J. Comp. & Math. Sci. Vol.3 (2), 217-220 (2012)
A Study of Detonation Waves in Magnetogasdynamics AMIT K. RAY Department of Mathematics M.G.P.G. College, Gorakhpur-273001 Uttar Pradesh – India. (Received on : March 29, 2012) ABSTRACT In the present paper, we have studied about the flow field behind plane, cylindrical and spherical detonation wave. We used similarity solutions for our investigation. The constant amount of heat is produced during the detonation process. The results have been illustrated through tables. The effect of magnetic field also taken in consideration. Keywords: Detonation waves, similarity method, azimuthal magnetic field.
1. INTRODUCTION
field propagating atmosphere.
into
non-uniform
The problems of propagation of detonation waves are much useful in deferent field of engineering and technology. We have considered the problem of detonation wave allowing for the effect of magnetic field into a gas of varying density. Zeldovich and Raizer7 have assumed the imploding shock waves using similarity method. Welsh6, Nigmatulin1, Teipel2, Verma and Singh5 have shifted the shock front by contracting detonation wave front. Self similar solutions have been investigated for the flows behind plane, cylindrical and spherical detonation waves with magnetic
Verma and Vishwakarma3,4 have also given their views about the detonation problems. 2. FLOW GOVERNING EQUATION AND BOUNDARY CONDITION The equation of continuity, motion and energy for the unsteady flow are
∂ρ ∂ρ ∂u jρu +u +ρ + =0 ∂t ∂r ∂r r
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
(1)
218
Amit K. Ray, J. Comp. & Math. Sci. Vol.3 (2), 217-220 (2012)
∂ u ∂ t
+ u
∂ u ∂ r
H
∂ H ∂ r
+
1
+
ρ 2
H r
∂ p ∂ r
+
µ ρ
= 0
∂H ∂H ∂u +u +H =0 ∂t ∂r ∂r ∂S ∂S +u =0 ∂t ∂r
(2)
(3)
(4)
converging detonation wave a strong waves are generated in the neighborhood of the line of symmetry. Therefore, equation (6) may be simplified by retaining only the largest term for the region. The discontinuities for the density, pressure, velocity and magnetic field are given by
γ γ D2 ρ1 = ρ0 ; p1 = 2 p0 γ + 1 γ + 1 c0
Here j = 0,1 and 2 for plane, cylinder and sphere
1 γ u1 = D ; H1 = H 0 γ + 1 γ + 1
The effects of viscosity and heat conduction have been ignored. The distributions of density and magnetic field ahead of detonation front are given by
Here γ is the specific heat ratio. The suffix ‘1’ refer to the condition just behind the detonation front and ‘0’ just ahead of the shock front respectively.
ρ0 = ρc r −ω and H 0 = H c r −1
3. SIMILARITY SOLUTIONS
(5)
where u, ρ, p, S and H are velocity, density, pressure, entropy, and magnetic field respectively. The amount of heat released q per unit mass of a gas for unsteady flow by Teipel4 is
c02 c04 q 1D = 1 − 2 2 + 4 c pT0 2 c02 D D
The solution of the problem exist in the similarity form
ρ = ρ0 F (η) , p = ρ0
2
(6)
where c0 is the speed of sound, D is the detonation front velocity. t and cp are the temperature and the specific heat at constant pressure. The suffix 0 is related to the undisturbed gas. In case of cylindrical
(7)
r2 t
2
r u = U (η) t
P(η) , µ ½ H = ρ ½ r G (η )
(8)
0 t
Initially taking t = 0 and R = 0. Thus the time up to the instant of collapse is negative. We take
R = (− c0t )m , η =
r r = R (− c0t )m
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
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219
Amit K. Ray, J. Comp. & Math. Sci. Vol.3 (2), 217-220 (2012)
where m is the similarity exponent. The front velocity is given by
D=
mR t
(10)
Using (8) and (9) into the basic equations (1) to (4) we get following set of equations,
F' + ηU '+ (2 − w )U = 0 (11) F
η(U − m )
η (U − m )U ' + U (U − 1 ) + (2 − ω ) ω + 2 − 2
P η + P' F F
G2 G'G F +η F = 0
P'
(12)
η(U − m )
(13)
(14)
where F', U', P' and G' are the differentiation of F, U, P and G with respect to η. At detonation front when (η = 1) from equation (7) we have
U (1) =
m γ +1 ; F (1) = γ +1 γ
m(γ + 1) m2 P(1) = and G(1) = γ +1 γM A
(15)
where MA is the Alfven Mach number. The similarity exponent m is obtained, following Verma and Singh5, by the relation
F'
η (U − m ) −γ F P + {2 + ω (γ − 1 )}U − 2 = 0
G' ω + ηU '+ 2 − U − 1 = 0 G 2
m=
3( γ + 1) (3 − ω)(γ + 1) + γ
(16)
Table : MA = 10, K = 0
η
ρ/ ρL
µ/µ L
p/pL
H/HL
1.00
1.0000
1.0000
1.0000
1.0000
1.02
1.0194
1.0070
1.0132
0.9739
1.04
1.3454
1.0089
1.0212
0.9435
1.06
1.0443
1.0041
1.0222
0.9050
1.08
1.0528
0.9980
1.0215
0.8661
1.10
1.0603
0.9914
1.0200
0.8281
1.12
1.0675
0.9848
1.182
0.7913
1.14
1.0740
0.9781
1.0160
0.7558
1.16
1.0803
0.9716
1.0137
0.7215
1.18
1.0862
0.9651
1.0112
0.6884
1.20
1.0919
0.9588
1.0086
0.6564
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
220 4.
Amit K. Ray, J. Comp. & Math. Sci. Vol.3 (2), 217-220 (2012)
RESULTS
REFERENCES
The flow variables in the nondimensional form written as
u u1
=
η
U (η) ; U (1)
ρ ρ1
p –ω+2 P(η) =η ; P(1) p1
=
η– ω
F (η) F (1)
–ω+2
G(η) H =η 2 G(1) H1
For numerical calculation we take the parameters γ =1.4, MA = 10, ω = 0. The change in variables density velocity, pressure and magnetic field are shown through the tables. The problem can also be solved for different values of the above parameters.
1. Nigmatulin, R.I., J. Appl. Math, 31, 171177 (1967). 2. Teifel, I, Mech, Rescomm, 3, 21-16 (1976). 3. Verma, B.G. and Vishwakarma, J.P. Ind. J. Pure Appl. Math, 10,715 - 725 (1979). 4. Verma, B.G. and Vishwakarma, J.P. Ind. J. Pure Appl. Math, 15, 685-694 (1984). 5. Verma, B.G. and Singh, J.B. Def. Scie., 31, 1-6 (1981). 6. Welsh, R.L. J. Fluid Mech., 29, 61-79 (1967). 7. Zeldvoich, Y.B. and Razer, Y.P., Physics of shock waves and High temperature Hydrodynamics phenomenon Vol. 2, Academic Press. London, (1967).
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)