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Properties and exact solutions of the equations of motion of shallow water in a spinning paraboloid
Journal of Applied Mathematics and Mechanics 75 (2011) 350–356
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Properties and exact solutions of the equations of motion of shallow water in a spinning paraboloid夽 A.A. Chesnokov Novosibirsk, Russia
a r t i c l e
i n f o
Article history: Received 15 June 2009
a b s t r a c t A transformation is found and, using this, the non-linear system of equations describing the spatial oscillations of a thin layer of liquid in a spinning circular parabolic basin is reduced to the conventional equations of the model of shallow water over a level fixed bottom. This transformation is obtained by analyzing the properties of the symmetry of the equations of motion of spinning shallow water. The existence of non-trivial symmetries in the case of the model considered enabled group multiplication of the solutions to be carried out. Using the known steady-state rotationally symmetric solution, a class of time-periodic solutions is obtained that describes the non-linear oscillations of the liquid in a circular paraboloid with closed or quasiclosed (ergodic) trajectories of the motion of the liquid particles. © 2011 Elsevier Ltd. All rights reserved.
The modelling of large-scale atmospheric and ocean currents at central latitudes 1,2 is an important example of the geophysical application of the results of the analysis of the non-linear wave motions of a fluid in differently-shaped spinning basins using shallow water theory. General results of a study of the wave motion of a liquid in a rotating paraboloid have been obtained using the non-linear shallow-water model, equations for the centre of mass, the moment of inertia and the total energy of the moving liquid have been derived and exact solutions have also been found using a model with a linear velocity field 3,4 . The existence of time-periodic solutions of the equations of spinning shallow water has been established numerically.5 The non-linear axisymmetric oscillations of a liquid in a paraboloid have been studied, classes of exact solutions of the equations of motion have been constructed, including time-periodic solutions, 6–9 and a special feature of them is the linear dependence of the radial component of the velocity on the radius. Group analysis 10 of the equations of motion of spinning shallow water has been used 11 in calculating the symmetries and the group-theoretical interpretation of known results.3 An isomorphism of the Lie algebras of the admissible operators has been established 12 in the case of the systems of equations of a thin liquid layer on a level bottom, both with and without taking account of Coriolis forces. The symmetry properties of the equations of spinning shallow water 11,12 enables us to obtain the fundamental result of this paper that involves the conversion of the equations describing the spatial motions of a thin liquid layer in a spinning circular paraboloid to the conventional shallow-water equations.
1. Transformation of the shallow-water The non-linear spatial oscillations of an ideal homogeneous heavy fluid in a bounded basin, spinning with a constant angular velocity f/2 about the vertical z axis, are studied in the long-wave approximation. In a cylindrical system of coordinates (r, , z), spinning together with the basin, the fluid motion is described by the system of equations
(1.1)
夽 Prikl. Mat. Mekh. Vol. 75, No. 3, pp. 496–504, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.07.013
A.A. Chesnokov / Journal of Applied Mathematics and Mechanics 75 (2011) 350–356
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Here U and V are the radial and circumferential components of the velocity vector, h is the depth of the fluid layer, the constants g and f are the gravitational acceleration and the Coriolis parameter, and the relief of the bottom (the basin) is given by the equation z = Z(r, ). The methods of group analysis 10 have been used 11 to investigate the equations of motion of a spinning liquid using the shallow-water model (1.1). A 9-dimensional Lie algebra of the operators Lω 9 , which are admitted by Eqs (1.1), has been calculated 11 in the case of a basin having the shape of a paraboloid of revolution
(1.2) It can be shown that the highest 9-dimensional symmetry group is only admitted by system of equations (1.1) when the relief of the bottom has the shape (1.2). The Lie algebra of the operators L 9 decomposes into a direct sum of a 6-dimensional radical and a simple sl(2) Lie algebra. The structure of the known L9 Lie algebra of the symmetries of the equations of motion of shallow water over a flat fixed bottom (Eqs (1.1), when f = 0 and Z = const) is analogous to that mentioned earlier.10,13 In particular, the simple sl(2) Lie algebra consists of a time shift operator, a projection operator and a dilatation:
The corresponding simple sl(2) Lie algebra of the symmetries of Eqs (1.1) and (1.2) is formulated by the following operators
which will be used below for the group multiplication of the solutions. It can be shown that the Lie algebras L9 and L 9 are isomorphous and, by virtue of this, the question arises of the equivalence of the realization of the admissible operators, that is, concerning the existence of a change of variables which transfers the symmetries (admissible operators) of Eqs (1.1) and (1.2) into the symmetries of Eqs (1.1) when f = 0 and Z = const. It is established that such a change of variables exists and it converts Eqs (1.1) and (1.2) into the conventional shallow-water equations. Omitting the lengthy intermediate calculations, we will formulate a theorem, whose validity can be ascertained by direct calculations. Theorem 1. System of equations (1.1), describing the motion of an ideal fluid in the spinning basin (1.2) in the long-wave approximation and the conventional shallow-water equations (Eqs (1.1) when f = 0 and Z = const) are related by the transform
(1.3) Actually, if the set of functions U(t, r, ), V(t, r, ), h(t, r, ) satisfies Eqs (1.1) and (1.2), then the functions U (t , r , ), V (t , r , ), h (t , r , ), defined by formula (1.3), are the solution of the conventional shallow-water equations. We will now turn our attention to a number of properties of the transformation obtained. 1. Formulae (1.3) become identical when the Coriolis parameter f and the parameter for the relief of the bottom tend to zero. 2. Transformation (1.3) is linear with respect to the spatial variables and the corresponding projections of the velocity vector. 3. The change of variables (1.3) transfers any solution of the shallow-water equations, defined for all time values, into the solution of the equations of motion of spinning shallow water, defined in a finite time interval t ∈ (− /ω, /ω). 4. By virtue of relations (1.3), the solution of Eqs (1.1) and (1.2) is bounded when t → ± /ω if the corresponding solution of the conventional shallow-water equations decay sufficiently rapidly when t→ ∞ such that
5. Transformation (1.3) converts trajectories into trajectories. Suppose r˙ = U and ˙ = V/r. Then, by virtue of this transformation, we have
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By means of the change of variables (1.3), it is possible to obtain exact solutions of system of equations (1.1), (1.2), using the known solutions of the conventional shallow-water equations and conversely. For example, the known solution of the shallow-water equations
describing the spreading out of a liquid on a plane under the action of gravity, 13 corresponds to the equilibrium solution of system (1.1), (1.2)
Conversely, the following solution of Eqs (1.1) and (1.2)
defined only in a finite time interval, corresponds to the constant solution (written in Cartesian coordinates) within the limits of the shallow-water model
When t → ± /ω, the solution fails: the depth of the liquid layer increases without limit, which contradicts the assumption in shallow water theory. Remark. In the axisymmetric case, the change of variable V = V − fr/2 eliminates the terms responsible for the Coriolis force and the centrifugal force from Eqs (1.1). This can also be achieved in the spatial case by the change of variables (1.3) when = f, supplemented by the relation
which transforms the level of the bottom. Making this change of variables in Eqs (1.1), we obtain (in the variables with primes) the usual equations of shallow-water theory over an uneven bottom in a fixed system of coordinates, that is, model (1.1) when f = 0. The results of the analysis of the symmetry properties presented above show that it is only possible, by means of a change of variables, to eliminate the terms associated both with the spinning and the unevenness of the bottom from Eqs (1.1) in the case of a parabolic basin (1.2). 2. Group multiplication of the solutions It is well known 10 that a one-parameter group of transformations, defined by finite relations of the form x¯ i = x¯ i (x1 , . . . , xN , a), where a is a real parameter, can be matched to each infinitesimal operator X = i ∂xi . To do this, it is necessary to integrate the system of Lie equations
(2.1) We now find the finite transformations for the operators F7 , F8 and F9 . Integration of the Lie equations (2.1) for the operator F9 (when t = / (2n + 1)/ω, where n is an integer) gives the following result
(2.2) Here,
(2.3) where k is an integer such that t = t¯ |a=0 belongs to the interval
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353
The solution of Eqs (2.1) for the operator F9 in the case when t = t¯ | = (2n + 1)/ω has the form
(2.4) Unilateral limits of expressions (2.2) exist when t → (2n + 1)/ω ± 0 and they are equal. This enables us to complete the definition of relations (2.2) with respect to continuity at the points t = (2n + 1)/ω at which the parameter becomes infinite in accordance with equalities (2.4). The resultant mapping is continuously differentiable with respect to all arguments and will be used below in the group multiplication of the solutions. The finite transformations for the operator F8 are given by the formulae
(2.5) The functions (t) and (t) are defined below. When t = (2n + 1)/ω, the definitions of relations (2.5) have to be completed with respect to continuity. The finite transformations of the operator F7 are also expressed by formula (2.5) in which the functions (t) and (t) have to be replaced by (t) = − ctg(ωt/2) and 1 (t) = (2k + 1)/ω and the integer k is determined from the condition
According to the general theory of group analysis, 10 equations do not change when the variables change, corresponding to finite transformations of the admitted operators. Transformations (2.2) and (2.5) can therefore be used to multiply the solutions of system of equations (1.1), (1.2). The theorem presented below is obtained using transformation (2.2). Relations (2.5) can be used in the same way. Note that a similar approach has been used 14 to construct the solutions of the equations of motion of an ideal monatomic gas which allow of a projective transformation. Theorem 2.
If the set of functions
satisfies system of equations (1.1), (1.2), the set of functions
(2.6) satisfies the same system, where
¯ a is an arbitrary positive number, and the functions t¯ = t¯ ( , t), r¯ = r¯ ( ), ¯ = ( ) and = (t) are defined by the first three equalities of (2.2) and relations (2.3). 3. Time-periodic solutions The theorem formulated above enables us to obtain new exact solutions of Eqs (1.1) and (1.2) using known solutions. As an example, we carry out the group multiplication of a class of steady-state rotationally-symmetric solutions
(3.1) Here, V¯ (r)− is an arbitrary smooth function and h0 is a positive constant. Application of transformation (2.6) to solution (3.1) leads to the class of axisymmetric time-periodic solutions of the system of equations (1.1), (1.2) (3.2)
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A.A. Chesnokov / Journal of Applied Mathematics and Mechanics 75 (2011) 350–356
Fig. 1.
The functions on the right-hand sides of the first two equations, as well as the functions r¯ = r¯ ( , r) and = (t), are given by equalities (2.3). √ The period of the solution T = 2/ω = / g is solely determined by the curvature of the paraboloid at the pole. Similar solutions have 7,9 been obtained in investigations of the axisymmetric oscillations of a liquid in a paraboloid of revolution. We shall only dwell on the analysis of the trajectories of the motions of the particles and the evolution of the material volume. The trajectories of motion of the particles for solution (3.2) have the form
(3.3) The constants r0 and 0 give the initial position of a particle. It follows from relations (3.3) that the trajectories of the liquid particles, located on circles of radius r0 when t = 0, are closed when the equality
is satisfied, where n and m are integers. In the general case, it can be asserted that the trajectories of the liquid particles for the class of solutions considered are ergodic (or quasiclosed), that is, for any > 0, an instant tε > 0 can be found such that, when t = tε , a particle will be at a distance of not more than from its position when t = 0. We next consider a characteristic example of a solution from the class (3.2). Suppose in solution (3.1) the circumferential velocity has the form (3.4) the corresponding multiplied solution is given by formulae (3.2) and (3.4). The character of the motion of the liquid is illustrated by the graphs obtained for the following parameter values
(the qualitative form of the graphs is preserved when these values are changed). The level of the liquid z = h + Z(r), lying in the paraboloid of revolution z = Z(r) = ω2 r2 /(8g) at the instants 2k/ω and(2k + 1)/ω, corresponding to the greatest and least localization of the liquid, is shown in Fig. 1. The closed trajectories of the motion of the particles, lying,
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355
Fig. 2.
Fig. 3.
√ when t = 0, on circles of radius r0 = nω/ ml ˛ ≈ 1.51, 0.97, 0.49 that corresponds to the choice of the pairs of numbers (n, m) = (2, 9), (1,7), (1,14) are shown in Fig. 2. The deformation of the material contour during the evolution of the flow (3.2), (3.4) is shown in Fig. 3. At the initial instant, the selected liquid contour is a circle of radius with centre at the point (0.4, 0.5). This contour is shown when t = 3k/ω(k = 0, 1, 2, 3, 4) in Fig. 3. The trajectory of a particle, emerging from a point with coordinates (0.7, 0.5) at t = 0 is depicted by dashes. As time increases, the contour is twisted into a spiral. Its deformation after a longer period of time as well as the deformation of another contour (the dot-dash curve) are shown in Fig. 3, b. Here, the circles are material contours when t = 0 and the spirals are material contours when t = 25/ω ≈ 35.12. It is interesting to note the qualitative agreement between the deformation of the material contour in the solution considered and the result of experiments on the transport of an impurity in a spinning liquid.15 The experiments were performed in a cylindrical container
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A.A. Chesnokov / Journal of Applied Mathematics and Mechanics 75 (2011) 350–356
with a spinning disc on the bottom, which formed an axisymmetric vortex flow of the liquid. When spots of dye were introduced into the spinning liquid, the formation of spiral sleeves, similar to those shown in Fig. 3, was observed with time. Acknowledgements This research was financed by the Ministry of Education and Science of the Russian Federation (2.1.1/3543) and within the framework of the Programme of the Division of Power Engineering, Machine Building, Mechanics and Control Processes of the Russian Academy of Sciences, No. 2. 14.1. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Pedlosky J. Geophysical Fluid Dynamics. Berlin: Springer; 1987, 710 p. Gill A. Atmosphere-Ocean Dynamics. New York: Academ. Press; 1982. Ball FK. Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J Fluid Mech 1963;17(Pt. 2):240–56. Ball FK. The effect of rotation on the simpler modes of motion of a liquid in an elliptic paraboloid. J Fluid Mech 1965;22(pt. 3):529–45. Sachdev PL, Palaniappan D, Sarathy R. Regular and chaotic flows in paraboloid basin and eddies. Chaos, Solution and Fractals 1996;7(5):383–408. Ingel’ LKh. A class of exact non-steady-state solutions of the shallow-water equations with rotation. Izv Ross Akad Nauk Fizika Atmosphery in Okeana 1994;30(5):718–20. Svirkunov PN. Unsteady axisymmetric flows in the approximation of shallow-water theory. Prikl Mat Mekh 1996;60(3):520–2. Dotsenko SF, Rubino A. Exact analytical solutions of the non-linear equations of long waves in the case of axisymmetric oscillations of a liquid in a spinning parabolic basin. Izv Ross Akad Nauk MZhG 2003;2:158–64. Kalashnik MV, Kakhiani VO, Lominadze DG, et al. Non-linear isochronous oscillations of a liquid in a paraboloid: theory and experiment. Izv Ross Akad Nauk MZhG 2004;5:131–42. Ovsyannikov LV. Group Analysis of Differential Equations. Moscow: Nauka; 1978. Rogers C, Ames WF. Nonlinear Boundary Value Problems in Science and Engineering. New York: Academic; 1989, 416 p. Chesnokov AA. Symmetries of the equations of the theory of shallow water on a spinning plane. Sib Zh Industr Matem 2008;11(3):135–46. Ibragimov NKh. Groups of Transformations in Mathematical Physics. Moscow: Nauka; 1983. Nikol’skii AA. Invariant transformations of the equations of motion of an ideal monatomic gas and new classes of exact solutions. Prikl Mat Mekh 1963;27(3):496–508. Stepanova YeV, Chashechkin YuD. Anisotropic transport of an impurity in a compound vortex. Dokl Ross Akad Nauk 2008;423(4):474–8.
Translated by E.L.S.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Properties and exact solutions of the equations of motion of shallow water in a spinning paraboloid夽 A.A. Chesnokov Novosibirsk, Russia
a r t i c l e
i n f o
Article history: Received 15 June 2009
a b s t r a c t A transformation is found and, using this, the non-linear system of equations describing the spatial oscillations of a thin layer of liquid in a spinning circular parabolic basin is reduced to the conventional equations of the model of shallow water over a level fixed bottom. This transformation is obtained by analyzing the properties of the symmetry of the equations of motion of spinning shallow water. The existence of non-trivial symmetries in the case of the model considered enabled group multiplication of the solutions to be carried out. Using the known steady-state rotationally symmetric solution, a class of time-periodic solutions is obtained that describes the non-linear oscillations of the liquid in a circular paraboloid with closed or quasiclosed (ergodic) trajectories of the motion of the liquid particles. © 2011 Elsevier Ltd. All rights reserved.
The modelling of large-scale atmospheric and ocean currents at central latitudes 1,2 is an important example of the geophysical application of the results of the analysis of the non-linear wave motions of a fluid in differently-shaped spinning basins using shallow water theory. General results of a study of the wave motion of a liquid in a rotating paraboloid have been obtained using the non-linear shallow-water model, equations for the centre of mass, the moment of inertia and the total energy of the moving liquid have been derived and exact solutions have also been found using a model with a linear velocity field 3,4 . The existence of time-periodic solutions of the equations of spinning shallow water has been established numerically.5 The non-linear axisymmetric oscillations of a liquid in a paraboloid have been studied, classes of exact solutions of the equations of motion have been constructed, including time-periodic solutions, 6–9 and a special feature of them is the linear dependence of the radial component of the velocity on the radius. Group analysis 10 of the equations of motion of spinning shallow water has been used 11 in calculating the symmetries and the group-theoretical interpretation of known results.3 An isomorphism of the Lie algebras of the admissible operators has been established 12 in the case of the systems of equations of a thin liquid layer on a level bottom, both with and without taking account of Coriolis forces. The symmetry properties of the equations of spinning shallow water 11,12 enables us to obtain the fundamental result of this paper that involves the conversion of the equations describing the spatial motions of a thin liquid layer in a spinning circular paraboloid to the conventional shallow-water equations.
1. Transformation of the shallow-water The non-linear spatial oscillations of an ideal homogeneous heavy fluid in a bounded basin, spinning with a constant angular velocity f/2 about the vertical z axis, are studied in the long-wave approximation. In a cylindrical system of coordinates (r, , z), spinning together with the basin, the fluid motion is described by the system of equations
(1.1)
夽 Prikl. Mat. Mekh. Vol. 75, No. 3, pp. 496–504, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.07.013
A.A. Chesnokov / Journal of Applied Mathematics and Mechanics 75 (2011) 350–356
351
Here U and V are the radial and circumferential components of the velocity vector, h is the depth of the fluid layer, the constants g and f are the gravitational acceleration and the Coriolis parameter, and the relief of the bottom (the basin) is given by the equation z = Z(r, ). The methods of group analysis 10 have been used 11 to investigate the equations of motion of a spinning liquid using the shallow-water model (1.1). A 9-dimensional Lie algebra of the operators Lω 9 , which are admitted by Eqs (1.1), has been calculated 11 in the case of a basin having the shape of a paraboloid of revolution
(1.2) It can be shown that the highest 9-dimensional symmetry group is only admitted by system of equations (1.1) when the relief of the bottom has the shape (1.2). The Lie algebra of the operators L 9 decomposes into a direct sum of a 6-dimensional radical and a simple sl(2) Lie algebra. The structure of the known L9 Lie algebra of the symmetries of the equations of motion of shallow water over a flat fixed bottom (Eqs (1.1), when f = 0 and Z = const) is analogous to that mentioned earlier.10,13 In particular, the simple sl(2) Lie algebra consists of a time shift operator, a projection operator and a dilatation:
The corresponding simple sl(2) Lie algebra of the symmetries of Eqs (1.1) and (1.2) is formulated by the following operators
which will be used below for the group multiplication of the solutions. It can be shown that the Lie algebras L9 and L 9 are isomorphous and, by virtue of this, the question arises of the equivalence of the realization of the admissible operators, that is, concerning the existence of a change of variables which transfers the symmetries (admissible operators) of Eqs (1.1) and (1.2) into the symmetries of Eqs (1.1) when f = 0 and Z = const. It is established that such a change of variables exists and it converts Eqs (1.1) and (1.2) into the conventional shallow-water equations. Omitting the lengthy intermediate calculations, we will formulate a theorem, whose validity can be ascertained by direct calculations. Theorem 1. System of equations (1.1), describing the motion of an ideal fluid in the spinning basin (1.2) in the long-wave approximation and the conventional shallow-water equations (Eqs (1.1) when f = 0 and Z = const) are related by the transform
(1.3) Actually, if the set of functions U(t, r, ), V(t, r, ), h(t, r, ) satisfies Eqs (1.1) and (1.2), then the functions U (t , r , ), V (t , r , ), h (t , r , ), defined by formula (1.3), are the solution of the conventional shallow-water equations. We will now turn our attention to a number of properties of the transformation obtained. 1. Formulae (1.3) become identical when the Coriolis parameter f and the parameter for the relief of the bottom tend to zero. 2. Transformation (1.3) is linear with respect to the spatial variables and the corresponding projections of the velocity vector. 3. The change of variables (1.3) transfers any solution of the shallow-water equations, defined for all time values, into the solution of the equations of motion of spinning shallow water, defined in a finite time interval t ∈ (− /ω, /ω). 4. By virtue of relations (1.3), the solution of Eqs (1.1) and (1.2) is bounded when t → ± /ω if the corresponding solution of the conventional shallow-water equations decay sufficiently rapidly when t→ ∞ such that
5. Transformation (1.3) converts trajectories into trajectories. Suppose r˙ = U and ˙ = V/r. Then, by virtue of this transformation, we have
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By means of the change of variables (1.3), it is possible to obtain exact solutions of system of equations (1.1), (1.2), using the known solutions of the conventional shallow-water equations and conversely. For example, the known solution of the shallow-water equations
describing the spreading out of a liquid on a plane under the action of gravity, 13 corresponds to the equilibrium solution of system (1.1), (1.2)
Conversely, the following solution of Eqs (1.1) and (1.2)
defined only in a finite time interval, corresponds to the constant solution (written in Cartesian coordinates) within the limits of the shallow-water model
When t → ± /ω, the solution fails: the depth of the liquid layer increases without limit, which contradicts the assumption in shallow water theory. Remark. In the axisymmetric case, the change of variable V = V − fr/2 eliminates the terms responsible for the Coriolis force and the centrifugal force from Eqs (1.1). This can also be achieved in the spatial case by the change of variables (1.3) when = f, supplemented by the relation
which transforms the level of the bottom. Making this change of variables in Eqs (1.1), we obtain (in the variables with primes) the usual equations of shallow-water theory over an uneven bottom in a fixed system of coordinates, that is, model (1.1) when f = 0. The results of the analysis of the symmetry properties presented above show that it is only possible, by means of a change of variables, to eliminate the terms associated both with the spinning and the unevenness of the bottom from Eqs (1.1) in the case of a parabolic basin (1.2). 2. Group multiplication of the solutions It is well known 10 that a one-parameter group of transformations, defined by finite relations of the form x¯ i = x¯ i (x1 , . . . , xN , a), where a is a real parameter, can be matched to each infinitesimal operator X = i ∂xi . To do this, it is necessary to integrate the system of Lie equations
(2.1) We now find the finite transformations for the operators F7 , F8 and F9 . Integration of the Lie equations (2.1) for the operator F9 (when t = / (2n + 1)/ω, where n is an integer) gives the following result
(2.2) Here,
(2.3) where k is an integer such that t = t¯ |a=0 belongs to the interval
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The solution of Eqs (2.1) for the operator F9 in the case when t = t¯ | = (2n + 1)/ω has the form
(2.4) Unilateral limits of expressions (2.2) exist when t → (2n + 1)/ω ± 0 and they are equal. This enables us to complete the definition of relations (2.2) with respect to continuity at the points t = (2n + 1)/ω at which the parameter becomes infinite in accordance with equalities (2.4). The resultant mapping is continuously differentiable with respect to all arguments and will be used below in the group multiplication of the solutions. The finite transformations for the operator F8 are given by the formulae
(2.5) The functions (t) and (t) are defined below. When t = (2n + 1)/ω, the definitions of relations (2.5) have to be completed with respect to continuity. The finite transformations of the operator F7 are also expressed by formula (2.5) in which the functions (t) and (t) have to be replaced by (t) = − ctg(ωt/2) and 1 (t) = (2k + 1)/ω and the integer k is determined from the condition
According to the general theory of group analysis, 10 equations do not change when the variables change, corresponding to finite transformations of the admitted operators. Transformations (2.2) and (2.5) can therefore be used to multiply the solutions of system of equations (1.1), (1.2). The theorem presented below is obtained using transformation (2.2). Relations (2.5) can be used in the same way. Note that a similar approach has been used 14 to construct the solutions of the equations of motion of an ideal monatomic gas which allow of a projective transformation. Theorem 2.
If the set of functions
satisfies system of equations (1.1), (1.2), the set of functions
(2.6) satisfies the same system, where
¯ a is an arbitrary positive number, and the functions t¯ = t¯ ( , t), r¯ = r¯ ( ), ¯ = ( ) and = (t) are defined by the first three equalities of (2.2) and relations (2.3). 3. Time-periodic solutions The theorem formulated above enables us to obtain new exact solutions of Eqs (1.1) and (1.2) using known solutions. As an example, we carry out the group multiplication of a class of steady-state rotationally-symmetric solutions
(3.1) Here, V¯ (r)− is an arbitrary smooth function and h0 is a positive constant. Application of transformation (2.6) to solution (3.1) leads to the class of axisymmetric time-periodic solutions of the system of equations (1.1), (1.2) (3.2)
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Fig. 1.
The functions on the right-hand sides of the first two equations, as well as the functions r¯ = r¯ ( , r) and = (t), are given by equalities (2.3). √ The period of the solution T = 2/ω = / g is solely determined by the curvature of the paraboloid at the pole. Similar solutions have 7,9 been obtained in investigations of the axisymmetric oscillations of a liquid in a paraboloid of revolution. We shall only dwell on the analysis of the trajectories of the motions of the particles and the evolution of the material volume. The trajectories of motion of the particles for solution (3.2) have the form
(3.3) The constants r0 and 0 give the initial position of a particle. It follows from relations (3.3) that the trajectories of the liquid particles, located on circles of radius r0 when t = 0, are closed when the equality
is satisfied, where n and m are integers. In the general case, it can be asserted that the trajectories of the liquid particles for the class of solutions considered are ergodic (or quasiclosed), that is, for any > 0, an instant tε > 0 can be found such that, when t = tε , a particle will be at a distance of not more than from its position when t = 0. We next consider a characteristic example of a solution from the class (3.2). Suppose in solution (3.1) the circumferential velocity has the form (3.4) the corresponding multiplied solution is given by formulae (3.2) and (3.4). The character of the motion of the liquid is illustrated by the graphs obtained for the following parameter values
(the qualitative form of the graphs is preserved when these values are changed). The level of the liquid z = h + Z(r), lying in the paraboloid of revolution z = Z(r) = ω2 r2 /(8g) at the instants 2k/ω and(2k + 1)/ω, corresponding to the greatest and least localization of the liquid, is shown in Fig. 1. The closed trajectories of the motion of the particles, lying,
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Fig. 2.
Fig. 3.
√ when t = 0, on circles of radius r0 = nω/ ml ˛ ≈ 1.51, 0.97, 0.49 that corresponds to the choice of the pairs of numbers (n, m) = (2, 9), (1,7), (1,14) are shown in Fig. 2. The deformation of the material contour during the evolution of the flow (3.2), (3.4) is shown in Fig. 3. At the initial instant, the selected liquid contour is a circle of radius with centre at the point (0.4, 0.5). This contour is shown when t = 3k/ω(k = 0, 1, 2, 3, 4) in Fig. 3. The trajectory of a particle, emerging from a point with coordinates (0.7, 0.5) at t = 0 is depicted by dashes. As time increases, the contour is twisted into a spiral. Its deformation after a longer period of time as well as the deformation of another contour (the dot-dash curve) are shown in Fig. 3, b. Here, the circles are material contours when t = 0 and the spirals are material contours when t = 25/ω ≈ 35.12. It is interesting to note the qualitative agreement between the deformation of the material contour in the solution considered and the result of experiments on the transport of an impurity in a spinning liquid.15 The experiments were performed in a cylindrical container
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with a spinning disc on the bottom, which formed an axisymmetric vortex flow of the liquid. When spots of dye were introduced into the spinning liquid, the formation of spiral sleeves, similar to those shown in Fig. 3, was observed with time. Acknowledgements This research was financed by the Ministry of Education and Science of the Russian Federation (2.1.1/3543) and within the framework of the Programme of the Division of Power Engineering, Machine Building, Mechanics and Control Processes of the Russian Academy of Sciences, No. 2. 14.1. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
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Translated by E.L.S.