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Journal of Science and Technology in the Tropics 47
where in the reference frame U0
is the shock wave speed. This leads us to write
Eq. (27) under the steady state condition ∂ / ∂ τ ′ = 0, as
(1) (1) 2 (1)
∂Φ (1) ∂Φ ∂ Φ
0 1 1 2
− U + A Φ = C
∂ζ ∂ζ ∂ζ
(28)
As shown in Paul et al. [38], the solution of the above equation, Eq. (28) describes
the shock waves, whose speed U0
is related to the extreme values Φ( −∞ ) and
Φ( ∞)
by Φ( ∞)
- Φ( −∞ ) = 2 U0 / A1
. Therefore Φ is bounded at ζ = ±∞ under
this condition, the shock wave solution of Eq. (28) is [16, 17]:
⎡ ⎛
( ν 0) 0 1 tanh ζ ⎞⎤
Φ = = Φ ⎢ − ⎜ ⎟
∆
⎥
(29)
⎣ ⎝ ⎠⎦
where Φ 0 = U0 / A1
is the height of the DNIA shock waves and ∆ = 2 C1 / U0
is
the thickness of the DNIA shock waves.
It is to be noted here that in the present case of the non-planar geometry,
an exact analytic solution of Eq. (27) is not possible. Therefore, we have
numerically solved Eq.(27) and have studied the effects of cylindrical
( ν = 1) and spherical ( ν = 2) geometries on time-dependent non-linear
structure for the typical dusty plasma parameters as in [38], namely
2 2
µ e = ne0 / zhnh0 = 0.2 − 0.4, µ i = ni 0 / zhnh0
= 1.0 − 1.4, σ e = Th / Te
= 0.125,
3
σ i = Th / Ti
= 0.1− 0.25, Te
~ T i = 0.2 eV, zh
= 1, zd
= 10 , rd
= 5 µ m,
mi
= 39mp
, Th
= 0.125 Ti
, mh
= 146mp
, where mp
is the proton mass.
Figure 1. Time evolution of the cylindrical ( ν = 1) shock
wave potential Φ versus spatial coordinateξ and time Γ for
β = 0.4 , µ = 0.5, µ = 0.8, σ = 0.125.
i
e
i