THE STABILITY OF A ROTATING AND HEATED FROM BELOW HORIZONTAL CYLINDRICAL LAYER OF A VISCOUS, INCOMPRESSIBLE LIQUID WITH FREE BOUNDARIES

Oksana L. Andreeva1,2*, Leonid A. Bulavin3, Viktor I. Tkachenko1,2 1National Science Center Kharkiv Institute of Physics and Technology 1, Academichna Str., 61108 Kharkiv, Ukraine 2V.N. Karazin Kharkiv National University 4, Svobody Sq., 61022 Kharkiv, Ukraine 3Taras Shevchenko National University of Kyiv 64/13, Volodymyrska Street, 01601 Kyiv, Ukraine *E-mail: andreeva@kipt.kharkov.ua Received September 12, 2019; revised October 28, 2019; accepted November 15, 2019

It is known that a periodic structure in the form of Benard cells [1] is formed in a horizontal layer of a viscous, incompressible, below-heated liquid. Rayleigh described in [2] the physical nature of the convection onset in such layers. He obtained analytical expressions for perturbed velocity and temperature in the Cartesian coordinate system. On the basis of this theory, it was possible to explain the threshold nature of the convective instability development, when the convection occurs only at a certain temperature difference. The obtained solutions describe the occurrence of convective rolls in a horizontal liquid layer, on the vertical common boundaries of which the velocity was directed periodically up / down and vice versa. However, these solutions did not describe the experimental fact of the availability of polygonal structures, the number of angles of which varied from four to seven, but with a predominance of six [1]. Therefore, to explain the appearance of hexagonal convective cells, geometric transformations of the found solutions were used. The description of cells with a different number of angles also means involving geometric manipulations. And if we continue the work and try to describe the entire set of polygonal convective cells, then this task turns out to be practically impossible.
In contrast to the method of geometric transformations described above, the energy principle of the formation of Benard cells was proposed in the work [3]. The principle is based on the fact that cells in the nucleation phase are few in number and have a cylindrical form [3]. As the temperature of the lower boundary of the layer increases, their number increases. Ideally, all cells are tightly packed in a liquid layer, and create a polygonal (hexagonal or other) structure, i.e. Benard cells.
As we see, initially the main element of Benard cells is an elementary cylindrical convective cell, whose perturbed parameters under free boundary conditions are described in the work [3]. At a uniform rotation of the liquid in the cell relative to the vertical axis, new possibilities of controlling the thermal convection described in [1 -3] emerge. The control parameters under the new conditions will be Coriolis and centrifugal forces [4,5].
In this work the onset of convection in a uniformly rotating and below-heated cylindrical tank with a viscous, incompressible liquid with free boundaries was investigated.
T (13) To the boundary conditions (10), (12), (13), we should add the condition for the perturbed velocity at the external boundary:     v , , v , , 0    r r z t r z t at 0  r R .
As an initial condition we assume, that at the time point 0 t = all the perturbations are either missing: or specified as: Thus, the problem of studying the stability of a rotating, heated from below, viscous, incompressible liquid of a cylindrical form with free boundaries is reduced to solving the eigen value problem for the system of equations (3), (8), (9) with boundary conditions (10), (12), (13) [4] and with initial conditions (14), (15).

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SOLUTIONS OF THE ORIGINAL SYSTEM OF EQUATIONS
We write equations (3), (8), (9) in a cylindrical coordinate system , ,  r z . We assume the perturbations are axisymmetric, i.e. such that the perturbations do not depend on the angular coordinate  . This means that all the perturbed values meet the condition ... / 0     . Rotation of the cylindrical volume of a liquid is a flow with an equilibrium azimuthal velocity       V r r [7,8], which is obtained from the Couette flow, if we set the radius and angular velocity of rotation of the inner cylinder to zero. Based on this we write the projections of the velocity vector in the cylinder: The equations for the perturbed velocity projections follow from (8): In cylindrical coordinates the continuity equation (3) is converted to the form: Equations (16) (19) should be supplemented by the equation of heat balance (9). We will seek solutions of the original task in the form:   From (20) a), c), and e) it follows that the spatial distribution of perturbations of horizontal and vertical velocities and temperature are similar to those implemented for a layer of heated from below, viscous, incompressible liquid without rotation in a cylindrical coordinate system [3].  (21), which describe a stable rotation of a viscous, incompressible and heated from below liquid of a cylindrical form.

THE STABILITY ANALYSIS OF A ROTATING, VISCOUS, INCOMPRESSIBLE, HEATED FROM BELOW LIQUID OF A CYLINDRICAL FORM
We determine the condition of a monotone instability of a rotating, viscous, incompressible and below-heated liquid of a cylindrical form from (21), similarly to [5].
where c R -critical Rayleigh number of monotone instability. It should be noted, that the expression (22) goes into the expression obtained in the Cartesian coordinate system by Rayleigh, if we replace 2 r k by 2 [4,5].
The condition of monotone instability (22) in graphic form is shown in Fig. 2

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The discriminant Q is positive for        [5,9] and the corresponding radial wave number -    It should be noted that the expression of monotone instability (22) is observed for the values of radial wave numbers:
To analyze the stability of perturbations of the boundary-value problem we turn to expressions (21), (22). Further, when describing the stability of perturbations in a medium, we will consider only the perturbed temperature, noting that the transition to other perturbed quantities is carried out in accordance with expressions (20), a) -e).
The fulfillment of the monotone instability condition (22) gives the following eigen values of the problem: The perturbed temperature (20), e) depends on time as a general solution of the characteristic equation (21), consisting of the sum of partial perturbations: where m C -arbitrary constants determined by initial conditions, For the case of the development of monotonous instability, the eigen value 1 Below, we consider the solution (21) for the case with lack of external temperature effect (the lower boundary of the layer is not heated), as well as for the case when this temperature effect occurs.

THE STABILITY ANALYSIS OF A ROTATING LIQUID CYLINDRICAL VOLUME
WITHOUT HEATING FROM BELOW Let us determine the characteristic numbers of the equation (21) without heating the liquid from below. Despite the fact that there is no heating of the liquid, the Rayleigh number is not equal to zero, but it is determined by the temperature difference between the lower 2, fl T and upper 1, fl T boundaries at the fluctuation level: The temperature fluctuation value of the upper boundary of the liquid cylindrical volume with radius V R and thickness of several molecular layers V h is determined by a number of water molecules in it and the average temperature of the medium * T [10]: For water the fluctuation level of temperatures is sufficiently small and is of the order of 8 7 10 10 Therefore, if the liquid is not heated from below, we assume that the Rayleigh number is equal to -the Rayleigh number for the fluctuation temperature difference between the lower and upper boundaries of the tank. Here it should also be noted that the temperature unit in (8) is no longer  as well fl  .
However, this value is clearly not included in the characteristic equation (21), and thus does not affect its solutions. At the above fluctuation level of temperature difference, the Taylor number is a fairly large value: In this case, the discriminant of the equation (21) takes the value corresponds to one real root and two complex conjugate roots Substituting the roots of the characteristic equation in (28) and assuming  

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If we set in (29), for example, 2 1 C C    and, 3 0 C  then the amplitude   B t will have the form: Further everywhere, without loss of generality, we can set 1 The temperature difference between the horizontal boundaries of the liquid is fixed and equal to zero As noted above, in case with no heating from below the temperature difference between the horizontal boundaries of the liquid is not equal to zero, but it is determined by the fluctuation level From (30) it follows, that the amplitude of the perturbed temperature as a result of action of viscous forces increases from zero at 0 t  to a maximum value at The temperature variation rate of the liquid (30) at the initial time (zero time) is equal to where T B -the liquid temperature in °С.
The temperature dependence upon the time for different values 2  is presented in Fig. 4. The temperature difference between the horizontal boundaries of the liquid is not fixed and not supported from the outside If the temperature difference between the horizontal boundaries of the liquid cylindrical volume  is specified at the fluctuation level and is not supported from the outside, then the system can be considered as isolated. In such a system according to (30) the temperature   B t will increase from zero to the maximum value  within the time Based on the principle of entropy increasing we will describe the dynamics of temperature variation in the system within the time interval The law of variation of the system entropy can be written using (30) in the form: where, At the third stage, the process of increasing the temperature will repeat, and the maximum temperature will now be equal to 2  . The overall increase of the temperature will be equal to   With multiple repeating of temperature rise stages, the liquid temperature will increase in increasing geometric progression to the value At that, the liquid temperature will increase, but its final value should not exceed the temperature of the phase transition. Otherwise, the original system of equations is not applicable.
As an example of heating a uniformly rotating cylindrical volume of a viscous, incompressible liquid can be used the heating of water in the Ranque -Hilsch vortex tubes [12,13].
For a pure water with an initial fluctuation level of the temperature difference 7 10    fl °С [10] and 1.8 heating to a temperature difference of the order of (8-9)·10 -2°С (temperature difference at which the Rayleigh number increases to a value c R R  ) corresponds to the number of temperature rise stages 23  n . Further increase of the difference in the water temperature from (8-9)·10 -2 С to a given temperature is a continuation of the previous multistage rise and will be described in the next Section (Stationary temperature perturbations at large Rayleigh numbers (  c R R )). As follows from the above analysis, heating of the rotating volume of water in Ranque -Hilsch tubes is due to the action of viscous forces. The spatial distribution of the vertical and horizontal velocity of water coincides with a similar distribution in the Rayleigh problem on convection in a layer of a viscous, incompressible liquid heated from below. Therefore, the problem of occurrence of a temperature gradient in a rotating volume of water, as well as the problem of heating water in Ranque -Hilsch tubes should be considered as the inverse Rayleigh problem.
Rayleigh's inverse problem involves searching the temperature difference between the boundaries of a viscous, incompressible liquid of a cylindrical form according to a given distribution of its vertical and horizontal velocities.

Analysis of the stability of a rotating cylindrical volume of liquid when heated from below
The expression for the perturbed temperature (30) will change if the rotating cylindrical volume of the liquid is heated from below with a certain rate. To show this, let us set a dependence of the perturbed temperature amplitude upon the time at heating in the form:

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where , , we will get the expression for the temperature of the rotating liquid with heating: T T temperatures of the lower and upper boundaries of the liquid layer, respectively. We will assume 1 1   C in (33), as before.
The dependence of the perturbed temperature amplitude upon the time (33) for different values 2  is presented in Fig.4. and oscillating with an exponentially decreasing amplitude tends to unity.
From this we can conclude, that in experiments with heating a rotating cylindrical volume of liquid, the value  ex determines the final temperature of the liquid heating, as well as the initial rate of temperature variation, i.e. , 1 It follows that the rate of variation in the temperature of a heated liquid is greater than that of a non-heated one.

STATIONARY TEMPERATURE PERTURBATIONS AT LARGE RAYLEIGH NUMBERS (  c R R )
As follows from the previous presentation, the temperature difference between the horizontal boundaries of the rotating cylindrical volume of a liquid increases. This leads to an increase in the Rayleigh number, which may possess eventually a value  c R R . In this case, the characteristic equation (22) where m D -arbitrary constants determined by initial conditions. From (35) it follows, that in a stationary state the general solution for the liquid perturbed temperature consists of the sum of three particular solutions. One summand with an eigen value 1 0 We use the expressions (33), (35) to describe the experimental data on the stability of a rotating cylindrical volume of water heated from below [5,9].
In some experiments (let's designate them as experiments I) the water was placed in glass cylinders with an outer diameter of 29.4 cm. An electric heating element was attached to the horizontal bottom of the cylinder, which provided controllable temperature variation. Heating was carried out at a low rate (heater power -1 W), medium rate (heater power -3 W) and high rate (heater power -10 W). The cylinder with water rotated around its axis using special tools. The layer of water with a depth of 3 cm and the rotation velocity of 10 rpm was studied in the experiments.
For experiments I) in (34), the relation  c w R R is satisfied and, thus, the instability with the parameters 1,2 In other experiments (experiments ІІ) [4,9] in water layers of 18 cm depth at a temperature difference of lower and upper boundaries -0.7 ° C and a rotation velocity of 5 rpm we have found by visual observations using a rotoscope, that for a heating rate of more definite value in a cylindrical volume of water rotating as a whole the convective cells with a diameter of 4.6 cm arose. The motion inside the cells becomes visible as a result of dispersing a small amount of aluminum powder over the surface of the water.
We will analyze the described experiments on the basis of the performed analytical calculations below.
Stationary temperature perturbation at a low rate of liquid heating At a low rate of heating the bottom of the tank, the process of water heating will take place in two stages. These stages, due to the low rate of liquid heating and the low level of saturation temperature, will not go into one another, but overlap, i.e. to the first stage of temperature variation, valid within the entire time interval, one should add the second stage of its variation, valid after the temperature is established of the order of (8 -9)·10 -2°С (when the equality  c R R is satisfied).
At the first stage the water temperature increases as a result of multi-stage rise, reaches values of the order of 8-9·10 -2°С and the Rayleigh number increases to a value  c R R . This stage of the multistage temperature rise will be approximated by the expression: A D A B C D t -constants describing the second stage of water heating. After summing up the solutions (37), (38) we obtain an expression for water temperature variation at the second stage of instability: where    x -Heaviside unit function.
Graphs of temperature versus time for the rotating cylindrical volume of water heated from below, obtained experimentally, and calculated by formulas (36), (39) are presented in Fig. 6. The experimental dependence of temperature upon time (solid line) was obtained as a result of digitization and graphic transformation to the Cartesian coordinate system. The beginning of the obtained experimental curve was brought into coincidence with the beginning of the  heating rate: solid line -experiment [4,9]; ○ -formula (36), -formula (39) In Fig. 6 the water heating rate at the initial time is determined by the quantity: it is a fairly small quantity. From Fig. 6 it follows, that the experimental data and theoretical dependences are quantitatively consistent. The data show, that in a rotating cylindrical volume of water with a low heating rate, a monotonic rise in temperature to a certain level and its stabilization at a slightly higher level are observed. Lack of convective vortices in the experiment indicates that temperature stabilization in a cylindrical volume of water occurs, apparently, due to mechanical heat and mass transfer, which occurs as a result of the liquid rotation.
The radial and vertical spatial distribution of the velocity of mechanical heat and mass transfer of a rotating liquid, as follows from (20), is determined by the same expressions as for convective heat and mass transfer in the below heated water layer with free boundaries [3].

Stationary temperature perturbation at high heating rate
At the tank heating rate of more than a definite value the process of water heating can be divided also into two stages. However, the second stage of temperature variation, due to the high rate of water heating and rapid establishment of the equality c R R  , will be described only by the sum of particular solutions of the characteristic equation.
Thus, at the first stage in the time interval max 0 t t   the water will be heated to the value  
The constants in (36), (40) can be determined by combining theoretical dependences with experimental data. Graphs of temperature versus time of a rotating cylindrical volume of water heated from below, obtained experimentally and calculated by formulas (36), (40) are presented in Fig. 7. The experimental temperature dependence on time was taken from [4,9], digitized, and graphically transformed to the Cartesian coordinate system. The beginning of the experimental curve thus obtained was brought into coincidence with the beginning of the Cartesian coordinate system.
In Fig. 7 the water heating rate at the initial time is determined by the value: It can be seen that in the case under consideration the water heating rate at the initial time is higher than the heating rate in the experiment described above. From Fig. 7 it follows, that at the first stage within the time interval max 0 t t   the temperature increases according to the exponential law (36) to a certain level. Then, at max t t  the first stage goes into the second, where the perturbed temperature variation is described by the expression (40). At that, the perturbed temperature experiences oscillations damped in time with respect to temperature 15.45 T  . It should be noted, that in Fig. 6, 7 and below, the part of the experimental curve, not marked with markers, describes the water temperature variation after the heater is turned off.
Let us consider another experiment in which the water heating rate was higher than the heating rate used in the experiment in Fig. 7. Fig. 8 shows experimentally obtained and calculated by formulas (36), (40) graphs of dependence of the disturbed temperature upon the time of a cylindrical volume of water heated from below at rate exceeding the heating rate in the experiment in Fig. 7.
As before, the experimental dependence of temperature difference on time [4,9] is digitized, and graphically transformed to the Cartesian coordinate system. The beginning of the experimental curve thus obtained was brought into coincidence with the beginning of the Cartesian coordinate system.
As a result of approximation of the experimental data using formulas (36) Fig. 8 the water heating rate at the initial time is higher than the heating rate in two previous experiments 0 = 2.9 h t dB dt  .
From Fig. 8 it follows that increasing the water heating rate compared with the data in Fig. 7 changes the temporal dynamics of the perturbed temperature: the first stage of heating exists for a shorter period of time, and the damped vibrations at the second stage have a shorter period and larger amplitude of oscillations. As to the rest, the amplitude dynamics of the perturbed temperature in Fig. 8 corresponds to the dynamics in Fig. 7: the amplitude of the first maximum and the saturation level are almost the same.

CONCLUSION
In this work, in neglect of the centrifugal convective force, the equations of heat and mass transfer in a rotating, viscous, incompressible liquid of a cylindrical form with free boundaries are analyzed. Solutions of the linearized original system of equations in cylindrical geometry are obtained that satisfy the boundary conditions of the problem. The spatial distributions of the projections of the perturbed velocity and the perturbed temperature of the liquid are determined with the accuracy to constants.
It is shown, that the spatial distribution of horizontal and vertical velocities and the temperature of a rotating, viscous, incompressible liquid of a cylindrical form with free boundaries are similar to those implemented for a layer of a viscous, incompressible liquid heated from below without rotation, which correspond to solutions of the Rayleigh problem in the cylindrical coordinate system.
The analysis of stability of a rotating, viscous, incompressible liquid of a cylindrical form heated from below was carried out. A stability condition for the rotating, viscous, incompressible liquid of a cylindrical form heated from below in the cylindrical coordinate system was obtained. For the case of its stable rotation the graphs of dependence of the The stability analysis of a rotating cylindrical volume of a liquid without heating from below was carried out. It has been shown that at a fixed, zero temperature difference between the horizontal boundaries of the cylindrical tank of a liquid, an increase in its temperature is observed starting from zero. Then it reaches a maximum value of the order 1.5,..., 2.0   , and after, oscillating with exponentially decreasing amplitude of oscillations, again decreases to zero. In the final state at large times the liquid will rotate as a whole without perturbations of velocity, pressure and temperature.
If the temperature difference between the horizontal boundaries of the cylindrical volume of the liquid  is set at a fluctuation level and is not supported from the outside, then the system can be considered as isolated. In this case, according to the law of entropy increasing in isolated systems, the liquid will successively go through only the stages of temperature rise with the rise coefficient at each stage of the order so, that its temperature eventually will increase in increasing geometric progression up to the value  c R R . In this case, the characteristic equation will have another eigen values and, consequently, another expression for increasing the liquid temperature. The water temperature will increase, but its final value should not exceed the phase transition temperature, since in this case the original system of equations is not applicable.
Based on performed calculations an example of heating a rotating cylindrical volume of water without heating from below is considered. This example shows that the problem of the occurrence of a temperature gradient in the volume of water, as well as its heating in Ranque-Hilsch tubes, should be considered as an inverse Rayleigh problem. In
the Rayleigh inverse problem it is required to find the temperature difference between the horizontal boundaries of a viscous, incompressible liquid of a cylindrical form according to a given distribution of its horizontal and vertical velocities.
The stability analysis of a rotating cylindrical volume of a liquid at heating from below has shown, that over time the temperature difference of the liquid increases from zero at 0  t , reaches the first maximum value of the order  and tends to unity oscillating with exponentially decreasing oscillation amplitude. In experiments with heating the rotating cylindrical volume of a liquid, the value of the temperature difference set from outside determines the final heating temperature of the liquid, as well as the initial rate of its variation. It is shown that the rate of temperature variation of a heated liquid is greater than that of a similar one without heating.
Stationary temperature perturbations are considered for different rates of water heating. In all cases it was proposed to consider two stages of the development of stationary perturbations. At the first stage the water, as a result of a multistage heating at a rate set from outside, acquires a temperature at which the Rayleigh number is small, but increases from the value  c R R to  c R R . At the second stage of heating, when  c R R , the eigen values of the characteristic equation and the type of particular solutions, of which the general solution for a temperature consists, change. The temperature of the system is described either by a superposition of general solutions for the first and second stages of the temperature rise (at a low heating rate), or only by a general solution for the case  c R R (at a high heating rate). Comparison of theory and experimental data for heating water shows a qualitative and quantitative agreement.