International Journal of Pure and Applied Mathematics

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 48 No. 1 2008, 83-90

ON INTEGRALS AND INVARIANTS FOR INVISCID, COMPRESSIBLE AND THREE-DIMENSIONAL FLOWS UNDER GRAVITY Dinesh Khattar1 , B.B. Chakraborty2 § , Poornima Mital3 , Arti Kaushik4 1,2 Department

of Mathematics Kirori Mal Collage University of Delhi Delhi, 110007, INDIA 1 e-mail: khattar [email protected] 3 School of Sciences Indira Gandhi National Open University New Delhi, 110068, INDIA 4 Department of Mathematics MAIT, GGSIP University Delhi, 110085, INDIA

Abstract: Conservation laws are obtained for three-dimensional, isentropic, compressible flows under gravity, which are extension to three dimensions of earlier results (Chakraborty, Khattar and Verma [3]) in two-dimensional flows. AMS Subject Classification: 58D30, 76N99 Key Words: inviscid, compressible, three dimensional, isentropic flow

1. Introduction Longuet-Higgins [4] obtained using a direct and simplified method, the eight conservation laws in two-dimensional, incompressible, inviscid flows. These results were obtained earlier by Benjamin and Olver [2] using a rather general Received:

April 3, 2008

§ Correspondence

author

c 2008, Academic Publications Ltd.

84

D. Khattar, B.B. Chakraborty, P. Mital, A. Kaushik

analysis. These results have been generalised (Chakraborty, Khattar and Verma [3]) to two-dimensional compressible flows. In these flows it is found that only seven conservation laws are valid. In this note we further extend these results for a compressible flow when this flow is three-dimensional. We find that in three-dimensional flows, eleven quantities are conserved.

2. Definitions We define the quantities M , M¯r, I, A and T by the equations Z M = ρdV, Z M¯r = ρrdV, Z I = ρvdV, Z A = ρr × vdV, Z T = ρ(v2 /2)dV,

(1) (2) (3) (4) (5)

where ρ and v are the density and velocity of the compressible fluid and integrals are over a volume V of the flow region. We also define E = T + P and P = M g¯ z.

(6)

In the above equations M is the total mass of the fluid in the region V being considered. ¯r is the center of mass, I is the momentum vector and A is the angular momentum of this fluid, T , P and E are the kinetic, potential and total energy of the fluid within V , respectively.

3. Some Theorems The equation of motion for a compressible, inviscid flow is dv = −∇p + ρg, ρ dt and the equation of continuity is dρ + ρ div v = 0, dt

(7)

(8)

ON INTEGRALS AND INVARIANTS FOR INVISCID... where g = (0, 0, −g) is the gravitational acceleration. The operator ∂ d = + v.∇ dt ∂t is the mobile operator

85

(9)

We shall derive some theorems, and in the process we shall use the following theorem (Becker [1]) of gas dynamics. Let Φ be a field function, which may be a scalar or a vector quantity, and let Ψ=

Z

ρΦdV.

dΨ = dt

Z

ρ

(10)

Then dΦ dV. dt

(11)

We also note that dr = v. (12) dt Using (10) and (11), we obtain from (1) dM = 0. (13) dt Equations (2) and (3), in view of (10)-(13) show that d¯r M = I. (14) dt Similarly, equation (3), in view of (10) and (11), gives us Z dv dI = ρ dV. dt dt dv Using (7) to eliminate ρ dt from the last equation we finally have, in view of (1), the relation Z dI = − ∇pdV + gM. (15) dt We also know (Weatherburn [5]) Z Z ∇pdV = (pn)dS, (16) where the surface integral on the right hand side of (16) is taken over the surface S bounding the volume V in the flow region, and n is the unit normal vector to the surface S. In view of (16), (15) finally gives Z dI = − (pn)dS + gM. (17) dt

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In view of (10) and (11), we obtain from (4), using (12), the equation Z dv dA = ρr × dV. dt dt We use (7) to eliminate ρ dv dt from the last equation and finally obtain the relation Z dA = − r × ∇pdV + M¯r × g. (18) dt Since curl (rp) = ∇p × r, equation (18) gives the result Z dA = curl(rp)dV + M¯r × g. (19) dt As (Weatherburn [5]) Z Z curl(rp)dV = n × (rp)ds, we finally obtain from (19) the result Z dA = n × (rp)dS + M¯r × g. (20) dt Using (5) and (13), we obtain from (6), in view of (10) and (11), the result Z dv d¯ z dE = ρv · dV + M g . dt dt dt Using (7) to eliminate ρ(dv/dt) from the last equation, we get Z d¯ z dE = v · (−∇p + ρg)dV + M g . (21) dt dt Using (8), we obtain the result p dρ v · ∇p = ∇ · (pv) − p divv = ∇ · (pv) + (22) ρ dt Since g = (0, 0, −g), we use (22) in (21) and apply Gauss’s Theorem to finally obtain the result Z Z p dρ d¯ z dE = − ρgw + (23) dV − pv · n dS + M g . dt ρ dt dt We assume the flow to be isentropic so that pressure p and density ρ are related as p = kργ , where k is a constant and γ is the ratio of specific heats of the gas. Using (10) and (11) and the last relation between p and ρ, we obtain the result Z Z p dρ d p dV = dV. (24) ρ dt dt (γ − 1) Equations (3) and (14) give Z d¯ z M = ρwdz, (25) dt

ON INTEGRALS AND INVARIANTS FOR INVISCID...

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where v = (u, v, w). Equation (24) and (25), when used in (23), finally give us the result Z Z d p dE =− dV − pv · ndS. (26) dt dt (γ − 1) Following Longuet-Higgins [4] we define, for a compressible potential flow with velocity potential φ, the quantity B as Z B = ρ∇.(rφ)dV . (27) We recall that ρ = 1 in the Longuet-Higgins’s discussion of two-dimensional flow of an incompressible fluid, and (27) reduces to the definition of B given by Longuet-Higgins [4] (cf. his equation (2.8)). In a compressible isentropic flow Bernoulli’s equation can be written as Z dp ∂φ 2 + (v /2) + + gz = 0, (28) ∂t ρ the arbitrary function of time being absorbed in noting that

∂φ ∂t .

Using (10) and (11), and

v = ∇φ, we obtain from (27) Z Z d d dB = ρ [∇ · (rφ)]dV = ρ (3φ + r · v)dV . (29) dt dt dt Since dφ ∂φ = + v2 , dt ∂t (29) gives us the equation Z dB ∂φ dv 2 = ρ 3 + 4v + r · dV. (30) dt ∂t dt The equation (30), in view of (7) and (28), can be written as Z Z Z dp 3p 5v2 dB = + + + r · g dV − ∇. (rp) dV . (31) ρ −3gz − 3 dt ρ ρ 2 For an isentropic flow, since p = kργ , where k is a constant, we have Z p dp =− . p−ρ ρ (γ − 1) We also note that g = (0, 0, −g). The equation (31), in view of (5) and (6), therefore takes the form Z Z p dB = 5T − 4P − 3 dV − (rp) · ndS . (32) dt (γ − 1)

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The relations (13), (14), (17), (20), (26) and (32) are the theorems proved in the present paper for a three-dimensional compressible flow under gravity.

4. Conserved Quantities If the pressure vanishes (p = 0) on the bounding surface S of the flow region V being considered, equations (13), (14), (17), (20) and (26) show that the following quantities are conserved. M = c1 ,

(33)

I − gc1 t = c2 ,

(34)

M¯r − gc1 t2 /2 − c2 t = c3 ,

(35)

A − (gc1 t3 /6 + c2 t2 /2 + c3 t) × g = c4 , Z p dV = c5 . E+ (γ − 1)

(36) (37)

In the case R of potential flow the relation (32) involving B shows that, since the integral pdV is present in this relation, no conserved quantity can be obtained from (32) when the fluid is compressible. In the limiting case of an incompressible fluid (γ → ∞), however, (32) becomes dB = 5T − 4P. (38) dt The equation (37) for incompressible fluid gives E = c5

(39)

and the z-component of (35) shows that M z¯ = (−gc1 t2 /2 + c2z t + c3z ) .

(40)

From (6), (39) and (40), we have P

= (−c1 gt2 /2 + c2z t + c3z )g,

(41)

T

= c5 − P.

(42)

Using (41) and (42), and integrating (38), we obtain the conserved quantity B − 5c5 t + 9g(−c1 gt3 /6 + c2z t2 /2 + c3z t) = c6 .

(43)

As we noted earlier, the result (43) is valid for an incompressible potential flow.

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5. Concluding Remarks In this paper, we have studied some integrals and invariants for an inviscid, compressible, three-dimensional flow under gravity. In the case of an incompressible two-dimensional flow under gravity, in which all quantities are independent of z and vz vanishes, Longuet-Higgins [4] and Benjamin and Olver [2] showed that eight quantities related to the fluid in a domain D of the flow region are conserved as D moves with the flow. These conserved quantities in the two-dimensional flow involved the total mass (M ), two components of the momentum (Ix , Iy ), two coordinates (¯ x, y¯) of the centre of mass, the angular momentum (A) about the z-axis, the total energy E of the mass of the fluid in the domain D of the flow and a quantity B suitably defined by them. This discussion was generalised (Chakraborty, Khattar and Verma [3]) to the case of a two-dimensional, compressible, isentropic flows. It was found that when the flow is compressible only seven quantities are conserved. However, no quantity involving B could be shown to be conserved. In this paper, we further extend the analysis to obtain conserved quantities in a three-dimensional, compressible, isentropic flow under gravity. We find that eleven quantities are conserved for a fluid mass in a region V of flow, as V moves with the flow, if the pressure p vanishes on the boundary S of V . These eleven quantities involve the total mass (M ), the three components of momentum (Ix , Iy , Iz ), the coordinates of the centre of mass (¯ x, y¯, z¯), the R three p components of the angular momentum (A) and the total energy E plus (γ−1) dV of the fluid in a region V of flow. As in the case of a two-dimensional, irrotational flow of an incompressible fluid, in the present discussion also, if the fluid is incompressible but the flow is irrotational and three-dimensional, we find that additional quantity involving B is conserved. References [1] Ernst Becker, Gas Dynamics, Academic Press, New York (1968). [2] T.B. Benjamin, P.J. Olver, Hamiltonian structures, symmetric and conservation laws for water waves, J. Fluid Mech., 125 (1983), 137-185. [3] Chakraborty, Khattar, Verma, On integrals and invariants for inviscid, compressible two dimensional flows under gravity, Fluid Dynamics Research, 26 (2000), 141-147.

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[4] Longuet M.S. Higgins, On integrals and invariants for inviscid, irrotational flows under gravity, J. Fluid Mech., 134 (1983), 155-159. [5] C.E. Weatherburn, Advanced Vector Analysis, G. Bell and Sons Ltd., London (1924).