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Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations
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ARTICLE IN PRESS Journal of Applied Mathematics and Mechanics xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
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Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations夽 A.I. Rylov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
a r t i c l e
i n f o
Article history: Received 7 October 2013
a b s t r a c t The problems of constructing and revealing the functional relations between conservation laws, and constructing and finding additional conservation laws for the conservation laws previously obtained for three-dimensional unsteady flows (Terent’ev and Shmyglevskii, 1975), and for an infinite set of conservation laws for plane potential flows (Rylov, 2002), are considered. A functional relation means here that the sum of three or more left-hand sides of the divergent equations, with variable coefficients to be determined, is equal to zero. © 2015 Elsevier Ltd. All rights reserved.
In Section 1 we analyse a number of well-known conservation laws for three-dimensional unsteady flows1,2 and we construct some fairly simple and previously mentioned functional relations, closely related to the definition of additional conservation laws.3 In Section 2, for plane steady flows on a potential plane, we construct new additional conservation laws using four or two initial conservation laws. The initial laws are represented by the left-hand sides of divergent equations and/or their potentials. 1. Three-dimensional unsteady flows of an ideal gas We will show the relations between conservation laws, constructed previously,1,2 i.e., between the corresponding homogeneous divergent equations. Hence, it is also logical to use a homogeneous system,1,2 as the initial system, which, for clarity, we will write in vector form:
(1.1) Here u = (u, v, w) is the velocity vector in a Cartesian system x, y, z, is the density, p is the pressure, J = p/[(∗ – 1)] is the internal energy. Conservation laws have been previously constructed,1,2 i.e., the divergent equations
∗ is the adiabatic exponent, and
(1.2) where A,..., D are functions of t, x, y, z, u, , w, , J, while the functions u, , w, , J, in turn, depend on t, x, y, z. As a result, for A,..., D with
∗ =/ 5/3 relations were obtained containing nine arbitrary constants. Assuming all but one of these constants to be equal to zero, we arrive, as previously stated,1,2 at eleven divergent equations of the form (1.2), of which, for subsequent analysis, we distinguish ten:
(1.3)
夽 Prikl. Mat. Mekh., Vol. 79, No. 2, pp. 242–250, 2015. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006 0021-8928/© 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
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The eleventh divergent equation,1,2 in the special case considered here, is identical with the divergent continuity equation L0 = 0 of system (1.1), which will also be used below. The expressions for L1 , L2 ,..., L10 , more accurately, the expressions for Ai ,..., Di , occurring in Li , occur in generally accessible sources (Ref. 1, pp. 1540 – 1541, formulae (49) or Ref. 2, pp. 24 and 25, formulae (1.49)). Moreover, the pdf.file in Ref. 1 can be found in the site math.net.ru. The existence of such a list1,2 of divergent equations (1.3) indicates that, at least, some of them are related to one another, but these relations have not previously been discussed.1,2 Without attempting to consider all forms of such relations, we will confine ourselves to the fairly simple relations, connecting the conservation laws given above: (1.4) The zero right-hand side denotes that all the mathematical expressions of the left-hand side cancel one another. This, in particular, eliminates from consideration such a combination as L2 + L3 + L4 = 0, from which no conclusion can be drawn regarding the dependence, say, of L2 on L3 and L4 apart from the obvious conclusion that when L3 = L4 = 0, L2 = 0 also. The number of terms on the left-hand side of Eq. (1.4) is three. But this is not essential. There can be four or more terms. The factors Li , Lj , and Lk depend on the independent and on the dependent variables. Naturally, we will assume that the dimensionality principles are satisfied. It is relevant to recall here that a relation, similar to (1.4), was used3 when discussing the relation between the energy conservation law with combinations of the three laws of conservation of the components of the momentum vector. The idea of an additional conservation law, according to Godunov, applicable to relation (1.4), denotes that each of the three conservation laws occurring here can be regarded as a supplement to the two remaining ones, since combinations of them exist; for example,
In practice this denotes that, when analysing gas dynamic flows or when constructing a conservative difference scheme from the three conservation laws in (1.4) one can use, in the best case, only any two conservation laws, i.e., two divergent equations corresponding to them. Some authors4,5 understand by the term “additional conservation law” a new conservation law, differing from the known ones, all the more from the generally recognised laws, in no way linking it with other conservation laws. The problem of unifying the terminology is therefore highly appropriate. Formalizing the problem of constructing relations (1.4) is obviously fairly complex. Nevertheless, the principles of dimensions and the “trial and error” method enable us to write the following relations of the form (1.4):
(1.5) Verifying the first of relations (1.5), we have
This check, like the check of the remaining relations of (1.5), apart from the last, is fairly obvious. Verification of the last of relations (1.5) is much more complex, due to the fact that here the factors in front of L2 , L3 and L4 are the dependent variables u, v and w, respectively. However, it is not necessary to make this check here, since the corresponding problem was considered previously.3 Thereby, the conservation laws Li = 0 (i = 1,..., 10) from previous publications,1,2 and also L0 are related to, at least, eight relations of (1.5), the last of which was discussed above,3 while the remaining ones are, to all intents and purposes, new. We also see from relations (1.5) that each of the conservation laws Li = 0 (i = 1,..., 10) will be supplementary (according to Godunov) with respect to some of these laws. 2. The potentials of the divergent equations and new additional conservation laws of plane potential flows Consider the non-linear Chaplygin equations, describing plane potential gas flows:
(2.1) Here and below q and are the modulus and angle of inclination of the velocity vector, , p, a and M = q/a are the density, pressure, sound velocity and the Mach number, u = qcos, = qsin, and and are the potential and stream function of the flow. System (1) is closely related to the linear Chaplygin equations in the hodograph plane (2.2) Obviously, here the second-order equation is a corollary of the first-order system of equations. Linear equations (2.2), as is well known, have an infinite number of exact solutions, obtained, for example, by separation of the variables. But it is necessary to note that, as a rare exception, we mean here by exact solutions, solutions which reduce to quadratures or to model ordinary differential equations. Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
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3
As was shown in Ref. 6, for each solution of the form
of Eqs (2.2), there is a non-linear analogue of it in the plane of the potential (2.3) Some mathematical aspects of the presence of an infinite number of conservation laws from system (2.3) were discussed previously in Ref. 7. The divergent equation from system (2.3) in the (x, y) plane, using the obvious divergent equations
can be rewritten as: (2.4) The presence of a divergent equation in system (2.3) enables us to speak of a potential of this equation R (not to be confused with the potential of the flow ), such that R = g and R = f. As has been shown, these potentials R may play an important role when constructing new additional conservation laws. For example, we can consider as R the integral (2.5) taken along the streamline. In fact, a check shows that
Note that one can use as R the integral R = fd, taken along the line = const, or the integral over the arbitrary curve
But, taking into account the fact that in the flow around the body, a typical situation for which the stagnation line will be the line of discontinuity of some potentials, we will use definition (2.5). We will demonstrate the possibility of constructing new additional conservation laws, or, which is practically the same, the possibility of constructing new functional relations between the conservation laws, using the potentials of the divergent equations. To do this we will consider four conservation laws and the divergent equations and potentials corresponding to them. We will have
We will consider a combination, equal to zero, in which the potentials R1 and R2 act as factors for the left-hand sides of the divergent equations with subscripts 3 and 4: (2.6) Further, carrying out the necessary calculations, we obtain from (2.6)
We thereby arrive at the following theorem. Theorem. The functional combination of four conservation laws (2.6), when the condition = 0 is satisfied, leads to a new additional conservation law (2.7) The operator can be rewritten in the form of the following determinants, equivalent to one another, the first of which can be regarded as the vector analogue of the Jacobian:
When calculating the determinants, the diagonal elements are multiplied by one another using the scalar product rules. Corollary. If R1 and R3 , and also R2 and R4 change places in functional combination (2.6), then, apart from the sign, the value of remains unchanged. In other words, if = 0, the combination
corresponds to conservation law (2.7), if R1 and R3 , and also R2 and R4 change places. Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
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To illustrate the theorem, we will first consider four analogues of Chaplygin’s equations for the separation constant , occurring in the ordinary differential equation
with two independent solutions h(1) and h(2) . For the conservation laws occurring in these analogues, fi and gi are written as:6,8
It can be seen from these relations that
and, consequently, according to the theorem, the following additional conservation law holds:
(2.8) According to the corollary of the theorem, the additional conservation law, corresponding to (2.8), holds if R1 and R3 , and also R2 and R4 , change places. These conservation laws were in fact guessed earlier.6 We will now consider four conservation laws, corresponding to the separation constant , related to the ordinary differential equation
We have
Consequently, according to the theorem and corollary we have the following additional conservation laws:
(2.9) A non-stationary analogue of conservation laws (2.9) was presented in Ref. 9. We will consider in more detail the case = 1, for which the functions h(1) and h(2) and their derivatives are written explicitly, and the potentials of the initial and additional conservation laws, as a rule, have a physical meaning. We have
(2.10) Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
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The potentials R1 ,..., R4 of the divergent equations have the form
In fact, we will consider the following integrals along the streamline from the point a to the point :
Here X – Xa is the horizontal force, produced by the section of the streamline (a, ) on the assumption that there is a vacuum above the streamline. We treat the quantity Y – Ya similarly. The following conservation laws correspond to additional conservation law (2.8), and also its corollary (i.e., to the same law (2.8), in which the potentials R1 and R3 , and also R2 and R4 and their derivatives change places, when = 1:
(2.11)
The first of these has been known for a long time as the law of conservation of angular momentum. For a displacement along the streamline from point a to the current point by an amount , there corresponds an increment of the potential
(2.12) which also corresponds exactly to the increment in the angular momentum. The increment in the potential of the second of conservation laws (2.11) is:
(2.13) The physical meaning of the potentials y, x, X, Y and M is quite clear. However, there is no obvious explanation for the potential M*. The second of the conservation laws was constructed previously,6 and then10 some relations were constructed, connecting both all the potentials y, x, X, Y, M and M*, and also certain new potentials based on them. We consider this problem below. Consider the special case, when, in combination (2.6), R3 = R2 and R4 = R1 . As a result, as can easily be verified, = 0 and, consequently, we have additional conservation law (2.7), with
The potential R of this additional conservation law is equal to the product of the potentials of the initial conservation laws, i.e., R = R1 R2 . In other words, here both initial potentials are equally represented, unlike laws (2.7)–(2.9) and (2.11), in which the four initial potentials are represented either directly in the form of factors or in the form of derivatives. In conclusion we will present an example of a relation, connecting several additional conservation laws, corresponding to law (2.7) for = 0 and = 1, where one of these laws corresponds to two initial laws while the other corresponds to four. To do this we add relations (2.12) and (2.13) and obtain10
Naturally, this relation does not indicate any physical meaning of the potential M*, but it gives a simple relation between the potentials M, M*, yX and xY. Acknowledgement This research was partially financed by the Russian Foundation for Basic Research (12-01-00648-a). References 1. 2. 3. 4.
Terent’ev YeD, Shmyglevskii YuD. The complete system of divergent equations of ideal gas dynamics. Zh Vychisl Mat Mat Fiz 1975;15(6):1535–44. Shmyglevskii YuD. Analytic Investigations of Fluid Dynamics. Moscow: Editorial URSS; 1999. Godunov SK. An interesting class of quasi-linear systems. Dokl Akad Nauk SSSR 1961;139(3):521–3. Kozlov VV. Symmetries, Topology and Resonances in Hamiltonian Mechanics. Izhevsk: Izd Udmurt Gos Univ; 1995.
Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
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6 5. 6. 7. 8. 9. 10.
Chirkunov YuA. Non-local conservation laws for the equations of irrotational isentropic plane-parallel gas motion. J Appl Math Mech 2012;76(2):199–204. Rylov AI. Chaplygin equations and an infinite set of uniformly divergent gas-dynamics equations. Dokl Physics 2002;47(3):173–5. Chirkunov YuA. Conservation laws and group properties of the equations of isentropic gas motion. J Appl Mech Tech Phys 2010;51(1):1–3. Rylov AI. Functional dependence between gas dynamics conservation laws corresponding to separation of variables. Dokl Math 2014;89(1):124–7. Rylov AI. A non-stationary analogue of Chaplygin’s equations in one-dimensional gas dynamics. J Appl Math Mech 2005;69(2):222–33. Rylov AI. Potentials of divergent equations and new additional conservation laws for two-dimensional steady gas flows. Dokl Physics 2012;57(12):483–6.
Translated by R.C.G.
Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
ARTICLE IN PRESS Journal of Applied Mathematics and Mechanics xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations夽 A.I. Rylov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
a r t i c l e
i n f o
Article history: Received 7 October 2013
a b s t r a c t The problems of constructing and revealing the functional relations between conservation laws, and constructing and finding additional conservation laws for the conservation laws previously obtained for three-dimensional unsteady flows (Terent’ev and Shmyglevskii, 1975), and for an infinite set of conservation laws for plane potential flows (Rylov, 2002), are considered. A functional relation means here that the sum of three or more left-hand sides of the divergent equations, with variable coefficients to be determined, is equal to zero. © 2015 Elsevier Ltd. All rights reserved.
In Section 1 we analyse a number of well-known conservation laws for three-dimensional unsteady flows1,2 and we construct some fairly simple and previously mentioned functional relations, closely related to the definition of additional conservation laws.3 In Section 2, for plane steady flows on a potential plane, we construct new additional conservation laws using four or two initial conservation laws. The initial laws are represented by the left-hand sides of divergent equations and/or their potentials. 1. Three-dimensional unsteady flows of an ideal gas We will show the relations between conservation laws, constructed previously,1,2 i.e., between the corresponding homogeneous divergent equations. Hence, it is also logical to use a homogeneous system,1,2 as the initial system, which, for clarity, we will write in vector form:
(1.1) Here u = (u, v, w) is the velocity vector in a Cartesian system x, y, z, is the density, p is the pressure, J = p/[(∗ – 1)] is the internal energy. Conservation laws have been previously constructed,1,2 i.e., the divergent equations
∗ is the adiabatic exponent, and
(1.2) where A,..., D are functions of t, x, y, z, u, , w, , J, while the functions u, , w, , J, in turn, depend on t, x, y, z. As a result, for A,..., D with
∗ =/ 5/3 relations were obtained containing nine arbitrary constants. Assuming all but one of these constants to be equal to zero, we arrive, as previously stated,1,2 at eleven divergent equations of the form (1.2), of which, for subsequent analysis, we distinguish ten:
(1.3)
夽 Prikl. Mat. Mekh., Vol. 79, No. 2, pp. 242–250, 2015. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006 0021-8928/© 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
G Model JAMM-2281; No. of Pages 6 2
ARTICLE IN PRESS A.I. Rylov / Journal of Applied Mathematics and Mechanics xxx (2015) xxx–xxx
The eleventh divergent equation,1,2 in the special case considered here, is identical with the divergent continuity equation L0 = 0 of system (1.1), which will also be used below. The expressions for L1 , L2 ,..., L10 , more accurately, the expressions for Ai ,..., Di , occurring in Li , occur in generally accessible sources (Ref. 1, pp. 1540 – 1541, formulae (49) or Ref. 2, pp. 24 and 25, formulae (1.49)). Moreover, the pdf.file in Ref. 1 can be found in the site math.net.ru. The existence of such a list1,2 of divergent equations (1.3) indicates that, at least, some of them are related to one another, but these relations have not previously been discussed.1,2 Without attempting to consider all forms of such relations, we will confine ourselves to the fairly simple relations, connecting the conservation laws given above: (1.4) The zero right-hand side denotes that all the mathematical expressions of the left-hand side cancel one another. This, in particular, eliminates from consideration such a combination as L2 + L3 + L4 = 0, from which no conclusion can be drawn regarding the dependence, say, of L2 on L3 and L4 apart from the obvious conclusion that when L3 = L4 = 0, L2 = 0 also. The number of terms on the left-hand side of Eq. (1.4) is three. But this is not essential. There can be four or more terms. The factors Li , Lj , and Lk depend on the independent and on the dependent variables. Naturally, we will assume that the dimensionality principles are satisfied. It is relevant to recall here that a relation, similar to (1.4), was used3 when discussing the relation between the energy conservation law with combinations of the three laws of conservation of the components of the momentum vector. The idea of an additional conservation law, according to Godunov, applicable to relation (1.4), denotes that each of the three conservation laws occurring here can be regarded as a supplement to the two remaining ones, since combinations of them exist; for example,
In practice this denotes that, when analysing gas dynamic flows or when constructing a conservative difference scheme from the three conservation laws in (1.4) one can use, in the best case, only any two conservation laws, i.e., two divergent equations corresponding to them. Some authors4,5 understand by the term “additional conservation law” a new conservation law, differing from the known ones, all the more from the generally recognised laws, in no way linking it with other conservation laws. The problem of unifying the terminology is therefore highly appropriate. Formalizing the problem of constructing relations (1.4) is obviously fairly complex. Nevertheless, the principles of dimensions and the “trial and error” method enable us to write the following relations of the form (1.4):
(1.5) Verifying the first of relations (1.5), we have
This check, like the check of the remaining relations of (1.5), apart from the last, is fairly obvious. Verification of the last of relations (1.5) is much more complex, due to the fact that here the factors in front of L2 , L3 and L4 are the dependent variables u, v and w, respectively. However, it is not necessary to make this check here, since the corresponding problem was considered previously.3 Thereby, the conservation laws Li = 0 (i = 1,..., 10) from previous publications,1,2 and also L0 are related to, at least, eight relations of (1.5), the last of which was discussed above,3 while the remaining ones are, to all intents and purposes, new. We also see from relations (1.5) that each of the conservation laws Li = 0 (i = 1,..., 10) will be supplementary (according to Godunov) with respect to some of these laws. 2. The potentials of the divergent equations and new additional conservation laws of plane potential flows Consider the non-linear Chaplygin equations, describing plane potential gas flows:
(2.1) Here and below q and are the modulus and angle of inclination of the velocity vector, , p, a and M = q/a are the density, pressure, sound velocity and the Mach number, u = qcos, = qsin, and and are the potential and stream function of the flow. System (1) is closely related to the linear Chaplygin equations in the hodograph plane (2.2) Obviously, here the second-order equation is a corollary of the first-order system of equations. Linear equations (2.2), as is well known, have an infinite number of exact solutions, obtained, for example, by separation of the variables. But it is necessary to note that, as a rare exception, we mean here by exact solutions, solutions which reduce to quadratures or to model ordinary differential equations. Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
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As was shown in Ref. 6, for each solution of the form
of Eqs (2.2), there is a non-linear analogue of it in the plane of the potential (2.3) Some mathematical aspects of the presence of an infinite number of conservation laws from system (2.3) were discussed previously in Ref. 7. The divergent equation from system (2.3) in the (x, y) plane, using the obvious divergent equations
can be rewritten as: (2.4) The presence of a divergent equation in system (2.3) enables us to speak of a potential of this equation R (not to be confused with the potential of the flow ), such that R = g and R = f. As has been shown, these potentials R may play an important role when constructing new additional conservation laws. For example, we can consider as R the integral (2.5) taken along the streamline. In fact, a check shows that
Note that one can use as R the integral R = fd, taken along the line = const, or the integral over the arbitrary curve
But, taking into account the fact that in the flow around the body, a typical situation for which the stagnation line will be the line of discontinuity of some potentials, we will use definition (2.5). We will demonstrate the possibility of constructing new additional conservation laws, or, which is practically the same, the possibility of constructing new functional relations between the conservation laws, using the potentials of the divergent equations. To do this we will consider four conservation laws and the divergent equations and potentials corresponding to them. We will have
We will consider a combination, equal to zero, in which the potentials R1 and R2 act as factors for the left-hand sides of the divergent equations with subscripts 3 and 4: (2.6) Further, carrying out the necessary calculations, we obtain from (2.6)
We thereby arrive at the following theorem. Theorem. The functional combination of four conservation laws (2.6), when the condition = 0 is satisfied, leads to a new additional conservation law (2.7) The operator can be rewritten in the form of the following determinants, equivalent to one another, the first of which can be regarded as the vector analogue of the Jacobian:
When calculating the determinants, the diagonal elements are multiplied by one another using the scalar product rules. Corollary. If R1 and R3 , and also R2 and R4 change places in functional combination (2.6), then, apart from the sign, the value of remains unchanged. In other words, if = 0, the combination
corresponds to conservation law (2.7), if R1 and R3 , and also R2 and R4 change places. Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
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To illustrate the theorem, we will first consider four analogues of Chaplygin’s equations for the separation constant , occurring in the ordinary differential equation
with two independent solutions h(1) and h(2) . For the conservation laws occurring in these analogues, fi and gi are written as:6,8
It can be seen from these relations that
and, consequently, according to the theorem, the following additional conservation law holds:
(2.8) According to the corollary of the theorem, the additional conservation law, corresponding to (2.8), holds if R1 and R3 , and also R2 and R4 , change places. These conservation laws were in fact guessed earlier.6 We will now consider four conservation laws, corresponding to the separation constant , related to the ordinary differential equation
We have
Consequently, according to the theorem and corollary we have the following additional conservation laws:
(2.9) A non-stationary analogue of conservation laws (2.9) was presented in Ref. 9. We will consider in more detail the case = 1, for which the functions h(1) and h(2) and their derivatives are written explicitly, and the potentials of the initial and additional conservation laws, as a rule, have a physical meaning. We have
(2.10) Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006
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The potentials R1 ,..., R4 of the divergent equations have the form
In fact, we will consider the following integrals along the streamline from the point a to the point :
Here X – Xa is the horizontal force, produced by the section of the streamline (a, ) on the assumption that there is a vacuum above the streamline. We treat the quantity Y – Ya similarly. The following conservation laws correspond to additional conservation law (2.8), and also its corollary (i.e., to the same law (2.8), in which the potentials R1 and R3 , and also R2 and R4 and their derivatives change places, when = 1:
(2.11)
The first of these has been known for a long time as the law of conservation of angular momentum. For a displacement along the streamline from point a to the current point by an amount , there corresponds an increment of the potential
(2.12) which also corresponds exactly to the increment in the angular momentum. The increment in the potential of the second of conservation laws (2.11) is:
(2.13) The physical meaning of the potentials y, x, X, Y and M is quite clear. However, there is no obvious explanation for the potential M*. The second of the conservation laws was constructed previously,6 and then10 some relations were constructed, connecting both all the potentials y, x, X, Y, M and M*, and also certain new potentials based on them. We consider this problem below. Consider the special case, when, in combination (2.6), R3 = R2 and R4 = R1 . As a result, as can easily be verified, = 0 and, consequently, we have additional conservation law (2.7), with
The potential R of this additional conservation law is equal to the product of the potentials of the initial conservation laws, i.e., R = R1 R2 . In other words, here both initial potentials are equally represented, unlike laws (2.7)–(2.9) and (2.11), in which the four initial potentials are represented either directly in the form of factors or in the form of derivatives. In conclusion we will present an example of a relation, connecting several additional conservation laws, corresponding to law (2.7) for = 0 and = 1, where one of these laws corresponds to two initial laws while the other corresponds to four. To do this we add relations (2.12) and (2.13) and obtain10
Naturally, this relation does not indicate any physical meaning of the potential M*, but it gives a simple relation between the potentials M, M*, yX and xY. Acknowledgement This research was partially financed by the Russian Foundation for Basic Research (12-01-00648-a). References 1. 2. 3. 4.
Terent’ev YeD, Shmyglevskii YuD. The complete system of divergent equations of ideal gas dynamics. Zh Vychisl Mat Mat Fiz 1975;15(6):1535–44. Shmyglevskii YuD. Analytic Investigations of Fluid Dynamics. Moscow: Editorial URSS; 1999. Godunov SK. An interesting class of quasi-linear systems. Dokl Akad Nauk SSSR 1961;139(3):521–3. Kozlov VV. Symmetries, Topology and Resonances in Hamiltonian Mechanics. Izhevsk: Izd Udmurt Gos Univ; 1995.
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Please cite this article in press as: Rylov AI. Additional conservation laws, functional relations between conservation laws, and potentials of divergent gas dynamics equations. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.09.006