SELF-SIMILAR FLOW UNDER THE ACTION OF MONOCHROMATIC RADIATION BEHIND A STRONG CYLINDRICAL SHOCK WAVE IN A NON-IDEAL GAS

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1 SELF-SIMILAR FLOW UNDER THE ACTION OF MONOCHROMATIC RADIATION BEHIND A STRONG CYLINDRICAL SHOCK WAVE IN A NON-IDEAL GAS *J. P. Vishwakarma and Vijay Kumar Pandey Deartment of Mathematics & Statistics, D.D.U. Gorakhur University Gorakhur-2739, India *Author for Corresondence ABSTRACT Similarity solutions are obtained for one-dimensional flow under the action of monochromatic radiation behind a strong cylindrical shock wave roagating in a non-ideal gas. The initial density of the medium is assumed to be constant. It is inferred that the effects of the non-idealness of the gas and of the monochromatic radiation on the shock roagation become more significant when the ratio of secific heats is increased. Key Word: Shock waves, Non-ideal gas, Monochromatic radiation, Self-similar solutions. INTRODUCTION The influence of radiation on a strong shock wave and on the flow-field behind the shock front has always been of great interest, for instance, in the field of nuclear ower and sace research. Consequently, similarity models [Sedov (959)] for classical blast wave roblems have been extended, taking radiation into account [Elliott (96), Wang (964), Helliwell (969), Ray and Bhowmick (976), Nicastro (97), Ghoniem et al. (982), Gretler and Steiner (993)]. Elliott (96) discussed the conditions leading to self-similarity with a secified functional form of the mean freeath of radiation and obtained a solution for self-similar exlosions. Wang (964), Helliwell (969) and Nicastro (97) treated the roblems of radiating walls, either stationary or moving, generating shock at the head of self-similar flow-fields. Assuming the shock to be isothermal and transarent the self-similar solution of the central exlosions in stars has been obtained by Ray and Bhowmick (976) including the effects of radiation. The self-similar solutions have been used by Khudyakov (983) to discuss the roblem of the motion of a gas under the action of monochromatic radiation. Khudyakov (983) has considered that a homogeneous gas at rest occuies a half-sace bounded by a fixed lane wall and assumed that a radiation flux moves through the gas in the direction of the wall with a constant intensity j er unit area. From the instant of arrival of the radiation at the wall a shock wave is assumed to roagate out from the wall in the direction oosite to the radiation flux. The radiation flux is absorbed in the zone between the shock wave and the wall, and it is not absorbed in the undisturbed medium. It is also assumed that the gas itself does not radiate. Zheltukhin (968) has develoed a family of exact solutions of one dimensional motion (lane, cylindrical or sherical symmetry) of a gas taking into account of the absortion of monochromatic radiation. Nath and Takhar (99) and Nath (998) have studied the roagation of cylindrical shock waves in a gas under the action of monochromatic radiation when the medium is rotating or non-rotating. In all of these works, the medium is assumed to be a gas obeying the equation of state of an ideal gas. Because of the extreme conditions that generally arise behind a shock wave, roduced by an exlosion, the assumtion that the gas is ideal is no more valid. The oular alternative to the ideal gas is a simlified van der Waals model. Roberts and Wu (996) and Wu and Roberts (23) adoted this model to discuss the shock wave theory of sonoluminescence. In the resent work, we too adot this as our model of a non-ideal gas to obtain the self-similar solutions for the flow under the action of monochromatic radiation behind a strong cylindrical shock wave roagating in a non-ideal gas. The initial density of the medium is assumed to be constant. Effects of a change in the arameter of non-idealness of the gas, in the ratio of secific heats and in the arameter of the flux of monochromatic radiation on the shock roagation are investigated. It is observed that the effects of non-idealness of the gas on the flow variables in the flow-field behind the 27

2 shock are significant when the ratio of secific heats are higher. The resent work may be considered as an extension of the work of Nath and Takhar (99) by taking the medium a non-ideal gas in lace of an ideal gas. Basic Equations and Boundary Conditions The fundamental equations for cylindrically symmetric motion of a non-ideal gas under the action of monochromatic radiation neglecting heat-conduction, viscosity, radiation of the medium, may be written as [Khudyakov (983), Nath (998), Zedan (22)] v v v, (2.) t r r r v v v, (2.2) t r r e e v v ( jr), (2.3) 2 t r t r r r j Kj, (2.4) r where, v,, e and j are the density, radial velocity, ressure, internal energy er unit mass and the flux of monochromatic radiation er unit area at radial distance r from the axis at time t, and K is the absortion coefficient. Most of the henomena associated with shock wave arise in extreme conditions under which the ideal gas is not a sufficiently accurate descrition. To discover how deviations from the ideal gas can affect the flow behind a shock wave, we adot a simle model. We assume that the gas obeys a simlified van der waals equation of state of the form [Roberts and Wu (996), Wu and Roberts (23)] RT ( b ),e CvT, (2.5) b ( ) where R is the gas constant, Cv R ( ) is the secific heat at constant volume and is the ratio of secific heats. The constant b is the van der Waals excluded volume, it laces a limit, max b, on the density of the gas. The absortion coefficient K is considered to vary as [Khudyakov (983), Nath (998), Nath and Takhar (99)] K K n m q s l j r t, (2.6) where the coefficient K is a dimensional constant and the exonents n, m, q, s, l are rational numbers. A diverging cylindrical shock is assumed to be roagating in the non-ideal gas with constant density. The jum conditions across the shock front are as 28

3 (D v ) D, (2.7) (D v ) 2 D 2, (2.8) 2 2 e (D v ) e D, (2.9) 2 2 j j, (2.) where the suffices and refer to conditions just behind and just ahead of the shock resectively, and D is the shock velocity. If the shock is strong, the boundary conditions (2.7)-(2.) take the form v ( )D, (2.), (2.2) 2 ( )D, (2.3) j j, (2.4) where the quantity ( ) is obtained by the relation 2 ( ) 2( b) 2b, (2.5) b b being the arameter of non-idealness. The dimensions of the constant coefficient K in equation (2.6) are given by nmq 3nmsl 2m3ql [K ] M L T. (2.6) Following the aroach of Sedov [], we get the conditions under with the formulated roblem will have self-similar solutions. The dimensional constants in the resent roblem will be,, j, and K in which, and j are deendent given by j [ ] [ ]. (2.7) For self-similarity the radiation absortion coefficient K must be deendent on the dimensions of j, which is equivalent to s l. The self-similar indeendent dimensionless variable is taken in the form r r, where r j 3 3 t, (2.8) r being the radius of the shock surface. The value of the constant is so chosen that at the shock surface. 29

4 Similarity Solutions We introduce the following similarity transformations to reduce the equations of motion into ordinary differential equations; v DV( ), R( ), 2 D P( ), j j J( ), (3. 3.4) where V, R, P and J are functions of the non-dimensional variable (similarity variable) Using the transformations (3.)-(3.4), the equations of motion take the form r. r where R ' (V ) R(V ' V), (3.5) P' RV '( V), (3.6) P P '(V )( br) ( V ' V) [ J' J], (3.7) J' s R n P m J q (3.8) K q n m 2(m ) s j D (3.9) is a dimensionless quantity and a quantity with a dash reresents the derivative of that quantity with resect to. The quantity is taken as the arameter which characterizes the interaction between the gas and the incident radiation flux [Khudyakov (983), Nath (998), Nath and Takhar (99)]. Solving equations (3.6)-(3.) for dv d, dr d, dp d dj and, we have d s n m q PV ( )( R P J J) V ', 2 R( V) ( br) P (3.) R(V ' V) R', ( V) (3.) P( V ' V) ( )( J' J) P ', (3.2) ( V)( br) J' s R n P m J q. (3.3) The shock conditions (2.)-(2.4) are transformed into V(), (3.4) 3

5 R(), (3.5) P() ( ), (3.6) J(). (3.7) At the inner boundary surface of the flow-field behind the shock the condition is that the velocity of the surface is equal to the normal velocity of the fluid on the surface. The kinematic condition, from equation (3.), can be written as where V( ), (3.8) is the value of at the inner surface. For exhibiting the numerical solutions, it is convenient to write the flow variables in the nondimensional form as v V( ), v V() R( ), R() P( ), P() j J( ). (3.9) j J() RESULTS AND DISCUSSION The set of differential equations (3.)-(3.3) are numerically integrated with the boundary conditions (3.4)-(3.7) to obtain the non-dimensional variables of the flow-field V, R, P and J against the similarity variable, by using the Runge-Kutta method of order four, for the values [Khudyakov (983), Nath (998), Nath and Takhar (99), Ranga Rao and Purohit (976)].,.2; n 2; m 3 2; q ; s ; 7 5,5 3; b,.5,.. The case b corresonds to the erfect gas case studied by Nath and Takhar (99). In figures to 4, we have lotted the radial velocity v v, the density, the ressure and the radiation flux j j against the radial distance r r in the flow-field between the inner exanding surface and the shock surface. Tables and 2 show, resectively, the density ratio across the shock and the osition of inner exanding surface for various values of the arameters, b,. From tables and 2 and figures -4 it is observed that the effects of an increase in the value of (ratio of secific heats) are (i) to decrease x, i.e. to increase the distance of inner exanding surface from the shock front. Physically, it means that the gas behind the shock is less comressed, i.e. the shock strength is reduced; (ii) to decrease the value of, i.e. to decrease the shock strength, which is the same as given in (i) above; (iii) to decrease the value of v v and ; at any oint in the flow-field behind the shock; and (iv) to enhance the effect of non-idealness of the gas on the rofiles of v v, and j j. Thus an increase in the ratio of secific heats decays the shock wave and enhances the effect of nonidealness of the gas on the rofiles of the flow variables behind the shock. 3

6 The effects of an increase in the value of the arameter of the non-idealness of the gas b are (i) to increase the distance of the inner exanding surface from the shock front (see table 2); (ii) to decrease the value of (table ), i.e. to decrease the shock strength. Therefore the nonidealness of the gas has decaying effect on the shock wave; (iii) to increase the value of, j j and to decrease the value of at any oint in the flow-field behind the shock (see figures 2, 3, 4). These effects are significant when is equal to.667 instead of.4. Thus the non-idealness of the gas decays the shock wave, and affects the variables in the flow-field behind the shock significantly, when the value of ratio of secific heats is higher. Effects of an increase in the radiation arameter are (i) to decrease x (table 2), i.e. to decrease the shock strength; and to decrease v v,, and j j at any oint in the flow-field behind the shock. This decrease in these flow-variables is somewhat significant when the value of the ratio of secific heats is larger. 32

7 Table. The density ratio across the shock front for different values of b and b Table 2. Position of inner exanding surface x for various values of b, and b x

8 34

9 35

10 Conclusion The resent work investigates the self-similar flow of a non-ideal gas under the action of monochromatic radiation behind a strong cylindrical shock wave. The density of the ambient medium is uniform. On the basis of the resent work, one may draw the following conclusions: (i) An increase in the ratio of secific heats decays the shock wave and enhances the effect of non-idealness of the gas on the rofiles of the flow-variables behind the shock. (ii) The non-idealness of the gas decays the shock wave, and affects the flow behind the shock significantly when the values of the ratio of secific heats is higher. (iii) An increase in affects the rofiles of the flow variables somewhat significantly when the ratio of secific heats is higher. 36

11 REFERENCES Sedov LI (959). Similarity and Dimensional Methods in Mechanics, Academic Press, London Elloitt LA (96). Similarity methods in radiation and hydrodynamics, Procceding Royal Society London A 258(3) Wang KC (964). The iston roblem with thermal radiation, Journal Fluid Mechanics 2(3) Helliwell JB (969). Self-similar iston roblem with radiative heat transfer, Journal Fluid Mechanics 37(3) Deb Ray G and Bhowmick JB (976). Similarity solutions for exansions in star, Indian Journal of Pure Alied Mathematics 7() Nicastro JR (97). Similarity analysis of radiative gasdynamics with sherical symmetry, Physics Fluids 3(8) Ghoniem AF, Kamel MM and Berger SA Oenheim AK (982). Effect of internal heat transfer on the structure of self-similar blast waves. Journal Fluid Mechanics 7(), Greter W and Steiner H (993). Blast waves in inhomogeneous atmoshere with counter ressure and heat transfer effects Shock wave 3(2) Khudyakov VM (983). The self-similar roblem of the motion of a gas under the action of monochromatic radiation, Soviet Physics Doklady 28() Zheltukhin AN (968). A family of exact solution of the equation of the one dimensional motion of a gas under the influence of monochromatic radiation, Journal of Alied Mathematics Mechanics 52(2) Nath O and Takhar HS (99). Proagation of cylindrical shock waves under the action of monochromatic radiation, Astrohysics and sace Science 66() Nath O (998). Proagation of cylindrical shock wave in a rotating atmoshere under the action of monochromatic radiation, IL Nuovo Cimento D 2() Roberts PH and Wu CC (996). Structure and stability of a sherical imlosion, Physics Letters A 23() Wu CC and Roberts PH (23). The shock wave theory of sonoluminescence, Shock Focussing Effect in Medical Science and Sonoluminescence (Editiors. R. C. Srivastava et al.), Sringer-Verlag, Berlin (-2) -27. Zedan Hassan A (22). Alication of the grou of equations of the one-dimensional motion of a gas under the influence of monochromatic radiation, Alied Mathematics and Comutation 32() Ranga Rao MP and Purohit NK (976). Self-similar istion roblem in non-ideal gas, International Journal of Engineering Science 4()