the Sixth Millennium Prize Problem to be solved. Soon after publication of [14], a team of authors published studies for two-dimensional hydrodynamics [15, 16]. In those studies, the ideas of O. A. Ladyzhenskaya were developed significantly further. It is important to note that the investigation of the Navier-Stokes equations by analytical and numerical methods is necessary for many applications. Therefore, the particular results published in [2, 17-27, 29-35] are of interest. In [17-28], the theoretical results have been developed which will prepare the scientific community for solving the Sixth Millennium Prize Problem. The scientific works [29-35] present interesting results for applied problems, in particular, for working with the equations describing turbulence and nonclassical strongly stratified media. We note that, for the Navier-Stokes equations and their numerous modifications, no
general reliable algorithm has been developed so far which guarantees the existence of a unique "mathematical" and "physical" exact or approximate solution.
The application of the Navier-Stokes equations allows one to describe the processes of heat transfer in a fluid both by energy dissipation and by heat gain [1, 2, 36]. The study of convective fluid flows is one of the central problems of hydrodynamics [1-3]. This is due to the fact that convection in a fluid can be induced by different reasons, but is most frequently caused by thermal factors [1-3]. To study the motions of nonisothermic fluid flows, the Oberbeck-Boussinesq system has been obtained from the Navier-Stokes equations. Integration of this system is much more difficult because it has heat conductivity equations [1-3, 37, 38].
Attempts at finding exact solutions to the equations of fluid motion were made by Euler, the founder of hydrodynamics [4, 5]. He sought solutions to equations, called later by his name, in the form of polynomials by restricting himself to the fifth degree. His attempts were reflected in the subsequent constructions of exact solutions to the Navier-Stokes equations. A store of exact solutions to the Navier-Stokes equations began to increase gradually. For the history of exact solutions of the Navier-Stokes equations, we refer the reader to the bibliographical reviews and monographs [1, 5, 10, 11, 37-40].
Finding exact solutions to the Oberbeck-Boussinesq system is much more difficult. The first exact solution to convection equations is described by the Ostroumov-Birikh class [41, 42]. This solution was later published again in [43, 44]. The Ostroumov-Birikh family, which describes one-directional fluid flows, turned out to be very useful in solving the problem of convective hydrodynamical stability [45-47]. In the scientific works [27, 28, 45-52], various modifications and generalizations of the class of exact Ostroumov-Birikh solutions are presented.
Regarding the classical exact Ostroumov-Birikh solution for an infinitely extensive horizontal layer, it should be noted that there exists an exact solution for the infinite vertical Gershuni-Batchelor flow. As far as the structure of these flows is concerned, they differ in the direction of the vector of free-fall acceleration, which leads to different exact solutions (from the viewpoint of mathematical tools and physical interpretation) for the Ostroumov-Birikh and Gershuni-Batchelor problems. In this paper we consider generalizations for the Ostroumov-Birikh family since the Gershuni-Batchelor class under the condition that the flow is closed describes up and down flows, which makes the work with the one-directional flow difficult. To be fair, it should be noted that in recent years the term one-dimensional flow has been used to refer to a one-dimensional velocity field. This fact unifies the classes of exact Ostroumov-Birikh and Gershuni-Batchelor solutions and allows them to be considered from a unified point of view.
An exact Ostroumov-Birikh type solution with a two-dimensional homogeneous velocity field and three-dimensional pressure and temperature fields which do not reduce to a one-directional flow, but are laminar was first presented in [53]. This result was a starting point for finding exact solutions that describe inhomogeneous shear flows. The first exact solution describing the inhomogeneous isobaric flow of Couette type for different boundary conditions was first announced in [54-57]. It was shown that taking into account the inhomogeneity of the velocity field entails the presence of a vertical swirl of fluid outside the Coriolis field. In this case the increase in fluid velocities can occur and there exist reverse flows. In [58-60], the influence of pressure gradients on the inhomogeneous flows of a vertical vortex flow was studied.
In [54-60], use was made of an inhomogeneous velocity field which satisfies exactly the Navier-Stokes equations, to solve the problem of convection of a vertical vortex flow. Using an exact solution for the velocity field [54-60], an inhomogeneous Couette flow was considered in [61] in specifying the temperature gradients on the upper boundary. It was shown that the motion of the boundary leads to intensification of heat conduction from the heated boundary.
In [62], the classical exact Napolitano solutions for the Marangoni convection were generalized. The papers [63, 64] are devoted to the study of convective flow in specifying tangential stresses on the known free boundary of an infinite thin horizontal layer.
In this paper we investigate the steady-state Bénard convection for the inhomogeneous shear flows of a vertical vortex flow. In [61-64], no analysis was made of the convection induced by heating a thin fluid layer from below. Given that an exact solution was constructed in [61-64] for a new velocity field different from the Ostroumov-Birikh class [41, 42] and from the Napolitano family [49], the investigation of the steady flow is necessary not only for theoretical and experimental needs, but also for the study of new types of hydrodynamical stability.
1. Equations describing the large-scale stationary flow
1.1. Formulation of the problem of the fluid flow in a flat layer
Consider the large-scale flow of a heat-conducting viscous incompressible fluid (Fig. 1). By the large-scale flow we mean the shear flow of a fluid with the geometric anisot.ropy index ô ^ Here 5 = j, where h and I are, respectively, the typical scale of flow along the vertical (layer thickness) and along the horizontal.
T = const, p = const
©-
Vr
K,
_A_T = Ax + By _
/777777o^777777777777777777777t
Fig. 1. Diagram of shear convective fluid motion The steady flow is described by the Navier-Stokes equation [1] in the vector form
(V ■ V)V = -VP + vA V + gßTk,
(1.1)
where V = + j-^ + is the Hamilton operator, i, j and k are the unit vectors of the
rectangular Cartesian coordinate system, A = j^r + Jp- + is the Laplace operator, V = Vxi.+
+ Vyj + Vzk is the velocity vector, P is the deviation of pressure from the hydrostatic pressure divided by the constant average fluid density p, v is the coefficient of kinematic (molecular) viscosity, T is the deviation from the average temperature, g is the free-fall acceleration, and /3 is the temperature coefficient of volumetric expansion of the fluid.
The temperature and velocity fields in the moving medium are a consequence of thermal and mechanical interactions and, strictly speaking, cannot be considered separately from each other. Furthermore, the temperature field always depends on the velocity field. We will describe the heat exchange in a fluid using the heat conductivity equation
V -VT = x^T,
(1.2)
where % is the coefficient of thermal conductivity of the fluid. In Eq. (1.2) we neglect energy dissipation.
We close the system for the incompressible fluid by the continuity equation, which takes the
form
V-V = 0.
(1.3)
The scalar form of the equations of the system (1.1)-(1.3) describing the motion of a viscous incompressible fluid, which takes into account the influence of the temperature and pressure on its flow gives the following system of equations:
dVx dVx dVx
dx
dy
dVy dVy
2 dz
dV
y
dx
dy
dz dK
dVz dVz V —£ + V —£ + V dx y dy z dz
„ dT dT dT dVx dVy dVz
dP dx
dP dy
dP dz
dx
+
dy ^ dz
'd 2 Vx d2Vx d 2Vx
_ dx2
dx2 'd 2 Vz
+
+
dy2
d'% 2
dy
+
+
d2vz
H--£ +
X
0.
dx2 dy d 2 T d 2 T d2T\
dz2
dz2
cPV^ dz2
dx2 ^ dy2 + dz2 )
+
+ gpT,
(1.4)
Consider the steady convective shear flow of a viscous incompressible fluid in the horizontal planes (Vz = 0), which is due primarily to the horizontal components of the temperature gradient in contrast to the classical Couette flow, when a limitation of the flow of the moving fluid in one direction takes place. We will find an analytic exact solution of the system of equations (1.4) in the case where the velocity and the deviations of the temperature and pressure of steady motion are given as follows [54]:
Vx = U (z)+ yu(z),
(1.5)
Vy = V (z),
P = Po(z)+ xPi(z)+ yP2(z), T = To(z)+ xTi(z)+ yT2(z).
All coefficient functions of linear forms (1.5): U(z), u(z), V(z), P0(z), P\(z), P2(z), T0(z), Tl(z) and T2(z) depend only on the transverse coordinate z. Such a structure of hydrodynamical fields makes it possible to construct an exact solution to the system of equations (1.4) which describes the convective flows of a swirling dissipative medium. In this case, account is taken of the influence of inertial forces on the transfer of momentum and temperature in the fluid. The multiplier at the coordinate y is responsible for conservation of the inertial forces. When the condition u = 0 is satisfied, the convective derivative in the equations of motion is different from zero only in the heat conductivity equation [53]. Substituting the expressions (1.5) into the
system (1.4), we obtain
'(I2U{z) , __ d2u(z) ^ dz'2 d2v{z) dz2
" ( ) - p^ ~ =
- P2 (z) = 0,
(1.6)
f N m f \ m f \ \ ( dP0(z) dP, (z) dP2(z)
g/3(T0(z) + xT^z) + yT2(z)) - {—¡jr1 + + y^t^
,TT, ^ , , N id2T0(z) d2T,(z) d2TJz)\
(U(z) + yu(z))2\(z) + V(z)T2(z) - x + + =
As a result, we have a system of differential fifteenth-order equations for finding unknown functions:
d2u{z)
dz2
(PT^z)
dz2
dP^z) dz
= 0, = 0,
= gm (z),
X^f1 = u(z)T1(z),
d2T2(z)
dz2
dP2(z)
dz fVjz)
dz2
d2U(z)
dz2
= gT (z)), (1.7)
= P2(z),
= V (z)u(z) + Pl (z),
d%(z)
dz2
dPo(z)
= UWT^z) + V(z)T2(z),
dz
= gfiTQ (z).
The equations of the system (1.7) have been written taking into account the order in which their integration is performed. The general dimensional solution of the system (1.7) is written by polynomial functions. It is obvious that by the linear law only the functions u(z) and T,(z) are distributed. They are first-degree polynomials. P, (z) changes according to the quadratic law. T2(z), P2(z), V(z), U(z) and T0(z) are polynomials of degrees four, six, seven, ten and thirteen, respectively. The pressure P0 has the highest, fourteenth, degree.
1.2. Formulation of the boundary-value problem of shear flow
To determine the constants of integration of the exact solution (1.7) , it is necessary to define the boundary conditions (Fig. 1). For the velocities on the lower nondeformable boundary, when z = 0, the following adhesion condition is satisfied:
U (0) = 0, V (0) = 0, u(0) = 0. (1.8) _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2023, 19(2), 167 186_
On the upper boundary with layer thickness h, the following inhomogeneous velocity distribution is given:
z = h: U(h) = W cos <, V(h) = W sin <, u(h) = Q, (1.9)
where < is the angle between the velocity vector on the upper boundary and the abscissa axis. The pressure on the free boundary is
Po (h)= 5, Pi(h) = 0, P2(h) = 0, (1.10)
where S is the atmospheric pressure. Under the boundary conditions the pressure gradient is assumed to be equal to zero for the nondeformability condition of the upper free boundary to be valid.
Consider the heating/cooling on the lower boundary at zero reference temperature. On the upper boundary, the zero reference temperature and the zero horizontal temperature gradient of the fluid are given. Then for the temperature the boundary conditions can be written as
z = 0: To(0)=0, Ti(0)= A, T2(0)= B; (1.11)
z = h: T0(h) = 0, T1(h) = 0, T2(h) = 0. (1.12)
2. Hydrodynamical fields of a vertical vortex fluid
2.1. Functions defining hydrodynamical fields
The solution of the boundary-value problem posed in this paper is one of the exact solutions of the Navier-Stokes equations in the Boussinesq approximation. Although it has polynomial form, it is rather cumbersome. For ease of further analysis we write the functions defining the hydrodynamical fields by transforming to dimensionless coordinates: Z = Y =
Integration yields that the velocity field is defined both by the background velocity and by the spatial velocity gradient:
u(Z) = QZ,
= _ gf3AQh5 ^ , 6 _ 5 3 _ , _
v ; 10,080xv V ;
gfiBh 24v
3
Z (Z3 - 4Z2 + 6Z - 3) + W sin <Z,
U(Z) =--y>MQ2!1' Z (8Z9 _ 35Z8 + 150Z6 _ 756Z4 + 885Z3 _ 252) _
v ; 1814 • 103xt/2 v ;
- gf3mh2oZ (10Z6 - 56Z5 + 126Z4 - 105Z3 + 25) -10,080^2 v ;
gPAh3^ .3 Ar72 , ^ ^ , WQsin<h2 .3
Z (Z3 - 4Z2 + 6Z - 3) + " Z (Z3 - 1 )+W cos <pZ
24^ ' 12v
The components of the function which define the temperature field of the viscous fluid considered have the form
T0(Z) = 1Qiox2y2Z (1261 ■ 1q3z12 - 801 • 104Zn + 7824 • 103Z10 + 4098 • 104Z9-
-7172 • 104Z8 - 3319 • 105Z7 + 9607 • 105Z6 - 7254 • 105Z2 - 5164 • 105Z3 + 1033 • 106Z2-
-4106 • 105) + gl3A ^J,1 ,Z (3343 • 103Z12 - 2173 • 104Zn + 3319 • 104Z10 + 2825 • 104Z9-y 3312 • 1012 vx3
-1521 • 105Z8 - 1956 • 105Z7 + 716 • 106Z6 - 5129 • 105Z5 - 4213 • 104Z4 + 5129 • 105Z3-
-1176 • 105) + y/3BAQh z (4Zg - 33Z8 + 117Z7 - 198Z6 + 126Z5 + 75Z3 - 150Z2 + 59) + ' 362,900xt/ V '
gpABtth ,zg _ g5z8 27qzj _ 325z6 35Z? i8gz4 g5Z-3 _ 2g5z2 . _ 3,024,004v%2 v '
AW Q sin <h4 2
Z/
5040%v
(lOZ6 - 14Z5 - 35Z3 + 70Z2 - 31) + - (lOZ6 - 28Z5 + 35Z3 - 17)
'X
+ {A2 + B2)gf3h5z ,6 _ i4z5 + 42z4 _ 63z3 + 42z2 _ , _ 2016v% v y
W Q (A cos < + B sin <) h2 ,3 .
Z(Z3 - 2Z2 + 1),
12x
Ti(Z)= A(1 - Z), AQh2
T2(Z) = - 2Z'2 + 1) + B{1 - Z). (2.2)
The functions defining the pressure field in the case of the laminar flow of a viscous fluid in a thin layer are also defined by the combination of polynomials
g2 o2 A2Q2 h 10
Po{Z) = -9 P 2 (7955 • 106Z14 - 544 • 108Z13 + 5757 • 107Z12 + 329 • 109Zn-
394 • 10 V V
-4523 • 108Z10 - 3257 • 109Z9 + 106 • 1011Z8 - 915 • 1010Z7 - 9119 • 109Z5 + 228 • 1011 Z4-
g2 o2BAQh,8
-1813 • 10loZ2 + 6365 • 109) + —-r—T (80Z11 - 66Z10 + 2860Z9 - 5445ZS+
y 7983 • 104 xv2
+3960Z7 + 3300Z5 - 8250Z4 + 6490Z2 - 2269) - 9^Wilsm<f>h5 , 8_ gz7_ 5 ?oz4_
' 20,160%v v
—62Z2 + 23) + 9 ^ ^A + B ^ h (5Z8 - 40Z7 + 140Z6 - 252Z5 + 210Z4 - 90Z3 + 27) + ' 40,320xz/ V '
g2 o2a2Q2U0
-19-3- (3142 • 10nZ14 - 2419 • 1013Z13 + 3639 • 1012Z12 + 6758 • 1012Zn-
20 7 5 • 10 X V
-2002 • 1013Z10 - 2859 • 1013Z9 + 1338 • 1014Z8 - 964 • 1014Z7 - 1201 • 1014Z6+
Q2 132 AB Qh8
+1687 • 1014Z5 - 7734 • 1013Z2 + 3145 • 1013) + —-¡-=- (104Z11 - 836Z10+
y 2661 • 104x2v
+2640Z9 - 3575Z8 + 440Z7 + 2772Z6 + 1672Z5 - 6490Z4 + 4972Z - 1699) -
g(JAWSl sin (5Z§ _ igz7 + 28z5 _ 34z2 + i7) _
20,160x2 gOW (B sin < + A cos <) h3
120x
(2Z5 - 5Z4 + 5Z2 - 2),
pi(Z) = -MV- i)2.
P2(Z) = (4Z5 - 10Z" - 10Z2 + 4) + - l)2. (2.3)
In the analysis of the nonisothermic flow, hydrodynamical fields depend in inverse proportion both on the kinematic viscosity and on the coefficient of thermal expansion, and on the coefficient of thermal conductivity of the moving fluid.
3. The velocity field of a vertical vortex fluid
3.1. Dependence of the velocity field components on similarity numbers
Let us introduce the scale of velocity, W, choose as similarity parameters the Rossby number Ro = Reynolds number Re = Grashof numbers GrA = and GrB =
Prandtl number Pr = and reduce the resulting solution (2.1) for the velocity components of the flow to dimensionless form. We note that only the Prandl parameter is always positive. The signs of the other similarity numbers introduced above depend on the constants in the boundary conditions, which makes the analysis of the results much more difficult.
The velocity components of the flow are written in dimensionless form as follows:
Gr Rn2 Pr
VT{Y, Z) = - f ' Z (8Zg - 35Z8 + 150Z6 - 756Z4 + 885Z3 - 252) -1814 • 103 Re v '
Gr Rn
--s.-z (10Z6 - 56Z5 + 126Z4 - 105Z3 + 25) -
10,080-Re v ;
(Z3 - 4Z2 + 6Z - 3) + (Z3 - 1) + cos </>Z + |Vz, (3.1)
Vy(Y, Z) = -^Z (4Z6 - 14Z5 + 35Z3 - 84Z + 59) -Gr
-—-f Z (Z3 - 4Z2 + 6Z - 3) + sin <t>Z. 24Re v '
—►
Fig. 2. Diagram of the velocity distribution for the case where two stagnant points exist
The diagram of the velocity distribution in the flow of a viscous fluid (Fig. 2) reflects the existence of zero points defining the presence of reverse flows.
A change of the typical profile of the velocity component Vy(Z) is shown in Figs. 3-5 as a function of the Reynolds number, free-fall acceleration and the angle 0 given in the boundary conditions.
Fig. 3. The profile of the velocity component Vy as a function of the Reynolds number: Re = 50 (solid line), Re = 150 (dotted line), Re = 800 (dash-and-dot line)
The curves (Fig. 3) reflect the qualitative dependence of the profile of the velocity component Vy on the Reynolds number. The zero point is possible only for a positive value of the boundary constant W.
Fig. 4. The profile of the velocity component Vy as a function of free-fall acceleration g = 9.81 m/s2 (solid line), g =1 m/s2 (dash-and-dot line), g = 24.6 m/s2 (dotted line)
The curves (Fig. 4) reflect the qualitative dependence of the profile of the velocity component Vy on the free-fall acceleration. Free-fall accelerations on Earth at different heights and on
Jupiter are taken as numerical values. The zero point is absent only at small values of free-fall acceleration corresponding to a height of more than 10,000 km above the Earth.
Fig. 5. The profile of the velocity component Vy as a function of the angle $: $ = 0 rad (dotted line), $ = 0.7 rad (dash-and-dot line), $ = 0.4 rad (solid line)
The curves (Fig. 5) reflect the qualitative dependence of the profile of the velocity component. Vy on the angle (f) from the boundary conditions. When 0 G {0, 7r, 27r}, the zero values are reached only at Z = 0 and Z = 1.
If we set GrA = GrB = 0 in the formulae (3.1), we obtain an exact solution to the Navier-Stokes equations which was published in [54]. In this case the formulae (3.1) describe the isother-mic steady inhomogeneous Couette flow. If GrA = 0 and GrB = 0 or GrA = 0 and GrB = 0, then the formulae (3.1) describe an inhomogeneous convective flow of Couette type. The results obtained generalize the well-known exact solutions published in [37, 42, 48, 51]. Incorporation of the spatial acceleration into the expression for velocity Vx illustrates the complex interaction of horizontal velocities under thermal convection conditions.
3.2. Graph-analytical analysis of the velocity field
To study the characteristic properties of the velocity field, we represent the function Vy by combining the terms that correspond to the isothermic characteristics of motion and are responsible for convective heat transfer. The convective terms are characterized by the Grashof, Rossby, Prandtl and Reynolds similarity numbers. It is obvious that the velocity component Vy is defined by the additive interaction of the terms V1 and V2:
Vy (Y,Z) = Vi + V2, (3.2)
^Gr Ro Pr ^Gr
Vi =----Z (4Z6 - 14Z5 + 35Z3 - 84Z + 59)--(Z3 - 4Z2 + 6Z - 3),
1 10,080iîe v ; 2ARe v
V2 = sin 0Z. (3.3)
We examine in more detail the function V1. This polynomial can take a zero value at internal points of the domain of definition Z £ [0; 1] if the following inequality is satisfied:
59GrA RoPr\
Gl'B 1260 J
GrB —
Gr a RoPr\
7
'J
> 0.
(3.4)
A geometric interpretation of the law of distribution of the seventh- and fourth-degree polynomials of the components of V1 is presented in Fig. 6.
V11(Z)
Fig. 6. Qualitative form of the components of the function V1. Graph of the seventh-degree polynomial V1 (solid line), interpretation of the fourth-degree polynomial V12 (dashed line)
11
Hence, the velocity Vy depending on the value of sin 0 can have at least one stagnant point since the curves depicted in Fig. 6 intersect at one point. It is obvious that the parameters appearing in the expressions of velocity Vy can take values at which no stagnant points will exist. However, from a physical point of view this case is not interesting unless the velocity increases as the coordinate Z is varied.
The existence of one critical point is defined by the inequality
sin 01 sin 0 —
59GrA RoPr Gr
10,080Re
+
B
8Re
< 0.
(3.5)
Further analysis of the function of velocity shows that the possible existence of one or two roots of the polynomial Vy corresponds to one, two stagnant points or to their absence (Fig. 7).
In a similar way, we look at the velocity component Vx and split it into two components. The polynomial
V3 =
Ro sin 0
(Z4 — Z) + cos 0Z +
Ro
-YZ
(3.6)
12 v 7 ' """r ' Re'
corresponds to the inhomogeneous Couette flow. The other terms describe the convective contribution to the formation of the characteristic velocity profiles of the inhomogeneous convective Couette flow
v4 = -T^T-TT^tt (8zl° - 35Z° + 150z7 ~ 756z5 + 885z4 - 252Z) ~ 1814 • 103 Re
Gr Ro Gr
B (10Z7 - 56Z6 + 126Z5 - 105Z4 + 25Z) - (Z4 - 4Z3 + 6Z2 - 3Z). (3.7)
10,080Re
Fig. 7. The velocity profile Vy which characterizes the existence of reverse flows at certain angles: $ = = 0.2 rad (solid line); $ = 0.55 rad (dashed line); $ = 2.4 rad (dotted line); $ = 2n rad (dash-and-dot
The investigation of the spectral properties of the polynomial V4 by the graph-analytical method on the interval (0; 1) shows possible points of intersection of the graphs defining the constituent polynomials taking into account the weight coefficients: V41 is a polynomial of degree ten, V42 is a polynomial of degree seven, and V43 is a polynomial of degree four (Fig. 8).
V41(Z)
Fig. 8. Qualitative form of the components of the function V4. Images of the polynomials of degrees four (dash-and-dot line), seven (dashed line) and ten (solid line) taking into account the correction for an arbitrary choice of boundary conditions and physical characteristics of the fluid
The polynomial V3 corrects the resultant action of the polynomials making up the polynomial V4 and shows, under certain boundary conditions and physical constants of the fluid, stagnant points on the profile of velocity Vx (Fig. 9) as well.
Combination of the polynomials V3 and V4 forms a velocity field Vx which has no more than two values of the transverse coordinate Z, in which the velocity can vanish.
When two stagnant points exist simultaneously, the dimensionless hodograph of the velocity vector (Vx; Vy) represents the flow of a helical structure (Fig. 10). The existence of the vertical
Fig. 9. Velocity profile Vx in the planes Y = 0 (solid line), Y =10 (dashed line) and Y = 20 (dash-and-dot line) for existing stagnant points
Fig. 10. Hodograph of velocity in the presence of two stagnant points in the planes Y = 0 (solid line), Y =10 (dotted line) and Y = 20 (dash-and-dot line)
component of vorticity, the presence of convection and nonlinear inertia forces have the result that in the absence of a field of Coriolis forces the fluid flow is helical in the equatorial region.
4. Tangential stresses on a nondeformable boundary
The tangential stresses on a nondeformable boundary are defined by the relations
t jfi + ^jffl uu
xz 2 \0z Ox J 2 \dz J' yz 2 \dz dy J 2 V dz )' 1 ;
where n = vp is the dynamical viscosity.
In this case, the tangential stresses in dimensional form are
rj (dU du\ 2 \dz+Vdz)
a 2
g/3BQ 10,080z/2 V h2
gpAQ
80z 1814 • 103xt/2 70z6 336z5
315z l2"
+ 750h2z4 - 3540h3z3 - 252h6 ) -
3,3
h
+ 630z4 - 420hz3 + 25h4 -
«M _ 12l» + lihz _ „A _ E^m (g _ „)+E^îi+&
24v \ h ) 12v V h2 ) h h
n dV
Tyz 2 dz
gpAQ
28z
6
84z
5
10,080%v \ h2 g/3B f 4z3
h
+ 105z3h - 168zh3 + 59h4 I -
24v \ h
- 12z2 + 12hz - 3h +
W sin $ h
The tangential stresses on a nondeformable boundary with z = 0 are written as 63gpAQh6
453,500%v2
'yz
1 2
bgPBQh4 gfiAh2 Wsm^Qh W cos0 ytt 2016 v2 + 8v + 12Û + h + ~h~ 59gpAQh4 gf3Bh2 W sin </>
8v
h
(4.2)
(4.3)
10,080%^
As a result, the tangential stresses Txz and Tyz vanish on the lower boundary for values of fluid layer thickness defined from the algebraic equations
63 gpAQh6 453,500xv2
5gPBiïh4 g¡5Ah? W sin <f>iîh Wcos<f> yVt_ _ 2016z/2 + 8u + 12Û + h +~h~~ '
59gf3AQh4 gfiBh2 W sin <f>
(4.4)
8v
h
0.
10,080%^
These equations cannot be solved in general form. We mention particular cases where it is possible to calculate the fluid layer thickness for which there is no tangential stress on the lower nondeformable boundary. We first consider the case where W = 0 and take, without loss of generality, the section y = 0. In this case, the layer thickness is defined from the equations
126AQh4 5BQh2
113,375%v 252v 59AQh2
+ A = 0,
(4.5)
1260%
+ B = 0.
The solution of the first equation imposes restrictions on specification of boundary conditions of the boundary-value problem for the existence of fluid thicknesses for which rxz = 0. The condition on the discriminant of the biquadratic equation has the form
25B 2 Q2 504A2 Qv
63,504 113,375%
> 0.
(4.6)
In the process of solving the biquadratic equation, conditions imposed on the signs B and Q are derived. These values must be positive. The layer thickness is defined by the expression
h=
\
125B2Q2 504A2 Qv
566,875Bx 113,3 75% 63,504Av 252A<àv V 63,504 113,375*
(4.7)
xz
xz
if
566,875-Bx 113,375x /25B2Q2 _ 504A2Qv 63,504Av " 252AQu\l 63,504 ~ 113,375x'
If a strict inequality is satisfied under fixed boundary conditions and physical constants of the fluid, two values are determined for fluid layer thicknesses at which Txz = 0 vanish for z = 0. If the signs A and B coincide, there exists a fluid layer thickness from (4.5):
for which r..z = 0.
If we consider the case where Q = 0 in Eqs. (4.4), then a situation is possible where for the value of layer thickness there exists the following formula in finite form:
gfiAh3 + 8vW cos 6 = 0,
3 6 , (4.10)
gfiBh +8vW sin 6 = 0.
For the tangential stresses Txz and Tyz there exists a layer thickness
vWcoscf) vWsincf)
respectively, such that the tangential stresses on the lower boundary vanish when the conditions on boundary conditions of the following form are satisfied:
W cos é W sin é z, x
A B v y
At the same time, the tangential stresses will take zero values under the condition that relates the angle of direction of the velocity to the temperature gradients given on the upper boundary of the fluid layer:
B
tan (p = —. (4.13)
Discussion of results
The exact solutions (2.1)-(2.3) obtained above describe the convective flow of a vertical vortex flow heated from below. In the authors' opinion, the formulae (2.1)-(2.3) are important for several reasons. Firstly, a new exact solution has been obtained which describes the convective vortex of a vertical vortex flow without preliminary swirling of the flow. In other words, the vertical vorticity component determined by taking into account spatial acceleration, i.e., the inhomogeneity of the velocity field, manifests itself without uniform rotation of the fluid layer, but taking into account the convective flow of Couette type. Secondly, the solutions (2.1)-(2.3) can be useful in describing the nonisothermic flows of the World Ocean when the traditional approximation of geophysical hydrodynamics of an /-plane is used and a comparison is made with field observations or laboratory investigations of the ocean circulation. Third, the hydro-dynamical fields obtained, if they are taken as the main flow, will be useful for formulating and solving new problems of hydrodynamical stability. The authors of the paper have doubts concerning the possibility of applying the traditional method of normal modes for inhomogeneous
fluid flows. As a final remark, mention should be made of the reproducibility of the exact solution. Regarding the field observations, mention may be made of dry foliage or rubbish which go up and rotate in the air (which may be assumed to be incompressible) in the absence of an explicit field of the Coriolis force. The swirl occurs due to an inhomogeneous wind flow, which in the first approximation can be described by the velocity field (1.5) and by the corresponding boundary conditions. Field observations seem to provide a way to design an experimental facility for investigation of the unsteady-state and steady-state Rayleigh - Benard convection of a vertical vortex fluid while selecting a cocurrent inhomogeneous motion of air in order to reproduce the inhomogeneous wind flow subject to the boundary condition of a "hard" lid and to record the phenomena obtained theoretically.
Conclusion
A new exact solution is presented to the Navier-Stokes equations describing the stationary large-scale flow of a vertical vortex fluid for the case of an inhomogeneous steady-state Bénard convection induced by heating from below a thin layer of viscous fluid.
We have obtained polynomial hydrodynamical fields of a thin fluid layer which depend in inverse proportion on the coefficient of thermal conductivity, the coefficient of thermal expansion and kinematic viscosity in the analysis of the nonisothermic fluid flow.
The velocity field for inhomogeneous shear flows of a vertical vortex flow is investigated. By examining the spectral properties of polynomial solutions using the graph-analytical method, the qualitative and quantitative properties of exact solutions for velocities of the layers of incompressible fluid are studied. The existence of reverse flows, i. e., the existence of stagnant points under certain restrictions on physical constants of the fluid and on boundary conditions, is shown.
In particular cases, fluid layer thicknesses are determined for which the tangential stresses on the lower nondeformable boundary take zero values.
Conflict of interest
The authors declare that they have no conflict of interest.
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