Nonlinear Dynamics 36: 19–28, 2004. C 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Irrotational Barochronous Gas Motions∗ SERGEY GOLOVIN Lavrentyev Institute of Hydrodynamics SD RAS, Lavrentyev Prospect, 15, Novosibirsk 630090, Russia (e-mail:
[email protected]) (Received: 20 September 2003; accepted: 14 November 2003)
Abstract. A full description of irrotational ideal gas motions is given, in which pressure depends only on time. It is shown that under such motions the dependence of velocity vector on spatial coordinates is linear and of a special kind. Key words: barochronous motion, equivalence transformation, ideal gas, irrotational flow, multiple wave
1. Introduction Barochronous motions of an ideal gas, i.e., such motions in which pressure is a function of time only, are carefully studied in [1, 2]. These papers give the formulas of general solution (in an implicit form) as well as numerous examples of the analysis of specific exact solutions. At the same time, the description of special classes of barochronous solutions, which describe gas motions with some additional properties is of interest. In the present work, the potentiality of the velocity field is chosen as such a property. Exact solutions describing irrotational gas motions (especially in the case of twodimensional steady flows) present a classic subject of study. However, the problem of full description of barochronous irrotational ideal gas motions apparently has not yet been apparently stated. In this work it is completely solved. The explicit formulas of general solution are presented. The analysis essentially uses a classification result [3], reducing the investigation to the study of solutions of multiple-wave type.
2. Preliminary Information The equations describing motions of ideal gas are the following [4]: Du + ρ −1 ∇ p = 0,
Dρ + ρ div u = 0,
DS = 0,
D = ∂t + u∂x + v∂ y + w∂z
(1)
Here u is a velocity vector, p is pressure, ρ is density, and S is entropy. System (1) is closed by the state equation p = F(ρ, S). Further, we restrict our investigations to the case of barochronous potential gas motions. Barochronous motions of the gas are motions where pressure and density depend on time only: p = p(t),
ρ = ρ(t).
According to the state equation p = F(ρ, S) entropy S is a constant S = const.
(2)
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Remark 1. For simplicity we here treat the isentropic case S = const. Generalization to the nonisentropic case is straightforward for the gases with special state equations. In particular, all adduced facts related to the velocity field are also valid for a state equation of the form ρ = f ( p)g(S) (Prim’s gas or gas with split density). The complete theoretical investigations of barochronous motions of an ideal gas can be found in Chupakhin’s work [1, 2]. We adduce here some of Chupakhin’s results, which will be used further. 2.1. BAROCHRONOUS GAS MOTIONS According to (1) and (2) the barochronous motions of ideal gas are described by the following system of equations D u = 0,
div u = −
ρ (t) , ρ
D = ∂t + u · ∇.
(3)
The right side of the last equation of (3) is a function of t only. Thus system (3) is an overdetermined system for velocity vector u. Description of its compatibility conditions is given in terms of the Jacobi matrix J = ∂u/∂x. Theorem 1. The complete system of compatibility conditions of overdetermined system (3) has the form of the following relations between algebraic invariants jm of Jacobi matrix J D j1 + j12 − 2 j2 = 0, D j2 + j1 j2 − 3 j3 = 0, D j3 + j1 j3 = 0.
(4)
Here j1 = tr J, . . . , j3 = det J . The general solution of system (4) is given by the formulas jk =
1 Dk Q , k! Q
k = 1, 2, 3;
Q = 1 + j10 t + j20 t 2 + j30 t 3
(5)
Density ρ is determined from the equation of continuity as ρ = ρ0 /Q.
(6)
Constants jk0 , ρ0 are arbitrary initial values at t = 0 of invariants jk and density ρ, respectively. From the first equations of (3) it follows that j1 = tr J is a function of t only. Thus Equations (4) state that another two invariants jm , m = 2, 3 are also functions of t only. Corollary 1. The initial velocity field of barochronous motion has a Jacobi matrix J0 = ∂u0 /∂x0 with constant algebraic invariants. Conversely, any stationary vector field with constant algebraic invariants of its Jacobi matrix serves as initial velocity field for some barochronous gas motion.
Irrotational Barochronous Gas Motions
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In view of these results it is possible to state a Cauchy problem for systems (1) and (2) with initial data at t = 0 u(0, x) = u0 (x),
ρ(0, x) = ρ0 ,
S(0, x) = S0 .
(7)
The solution of a Cauchy problem is described in terms of Lagrangian variables. New independent variables are introduced (t, x) → (t, ξ = x − tu),
u = v(t, ξ).
(8)
The Jacobi matrix N = ∂v/∂ξ is related to J by the equation (E − t J )(E − t N ) = E.
(9)
Here and below E is an identity matrix. Denote D = D(t, ξ) = det(E + t N ). Rewriting (1) and (2) in variables (8) we obtain vt = 0,
(ρD)t = 0,
St = 0.
(10)
From these we have N = N (ξ); D = D(t). Theorem 2. Solution of the Cauchy problem (7) for systems (1), (2) is given by the formulas u = u0 (ξ),
ρ = ρ0 /D,
S = S0 .
(11)
From the above it follows that any barochronous solution of a gas dynamics equations is completely characterized by its initial velocity field. The latter can not be arbitrary, namely, algebraic invariants for the Jacobi matrix initial velocity field are constants. All vector fields with such properties are described in [1]. For any admissible initial velocity field u0 = v(ξ), the solution at any moment of time is restored by transformation (8), which is equivalent to substitution ξ → x − tu,
u0 → u.
(12)
Barochronous motions of an ideal gas have many interesting properties. The trajectories of particles in such motions are straight lines. However, the whole motion is non-trivial. The typical feature of barochronous motions is the collapse of density at a finite moment of time. At that time all gas particles simultaneously come to some manifold of lower dimension than the dimension of motion. The behavior of sonic and contact characteristics of gas dynamics equations in the neighborhood of collapse is already known [2]. 2.2. IRROTATIONAL GAS MOTIONS The irrotational (potential) motions of a gas are a much more classical object of investigation. Irrotational gas motions are distinguished by a special kind of the velocity field: u = ∇ϕ
(13)
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with some potential ϕ(t, x, y, z). The state equation in the isentropic case S = const reads p = f (ρ). Integration of momentum equations of (1) gives the Cauchy–Lagrange integral 1 ϕt + |∇ϕ|2 + i( p) = 0. 2 Here i( p) = ρ −1 d p is a specific enthalpy. The continuity equation provides i t + ∇ϕ · ∇i + a 2 ϕ = 0.
(14)
(15)
In this paragraph a 2 = f (ρ) is the square of sound speed. Equations (14) and (15) serve for description of irrotational gas motions. One can obtain a single second-order equation for ϕ by substitution of specific enthalpy i from (14) into (15).
2.3. IRROTATIONAL BAROCHRONOUS GAS MOTIONS In present paper we combine two properties described above. We look for solutions which are simultaneously irrotational and barochronous. There are two ways of solving the problem. The first is to start from Equations (14) and (15) and to demand the solution to be barochronous. Equations (2) imply that all thermodynamic functions depend on time only: i = i(t), a = a(t). Hence we obtain an overdetermined system of two equations (14) and (15) for one function ϕ(t, x, y, z). This system must be completed to involution. The second way is to start from the barochronous solution taking property (13) into consideration. As noted above, the description of barochronous gas motions is reduced to the investigation of the equations for the initial velocity field (hereafter we omit zeros at u0 and replace ξ by x) u x + v y + wz = j1 , u x u y v y vz wz w x + + vx v y w y wz u z u x = j2 , ux u y uz vx v y vz = j3 . w w w x y z
(16)
The constants j1 , j2 , j3 can take arbitrary real values. Since Equation (13) is valid during the whole time of motion it is also valid for the initial time t = 0. Thus, to describe potential barochronous gas motions it is necessary to study system (16) with the substitution of the velocity field (13). In other words, the investigation of potential barochronous gas motions is reduced to description of all functions ϕ(x, y, z), which have a Hesse matrix with constant algebraic invariants. Below we use this second approach.
3. Equivalence Transformations To simplify the analysis of system (16) it is convenient to transform vector j = ( j1 , j2 , j3 ) to some canonical form. The group of equivalence transformations for system (16) is known [3]. This group is
Irrotational Barochronous Gas Motions
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18-dimensional and generated by the following set of operators: X k = ∂x k ,
X k+3 = ∂u k ,
Mlk = x k ∂x l + u k ∂ul ,
T1 = x k ∂u k + 3∂ j1 + 2 j1 ∂ j2 + j2 ∂ j3 , T2 = u k ∂u k − x k ∂x k + 2 j1 ∂ j1 + 4 j2 ∂ j2 + 6 j3 ∂ j3 , T3 = u k ∂x k + (2 j2 − j12 )∂ j1 + (3 j3 − j1 j2 )∂ j2 − j1 j3 ∂ j3 .
(17)
Here k, l = 1, 2, 3; the summing is performed over repeated indexes. For brevity, index notation of coordinates and components of velocity is used: x = (x 1 , x 2 , x 3 ), u = (u 1 , u 2 , u 3 ). Note that the transformations generated by the operators X i (translations and Galilean translations along the axes of coordinates) are inherited from the transformations admitted by the initial gas dynamics equations j [5]. Besides, combinations M ij − Mi generate matched rotations in spaces IR 3 (x) and IR 3 (u). They are also inherited from the group, admitted by the initial equations. Below we use the enumerated transformations to eliminate insignificant constants in solutions obtained. The transformations generated by operators Ti are classifying because they nontrivially act in the space of constants IR 3 (j). It is known [3] that due to transformations Ti , any system (16) is equivalent to one of the four canonical systems with vector j of the form 10 (0, 0, 0);
20 (1, 0, 0);
30 (0, 1, 0);
40 (0, −1, 0).
(18)
The principal moment is that in all four cases j3 = 0. This means that there exists a perfect relationship between functions u, v and w for system (16) in a canonical form. Thus, the initial velocity field of barochronous motion is equivalent to some double u = u(v, w) or ‘sesquilateral’ u = u(v), w = w(x, y, z) wave. Thus we have to investigate system (16) with substitution of relation (13) for all the cases (18). Below, we do it step-by-step for the double wave, two-dimensional gas motions and ‘sesquilateral’ wave. The result is presented in the last paragraph. Enumeration of all possible initial velocity fields of potential barchronous motion is given in Theorems 3 and 4. Both general form of solution for the arbitrary moment of time and dependence ρ(t) are specified in Theorem 5. 4. Double Wave Let the derivatives of function ϕ be related to the expression ϕx = u(ϕ y , ϕz ).
(19)
After finding the absolute integral [6] of Equation (19), its general solution is represented in a parametric form: ϕ = u( p1 , p2 ) x + p1 y + p2 z + ψ 0 ( p1 , p2 ).
(20)
Here ψ 0 ( p1 , p2 ) is the arbitrary function; p1 , p2 are parameters, which are related to initial variables by the equalities 0=
∂u ∂ψ 0 x+y+ , ∂ p1 ∂ p1
0=
∂u ∂ψ 0 x+z+ . ∂ p2 ∂ p2
(21)
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Next, it is convenient to move from independent variables (x, y, z) to independent variables (x, p1 , p2 ) according to formulas (21). The Jacobian of such transformation
∂(x, y, z) = = ∂(x, p1 , p2 )
∂ 2ψ 0 ∂ 2u x+ 2 ∂ p1 ∂ p12
∂ 2u ∂ 2ψ 0 x+ 2 ∂ p2 ∂ p22
−
∂ 2u ∂ 2ψ 0 x+ ∂ p1 ∂ p2 ∂ p1 ∂ p2
2 (22)
differs from zero due to the arbitrariness of the choice of function ψ 0 ( p1 , p2 ). According to (20) and (21) the derivatives of function ϕ are given by the following formulas ϕx = u( p1 , p2 ), ∂ p1 1 = ∂x
ϕ y = p1 ,
ϕ z = p2 .
∂u ∂u 2 − 3 , ∂ p2 ∂ p1
∂ p1 3 =− , ∂y
∂ p2 2 = , ∂y
∂ p2 1 = ∂x ∂ p1 2 = , ∂z
(23)
∂u ∂u 2 − 1 , ∂ p1 ∂ p2 ∂ p2 1 =− . ∂z
(24)
The following notation is introduced 1 = x
∂ 2u ∂ 2ψ 0 + , 2 ∂ p1 ∂ p12
2 = x
∂ 2u ∂ 2ψ 0 + , ∂ p1 ∂ p2 ∂ p1 ∂ p2
3 = x
∂ 2u ∂ 2ψ 0 + . 2 ∂ p2 ∂ p22
(25)
The substitution into the first two equations (16) gives ∂u 2 ∂u ∂u ∂u 2 − 1+ 1 + 2 3 = c1 , 2 − 1 + ∂ p2 ∂ p1 ∂ p2 ∂ p1 1+
∂u ∂ p1
2
+
∂u ∂ p2
2 = c2 .
(26)
From the last equation (26) it follows that c2 = 0, i.e., cases 10 and 20 from classification (18) are not considered here. Let us consider the remaining cases 30 and 40 . Here c1 = 0, c2 = ±1. After the splitting of Equations (26) on independent variable x we obtain five equations. Among them the following ones are interesting. Linear with respect to x term in the first equation (26) gives the equation of minimal surfaces,
1+
∂u ∂ p2
2
∂ 2u ∂u ∂u ∂ 2 u ∂u 2 ∂ 2 u −2 + 1+ = 0. ∂ p1 ∂ p2 ∂ p1 p2 ∂ p1 ∂ p12 ∂ p22
(27)
The coefficient of x 2 in the second equation (26) gives the Monge–Amp`ere equation ∂ 2u ∂ 2u = ∂ p12 ∂ p22
∂ 2u ∂ p1 p2
2 .
(28)
Thus, surfaces z = u(x, y) with function u satisfying Equations (27) and (28) are enveloping and minimal. A set of such surfaces turns out to become exhausted by planes.
Irrotational Barochronous Gas Motions
25
Indeed, from Equation (28) follows a perfect relationship of derivatives u p1 = g(u p2 ). Let us calculate the second derivatives: u p1 p1 = g u p2 p2 ,
u p1 p2 = g u p2 p2 ,
u p2 p2 = u p2 p2 .
(29)
1 + u 2p2 (g )2 u p2 p2 − 2u p2 gg u p2 p2 + (1 + g 2 ) u p2 p2 = 0.
(30)
Substitution into (27) gives
From Equation (30) follows u p2 p2 = 0 as (1 + ξ 2 )g 2 − 2ξ gg + (1 + g 2 ) = 1 + g 2 + (ξ g − g)2 = 0.
(31)
Together with (29) this means that u( p1 , p2 ) is a linear function. Returning to the initial notation we obtain linear relation between the derivatives of function ϕ: ϕx = aϕ y + bϕz + c,
a, b, c = const.
(32)
Equation (32) is integrated in the form ϕ = cx + ϕ 0 (ax + y, bx + z). Accurate to the transformations of rotation and Galilean translation we can assume that a = b = c = 0. Thus, function ϕ is equivalent to the following one: ϕ = ϕ(y, z). We will consider this possibility separately. 5. Two-Dimensional Motions Let ϕ = ϕ(x, y). Due to (16) we have ϕxx + ϕyy = c1 ,
ϕxx ϕyy − (ϕxy )2 = c2 .
(33)
Consider all the possibilities (18) step-by-step. 10 · c1 = c2 = 0. Then ϕx = g(ϕ y ). From the first equation (33) is follows (1 + g 2 )ϕyy = 0. It means that function ϕ is linear with respect to variables, i.e., equivalent to the constant accurate to Galilean translation. 20 · c1 = 1, c2 = 0. Let us find the second derivatives: ϕxx =
g 2 , 1 + g 2
ϕxy =
g , 1 + g 2
ϕyy =
1 . 1 + g 2
(34)
The calculation of mixed derivative ϕx yy due to (34) by two ways gives g = 0, which means g = a = const. Function ϕ is found explicitly in the form ϕ=
a2 1 a 1 (ax + y)2 2 2 x y ∼ x 2. + x y + = 2 2 2 2 2(1 + a ) 1+a 2(1 + a ) 2(1 + a ) 2
(35)
Here and below, sign ‘∼’ denotes the equivalence due to the transformations of translation, Galilean translation and rotation. 30 , 40 · c1 = 0, c2 = ±1. Let us express derivative ϕyy from the first equation (33) and substitute into the second one. We have 2 2 −(ϕxx + ϕxy ) = ±1.
(36)
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Sign ‘+’ is not suitable, i.e., case 30 does not give solutions. In case 40 we have a representation ϕxx = cos α,
ϕxy = sin α,
ϕyy = − cos α.
(37)
With the help of cross differentiation we obtain α = const. Hence, we find the expression for function ϕ: ϕ=
1 1 1 cos α x 2 + sin α xy − cos α y 2 ∼ (x 2 − y 2 ). 2 2 2
(38)
6. ‘Sesquilateral’ Wave In this case the relation of derivatives is given by the formula ϕx = u(ϕ y ),
ϕz —arbitrary function.
(39)
The substitution in (16) gives (1 + u 2 ) ϕyy + ϕzz = c1 ,
(40)
2 (ϕyy ϕzz − ϕyz ) (1 + u 2 ) = c2 .
First consider cases 10 , 20 according to classification (18). Then c2 = 0 which means that there is a relation of derivatives ϕz = h(ϕ y ). From the first equation (40) follows (1 + u 2 + h 2 ) ϕyy = c1
(41)
The expressions of the second derivatives are: ϕxx = ϕyz =
u 2 , 1 + u 2 + h 2 h 1 + u 2 + h
, 2
ϕxy = ϕzz =
u
,
ϕyy =
h 2 , 1 + u 2 + h 2
ϕx z =
1+
u 2
+
h 2
1 1+
u 2
+ h 2
,
u h . 1 + u 2 + h 2
(42)
In case 10 we obtain ϕyy = 0 which means that ϕ is linear with respect to (x, y, z). In case 20 the calculation of mixed derivative ϕx yz by three ways gives u h = u h = u h + u h . Hence the following situations are possible: a) u = const, b) h = const, c) u, h are linear. In case a) we have ϕ = ax + ϕ0 (y, z) ∼ ϕ0 (y, z). It is reduced to two-dimensional motions. Case b) is analogous. Case c) gives two linear equations for ϕ: ϕx = a ϕ y + b,
ϕz = c ϕ y + d,
a, b, c, d = const.
(43)
Hence, ϕ is linear, which means that it is equivalent to the constant. The possibilities 30 , 40 remain. Here c1 = 0, c2 = ±1. Due to (40) we have ϕzz = −(1 + u 2 )ϕyy ,
2 2 −[(1 + u 2 )ϕyy + ϕyz ](1 + u 2 ) = ±1.
(44)
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Sign ‘+’ is not suitable. To linearize the first equation we perform the Legendre transformation (y, z, ϕ, ϕ y , ϕz ) → (ξ, η, ω, ωξ , ωη ) according to the formulas [6] ξ = ϕy ,
η = ϕz ,
y = ωξ ,
z = ωη ,
ω(ξ, η) + ϕ(y, z) = y ξ + z η. System (40) transforms here to the following one: (1 + u (ξ )2 )ωηη + ωξ ξ = 0,
ωξ2η − ωξ ξ ωηη = (1 + u (ξ )2 ).
As in the two-dimensional case the following representation is valid ωηη = cos α, ωξ η = 1 + u 2 sin α, ωξ ξ = −(1 + u 2 ) cos α.
(45)
(46)
Cross differentiation gives αξ = −
u u cos α, 1 + u 2
αη =
u u sin α. (1 + u 2 )3/2
(47)
From the compatibility conditions of Equations (47) follows
u u (1 + u 2 )3/2
sin α =
(u u )2 . (1 + u 2 )5
(48)
Hence and from (47) we find u = 0 and α = const. From (39) we obtain a linear equation for ϕ: ϕx = a ϕ y + b, which reduces the problem to the two-dimensional case investigated before. 7. Final Result Summing up all the calculations we can formulate the following statements. Theorem 3. The initial velocity field of irrotational barochronous gas motions is equivalent to one of the following: a) constant; b) u = x, v = w = 0; c) u = x, v = −y, w = 0. The formulation of the given statement contains canonic forms of solutions according to the classification (18). The action of equivalence transformations Ti allows us to obtain the solution of equations (16) and (13) for arbitrary set of constants j. The transformations of Ti retain the linearity of dependence u(x). Therefore, due to Theorem 3 the most general solution of Equations (16) and (13) is u = Ax + b with symmetrical real constant matrix A. With the help of rotations and Galilean translation one can transform matrix A to the diagonal form and make b = 0. Theorem 4. The initial velocity field of irrotational barochronous gas motions accurate to the transformations of rotations and Galilean translation has the form u = ax,
v = by,
w = cz; a, b, c = const.
(49)
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Knowledge of the initial velocity field of barochronous motion of a gas allows one to reconstruct a solution at an arbitrary moment of time via substitution (12). The value of density is then regenerated by the integration of the equation of continuity. Gas pressure is determined from the equation of state. Application of such a procedure to the initial field (49) allows us to formulate the following statement. Theorem 5. The irrotational barochronous gas motions accurate to inessential constants are described by the following formulas u=
ax , 1 + at
ρ=
ρ0 (1 + at)(1 + bt)(1 + ct)
v=
by , 1 + bt
w=
cz , 1 + ct
a, b, c, ρ0 = const.
(50)
Remark 2. For non-isentropic gas flows value ρ0 in (50) is an arbitrary function, which is preserved along the particle trajectory: Dρ0 = 0. Pressure p = p(t) and entropy S would be determined according to the state equation.
8. Conclusion The investigation of the set of Equations (13) and (16) has allowed us to describe irrotational barochronous ideal gas motions. This class of motions is completely described by formulas (50).
Acknowledgement The work is financially supported by RFBR grant 02-01-00550 and by Council of Support of the Leading Scientific Schools, grant Sc.Sch.-440.2003.1.
References 1. Chupakhin, A. P., ‘On barochronous gas motions’, Doklady Physics 42(2), 101–104, 1997 [translation of Doklady of Russian Academy of Science 352(5), 624–626, 1997]. 2. Chupakhin, A. P., ‘Barochronous gas motions. General properties and submodel of types (1,2) and (1,1)’, preprint, Lavrentyev Institute of Hydrodynamics, Novosibirsk. No. 4, 1998 [in Russian]. 3. Ovsyannikov, L. V., ‘Symmetry of the barochronous motions of a gas’, Siberian Mathematical Journal 44(5), 857–866, 2003 [translation of Sibirskij Matematicheskij Zhurnal 44(5), 1098–1109, 2003]. 4. Oswatitsch, K., Grundlagen der Gasdynamic, Springer, Vienna, New York, 1976. 5. Ovsyannikov, L. V., Group Analysis of Differential Equations, Academic Press, New York, 1982. 6. Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. II: Partial Differential Equations, Interscience Publishers, New York/London, 1962.
