 # 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 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.

# Full Text

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: = 0 (1 ), where 0 is the average density, 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 nonlinearity 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-Boussinesq system. The following system of equations of thermal shear convection in the Boussinesq approximation is considered [30, 31, 35, 36, 44, 45]: + = + , + = + , = , + = , + = 0. (1) Here, is pressure deviation from hydrostatic, divided by constant average fluid density ; is deviation from the average temperature; and are the coefficients of kinematic viscosity and thermal diffusivity of the fluid, respectively; 2 2 2 = 2 + 2 + 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 , , , and . 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 = (), = () (2) the incompressibility equation holds identically. The choice of the class (2) allows system (1) to be reduced to the form 2 2 = , = , = , 2 2 ( 2 2 2 ) = + . + + 2 2 2 (3) 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 . An exact solution of the form (2) generalizes the unidirectional Couette flow [50, 51] and the Birich-Ostroumov 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], = (, ), = (, ), (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: ( 2 2 ) = + + , 2 2 ( 2 2 ) = + + , 2 2 = , ( 2 2 2 ) + = + + . 2 2 2 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 Oberbeck-Boussinesq system (1). In contrast to class (4), for which all the vorticity components = rot V, = , = , = (5) are non-zero in the general case, the vertical component of vorticity (5) is always zero for the velocity field (2), = , = , = 0. 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] = () + (), = (), (6) which is a special case of class (4). The vorticity vector components calculated for it take the form Class (6) differs from class (2) in that the additional term () in the expression for the velocity is taken into account. Note that, when = 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 2 ) = + , 2 2 2 2 = , = , ( 2 2 2 ) + = + + . ( + ) 2 2 2 ( 2 + (7) It follows from the second equation of system (7) that the horizontal pressure gradient / depends only on the transverse (vertical) coordinate ; therefore, the pressure can be represented as = 1 (, ) + 2 (). Substituting the partial derivative / = 1 / into the first equation of system (7), we obtain that 1 / depends only on , i.e. the pressure proves to be linear in the coordinate . Finally, we arrive at the form = 0 () + 1 () + 2 () . (8) At the end, we substitute (8) into the third equation of system (7), 2 0 1 + + = and we find that the temperature is a linear function of the horizontal coordinates, i.e. = 0 () + 1 () + 2 () . (9) In view of the chosen structure of the temperature and pressure fields (8), (9), the equations of system (1) take the form ( 2 2 ) 2 = + , = 2 , 1 2 2 2 0 1 2 + + = (0 + 1 + 2 ), ( 2 2 1 2 2 ) 0 1 + 2 + 1 = + + . 2 2 2 + (10) The equations of system (10) are equalities of the form () + () + () = 0. (11) Applying the method of undetermined coefficients, we equate to zero the coefficients at the independent variables , and the free terms in the polynomial expressions (11). Inasmuch as all the required functions depend only on , we denote the derivatives with respect to the coordinate 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): = 0, = 2 , 1 = 0, 1 = 1 , = + 1 , 2 = 1 , 2 = 2 , 0 = 1 + 2 , 0 = 0 . (12) Note that only the first two equations in system (12) are isolated. After integrating the differential equations in order to determine the functions and 1 , the exact solutions for which are the linear functions = 1 + 2 , 1 = 3 + 4 , we arrive at a solution for the remaining functions involved in system (1). Hereinafter, we discuss the case of constant vertical twist, setting = = const. 2. Boundary value problem. As boundary conditions for the horizontal temperature gradients 1 and 2 , the horizontal pressure gradients 1 and 2 , the background temperature 0 , the background pressure 0 and the velocities and , we select the conditions described in [30, 31]. The absolutely solid bottom surface = 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, (, , 0) = 0. The velocity of the lower boundary = 0 is set as (0) = , (0) = 0. On the upper undeformed (free) boundary = , a constant atmospheric pressure is set and, by analogy with temperature setting, it is measured from zero, (, , ) = 0. We also assume that a homogeneous field of tangential stresses is specified on the upper boundary as = = 1 , = = 2 . 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 (, , 0) = + , (, , ) = + + . Taking into account the class of generalized solutions (6), (8), and (9), we write the selected boundary conditions as follows: (0) = (0) = 0, () = 1 , () = 2 , 0 (0) = 0, 1 (0) = , 2 (0) = , 0 () = , 1 () = , 2 () = , 0 () = 1 () = 2 () = 0. (13) 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 0 and 0 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 = ; 2 = = ) ( ; 1 = ( ) 2 + 2 ( + )2 ; 2! 2 = + ( )(2 + + ); 3! 1 = + ( )( + + )+ 2! + ( )2 (2 + 2 + 2 + 2 2 + 2 ); 4! ] [ 2 + (43 62 + 4 2 3 ) + (83 62 + 3 ) 4! [ (145 154 + 102 3 6 4 + 5 )+ 6! ] + (165 154 + 52 3 5 ) ; (14) ] [ (43 62 + 4 2 3 ) + (83 62 + 3 ) 6! [ (245 203 2 + 152 3 6 4 + 5 )+ 4! 2 ] + (665 403 2 + 152 3 5 ) + 2 [ + (5287 3925 2 + 2104 3 562 5 + 24 6 3 7 )+ 3 · 8! 2 ] + (6487 4485 2 + 2104 3 282 5 + 3 7 ) 1 (32 2 2 2 ) + . 3! = Note that, in view of the exact solution (14), the condition = 0 determining the degeneracy of class (6) into class (2), is equivalent to the condition = 0. Therefore, the effect of the parameter on the topology of the velocity field will be further studied in detail. 3. Velocity field analysis. We set the horizontal temperature gradients = = 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 . As we pass to the dimensionless coordinate = / [0, 1], the expressions for the velocities and assume the form = = 2 5 [ (14 15 + 10 3 6 4 + 5 )+ 6! ] + (16 15 + 5 3 5 ) , (15) ] 3 [ (4 6 + 4 2 3 ) + (8 6 + 3 ) + 6! 2 7 [ + (528 392 2 + 210 3 56 5 + 24 6 3 7 )+ 3 · 8! 2 ] + (648 448 2 + 210 3 28 5 + 3 7 ) 1 2 3 (3 2 ) + . (16) 3! The velocity field (15), (16) describes the convective flow of a viscous incompressible fluid, which cannot be reduced to unidirectional flow at = 0. Thus, the boundary value problem (12), (13) is essentially non-one-dimensional. Denote by 0 and 0 the velocities (15), (16) calculated in the absence of vertical twist ( = 0), 2 , 0 = ] 1 3 [ 0 = (4 6 + 4 2 3 ) + (8 6 + 3 ) + . (17) 6! Let us now study how the inclusion of the terms containing spatial acceleration in Eqs. (15), (16) changes the structure of the flow velocities in comparison with the velocity field 0 , 0 when different values of the temperature gradients and are specified. We start with the simplest case, namely, the case of a uniform heat source ( = = = = 0). When a uniform heat source 1 = 2 = 0, 0 = is set, the velocities and are determined by linear functions as = 0 = 1 , =0 = 2 . Thus, in the direction of both longitudinal axes, the flow is reduced to a combination of unidirectional flows of the Couette type , which correspond to a constant field of tangential stresses Moreover, the direction of the vortex remains unchanged everywhere inside the layer. Hereinafter, it is assumed that the flow is convective, i.e. that 2 + 2 = 0. We start by analyzing the velocity (15), since the structure of Eq. (15) is simpler than Eq. (16) determining the velocity . According to Eq. (15), the velocity 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 , , 2 , , but also depends on the thickness of the layer . By choosing , one can make the linear term 2 prevail over the non-linear terms in the velocity expression (15). Let us consider two limiting cases, = 0 and = 0, which allow us to reduce the number of streams contributing to the resultant flow. Assume that = 0, then the expression (15) for the velocity becomes 1 = 5 2 (16 15 + 5 3 5 ) = 6! 5 [ 5 6! 2 ] = 5 3 + 15 16 + , 6! 4 wherefrom it follows that the velocity 1 can have stagnant points only if the polynomial equation 5 5 3 + 15 + 1 = 0, with some 1 = 16 + 6!2 , 4 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 2 1 6 6 . 4 144 45 The dependence of the position of the stagnant point of the velocity 1 on the value of the parameter 1 is shown in Fig. 1 (curve 1). Note that in the case under consideration, for = 0, there is such a value 1 of the vertical coordinate that the tangential stress 1 = 1 4 = [6 5 20 3 + 30 + 1 ] 6! (18) vanishes. In this case, the stress changes its type (from tensile to compressive). Such 1 exists only for 1 [16, 0], i.e. when 06 2 1 6 . 4 45 The dependences of the coordinate 1 (the zeros of the polynomial (18)) on the parameter 1 are shown in Fig. 1 (curve 2). Figure 1. Dependencies of the position of the stagnant points of the velocities 1 , 2 , 10 , 20 1 2 01 01 and the critical point of the corresponding stresses , , , on the parameters 1 , 2 , 3 , 4 : curve 1 - the set of points satisfying the condition 1 = 0; curve 2 - the set of points 1 satisfying the condition = 0; curve 3 - the set of points satisfying the condition 2 = 0; 2 curve 4 - the set of points satisfying the condition = 0; curve 5 - the set of points satisfying 0 01 the condition 1 = 0; curve 6 - the set of points satisfying the condition = 0; curve 7 - the set of points satisfying the condition 20 = 0; curve 8 - the set of points satisfying the 02 condition =0 Consider another limiting case, assuming that = 0 in (15). Then the velocity 2 = =0 takes the form 2 = 2 5 (14 15 + 10 3 6 4 + 5 ) = 6! 5 [ 5 6! 2 ] = 6 4 + 10 3 15 + 14 . 6! 4 Obviously, the velocity 2 can have stagnant points only if the equation 5 6 4 + 10 3 15 + 2 = 0 has solutions inside the layer (0, 1) for some 2 = 14 6! 2 . 4 The tangential stress 2 = corresponding to the velocity 2 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 2 vanish. We have the following inequality velocity 2 and the tangential stress for the velocity 2 : 1 7 2 6 6 . 4 180 360 2 , it is written as follows: For the tangential stress 06 7 2 6 . 4 360 Note that, even in these limiting cases, the velocity can have one stagnant point. Next, we assume in (15) that = 0, = 0, = ] 5 [ 5 5 3 + 15 + 1 ( 5 6 4 + 10 3 15 + 14) . 6! Here, = / is a dimensionless parameter. The velocity can have two stagnant points (Fig. 2). Thus, in the chosen case of heating of the boundaries ( = = 0), the velocity (15), when the vertical twist ( = 0) is taken into account, can have up to two zero points. We now study in a similar way the velocity (16) of fluid flow along the axis. The expression for the velocity , in contrast to the velocity (15), is determined by the superposition of six streams: two streams caused by setting the tangential stresses 1 and 2 and four streams induced by the temperature gradients and . Besides, some of these streams are caused by the presence of a vertical twist in the fluid, characterized by the parameter. In the same way as Figure 2. The profile of the velocity for 1 = 15.2, = 1.1 in the analysis of the velocity (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 = 0 and the velocity is determined by the expression (17)), the type of velocities 3 1 10 = 0 =0 = (8 6 + 3 ) + = 6! 6!1 ) 3 ( 3 , 6 + 8 + = 6! 2 3 1 20 = 0 =0 = (4 6 + 4 2 3 ) + = 6! 3 ( 6! 1 ) = 3 + 4 2 6 + 4 + 6! 2 is similar to the form of the velocities 1 and 2 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 10 and 20 , as well as the corresponding tangential 01 = 0 02 0 stresses =0 , and = =0 , can vanish inside the layer. The location of the stagnation points depends on the combination of the parameters 1 1 and 2 (Fig. 1, curves 5 to 8). 2 Let us now study the effect of the vertical twist on the behavior of the velocity (16) in the limiting cases = 0 and = 0. If = 0, then the velocity 1 = |=0 takes the form 1 = [ 3 6! (8 6 + 3 ) 2 3 1 (3 2 ) + + 3! ] 2 7 2 3 5 7 (648 448 + 210 28 + 3 ) = 3 · 8! 2 2 7 [ 7 3 · 8! 2 1 5 3 2 = 3 28 + 210 448 + 648 + + 3 · 8! 2 2 6 ] 18 4 · 7! 2 2 + 2 4 (8 6 + 3 ) (3 ) 4 + by virtue of (16). The polynomials 3 7 28 5 + 210 3 448 2 + 648, 8 6 + 3 , and 3 2 involved in the expression of the velocity 1 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 1 with three stagnant points is shown in Fig. 3. When = 0, it follows from (16) that the velocity 2 = |=0 can be transformed in the same way Figure 3. The profiles of the velocities 1 and 2 when they have three stagnant points 2 = [ 3 (4 6 + 4 2 3 ) 2 3 1 (3 2 ) + + 3! 6! ] 2 7 2 3 5 6 7 (528 392 + 210 56 + 24 3 ) = + 3 · 8! 2 2 7 [ 3 7 + 24 6 56 5 + 210 3 392 2 + 528+ = 3 · 8! 2 168 4 · 7! 2 3 · 8! 2 1 ] 2 + 2 4 (4 6 + 4 2 3 ) (3 ) + , 4 2 7 whence it follows that 2 also can have up to three stagnant points (Fig. 3). Thus, in the considered limiting cases ( = 0 and = 0) with = 0, the velocity can have no more than one critical point, and when the vertical twist ( = 0) is taken into account, their number can reach three. In the case = 0, = 0, the structure of the velocity 0 (17) is similar to the structure of the expression (15) for the velocity ; namely, the expression of velocity 0 , as well as the velocity , 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 = . Consequently, the velocity 0 (17) can have up to two stagnant points. Let us now analyze how the velocity 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: ] 3 [ (4 6 + 4 2 3 ) + (8 6 + 3 ) + 6! 2 7 [ + (528 392 2 + 210 3 56 5 + 24 6 3 7 )+ 3 · 8! 2 ] 2 3 1 (3 2 ) + = + (648 448 2 + 210 3 28 5 + 3 7 ) 3! = [ 3 ( ) ) 2 7 ( 1 () + 2 () + 3 () + 4 () 2 720 120960 1 ] 2 3 5 () + . (19) 6 All the polynomials () 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 5 and the free term are independent due to the arbitrariness of the choice of the control parameters 1 and 2 ; it can be seen from (19) that the coefficients in front of the polynomials 1 , 2 , 3 , and 4 are related, 2 7 3 2 4 = · , 3 · 8! 2 6! 168 i.e. we have only three independent parameters for all these four coefficients, namely, , , , and this imposes additional restrictions on the behavior of the function if we consider the properties of the flow of a particular fluid in a horizontal layer of a fixed thickness . Thus, in the case = 0, the number of critical points of the velocity , as in the cases = 0 and = 0, increases by two when the vertical twist characterized by the parameter is taken into account. The qualitatively different profiles of the velocity are shown in Fig. 4. Figure 4. The profiles of the velocity 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.

# About the authors

### Natal'ya Vladimirovna Burmasheva

Ural Federal University named after the First President of Russia B. N. Yeltsin; Institute of Engineering Science, Urals Branch, Russian Academy of Sciences Candidate of technical sciences

Email: nat_burm@mail.ru
34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation; 19, Mira st., Ekaterinburg, 620002, Russian Federation
Candidate of technical sciences

### Eugenii Yurevich Prosviryakov

Ural Federal University named after the First President of Russia B. N. Yeltsin; Institute of Engineering Science, Urals Branch, Russian Academy of Sciences Candidate of technical sciences

Email: evgen_pros@mail.ru
34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation; 19, Mira st., Ekaterinburg, 620002, Russian Federation
Doctor of physico-mathematical sciences, no status

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