Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 On dynamics f imlding shck waves in a mixture f gas and dust articles R. K. Anand Deartment f Physics, University f Allahabad, Allahabad-1100, India E-mail address: anand.rajkumar@rediffmail.cm Abstract In this aer, the generalized analytical slutins fr ne-dimensinal adiabatic flw behind the imlding shck waves ragating in a dusty gas are btained using the gemetrical shck dynamics thery. The dusty gas is assumed t be a mixture f a erfect gas and sherically small slid articles, in which slid articles are cntinuusly distributed. Anand s shck jum relatins fr a dusty gas are taken int cnsideratin t exlre the effects due t an increase in (i) the ragatin distance frm the centre f cnvergence, (ii) the mass fractin f slid articles in the mixture and (iii) the rati f the density f slid articles t the initial density f the gas, n the shck velcity, ressure, temerature, density, velcity f mixture, seed f sund, adiabatic cmressibility f mixture and the change-in-entry acrss the shck frnt. The results rvided a clear icture f whether and hw the resence f slid articles influences the flw field behind the imlding shck frnt. PACS numbers: 47.40-X, 47.55.kf Keywrds: Analytical slutins. Imlding shck waves. Dusty gas, Gemetrical shck dynamics. Mach number. Tw-hase gas-article flw 1. Intrductin The gas-article tw-hase flws ccur in a variety f natural henmena and are invlved in many industrial rcesses. The natural henmena accmanied by the gasarticle tw-hase flws are tyified by exlsin f suernva, sand strms, mving Page 1 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 sand dunes, aerdynamic ablatin, csmic dusts, etc. The flw field, that devels when a mving shck wave hits a tw-hase gas-article medium, has a clse ractical relatin t industrial alicatins, e.g. slid rcket engine in which aluminium articles are used t reduce the vibratin due t instability, as well as industrial accidents such as exlsins in calmines and grain elevatrs. In heat transfer alicatins, the gas-article flws are invlved in nuclear reactr cling and slar energy transrt using grahite susensin flws. S far, a number f aers have been rerted n the shck waves in a mixture f gas and dust articles. The structure f gas-article tw-hase flws behind the nrmal shck waves under different cnditins was cnsidered in a large number f aers, starting frm the ineering wrks f Carrier [1], Rudinger [] and Kribel [3]. Exerimental study f the shck structure in a gas-article mixture was fulfilled by Outa et al. [4]. The main stages f the investigatins f tw-hase flws with shck waves are reviewed by Marble [5], Kraik et al. [6], Higashin and Suzuki [7], Ivandaev et al. [8], Igra and Ben-Dr [9], Pai and Lu [10] and Ben-Dr [11]. The wrk f Pai [1], Rudinger [13], Pai et al. [14], Miura and Glass [15], Steiner and Hirschler [16], Sait et al. [17], Naidu et al. [18] and Anand [19-0] is wrth mentining in the cntext f this aer. One f the main reasns fr cntinuing interest in shck fcusing is its ability t create extremely high temerature and ressure at the centre f cnvergence. Hwever as the strength f imlding shck waves increases the effects due t dust articles becme significant. These effects need t be accunted fr in rder t crrectly describe the st shck cnditins and acquire infrmatin n the attainable ressure and temerature by shck fcusing. The study f ragatin f shck waves in a dusty gas is f immense significance due t its wide alicatins t suersnic flights in a dusty gas envirnment, suersnic-vehicle mtin in desert sand strms r cluds f vlcanic dust, exhaust lumes f rcket mtrs relled by slid fuels, ensuring the exlsin safety f cal mines and industrial manufacture f wder materials, cating technlgies using suersnic tw-hase jets, etc. This has develed ur interest in studying the imlding shck waves ragating in a mixture f gas and dust articles. The urse f writing this aer is, therefre, t resent the analytical slutins fr ne-dimensinal adiabatic Page f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 flw behind the imlding shck waves in a mixture f gas and slid articles t ursue the alicatins f shck waves in a dusty gas envirnment. T ur best knwledge, s far there is n aer rerting the analytical slutins fr imlding shck waves in a dusty gas, btained by using the gemetrical shck dynamics [1-]. The gemetrical shck dynamics arach gives highly accurate results esecially, in the case f a sherical symmetry. Since an imlding shck is strengthened as it fcuses n the rigin. In the resent research aer, the influence f dust articles has been investigated n the flw quantities f the regin just behind the imlding shck frnt. Fr this urse, a mdel based n gemetrical shck dynamics is develed t rvide a simlified and cmlete treatment fr the ragatin f imlding shck waves in a dusty gas during the cnvergent rcess. Gemetrical shck dynamics was intrduced by Whitham [1] and it is imrtant t mentin that the riginal gemetrical shck dynamics methd des nt accunt fr the influence f the flw ahead f the shck frnt. T btain the analytical slutins fr ne dimensinal adiabatic flw behind the imlding shck waves in a dusty gas we make the fllwing assumtins and arximatins: (i) the dusty gas is a mixture f a erfect gas and small slid articles, (ii) the dust hase cmrises the ttal amunt f slid articles which are cntinuusly distributed in the erfect gas, (iii) the gas fllws the equatin f state f an ideal gas, (iv) the mtin f gas can be regarded as inviscid, s that the fluid viscsity and cnductivity are neglected excet in the interactin with the articles, (v) the slid articles are inert, rigid and sherical, (vi) the articles have cnstant heat caacity and unifrm temerature distributin, (vii) the article cllisin is the dminant mechanism f inter-article interactin. At the mdest article vlume fractin t be cnsidered here, article cmactin can be ignred, (viii) the effect f thermal radiatin is negligible. It is wrth mentining, hwever, that a clud f articles is usually a better emitter and absrber f radiatin than a ure gas. Thus, the ht articles dwnstream f the shck may reheat the cld articles ustream f the shck by radiatin. This effect becmes significant as the surface area f the articles increases, (ix) the inter-article heat transfer due t article cllisin is neglected, (x) the gas-article flw is ne-dimensinal, (xi) the medium is initially unifrm and at rest, and (xii) the disturbances due t the reflectins, wave interactins in the wake, etc. d nt vertake the imlding shck wave. It is nted Page 3 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 that with an increase in the article vlume fractin, article-article cllisins and ther tyes f article interactins becme increasingly imrtant and thus the assumtin f negligible article interactins is n lnger feasible. The analytical exressin fr the ragatin velcity f shck is btained by substituting the generalized shck jum relatins derived by Anand [0] int the negative characteristic equatin. The general nn-dimensinal frms f analytical exressins fr the distributin f ressure, temerature, density, velcity f mixture, seed f sund and adiabatic cmressibility f mixture just behind the imlding shck frnt are btained, assuming the medium t be initially unifrm and at rest. Mst f the rir studies have remained fcused n the ragatin f shck waves in an ideal r dusty gaseus media withut discussing the change-in-entry acrss the shck frnt. The exressin fr change-in-entry acrss the imlding shck frnt is als derived. Tw cases are cnsidered cylindrical and sherical imlding shcks t highlight the differences between the D and 3D cnvergence. The lanar case is nt f interest since n area cnvergence and shck amlificatin exist and it is simly the rdinary lanar blast wave rblem. The numerical estimatins f flw variables behind the imlding frnt with cylindrical and sherical shck symmetries are carried ut using MATHEMATICA and MATLAB cdes. The effects f the mass fractin (cncentratin) f slid articles in the mixture and the dust lading arameter, i.e. the rati f the density f the slid articles t the initial density f the gas are exlred as the imlding shck wave ragates twards the centre f cnvergence. This mdel arriately makes bvius the effects due t an increase in (i) the ragatin distance frm the centre f cnvergence, (ii) the mass cncentratin f slid articles in the mixture and (iii) the rati f the density f the slid articles t the initial density f the gas, n the ragatin velcity f shck, ressure, temerature, density, velcity f mixture, seed f sund, adiabatic cmressibility f medium and the change-in-entry acrss shck frnt. The results are dislayed grahically and discussed by cmarisn with thse fr the case f a erfect r dust-free gas flw. Thus, the results rvided a clear icture f whether and hw the resence f dust articles influences the flw field behind the imlding shck frnt. Page 4 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 The aer is rganized as fllws. The backgrund infrmatin is rvided in Sectin 1 as an intrductin. Sectin cntains general assumtins and ntatins as well as shck jum relatins. In Sectin 3 analytical slutins are resented. Sectin 4 mainly describes results with discussin n the imrtant cmnents f the resent mdel. The last sectin 5 resents the cncluding remarks.. Basic equatins and shck jum relatins The unsteady, ne-dimensinal flw field in a mixture f a erfect gas and small slid articles is a functin f tw indeendent variables; the time t and the sace crdinate r. The cnservatin equatins fr ne-dimensinal unsteady flw f a dusty gas can be exressed cnveniently in Eularian crdinates [0] as fllws: u u 1 + u + t r ρ r ρ ρ u + u + ρ + t r r e e + u t r ρ = 0, (1) u D r = 0, () ρ ρ + u = 0, (3) t r where u( r, t) is the velcity f mixture, ρ( r, t) the density f mixture, ( r, t) the ressure f mixture, e( r, t) the internal energy f mixture er unit mass, r is the distance frm the rigin O and t is the time crdinate. The dimensinality index D is defined by D = d ln A d ln r, where A D ( r) = π D r is the flw crss-sectin area. The index D = 1, r is fr ne-dimensinal cylindrical, r sherical gemetry, resectively. Dust articles entrained in nn-steady gas flw cannt immediately fllw the changes in velcity and temerature f the gas. Cnsequently behind a shck frnt in a dusty gas there is a relaxatin zne in which drag and heat transfer act t bring gas and dust velcities clse tgether again. Due t the cnditin f velcity and temerature equilibrium, the terms f drag frce and heat-transfer rate, which can be exressed via the drag cefficient and the Nusselt number, d nt aear in the right-hand sides f the Eqs. (1) and (3). These terms are, f curse, imrtant fr evaluating the extent f the Page 5 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 relaxatin zne behind the shck frnt, which is hwever, beynd the sce f this aer. It is wrth mentining that the dusty gas is a ure erfect gas which is cntaminated by small slid articles and nt as a mixture f tw erfect gases. The slid articles are cntinuusly distributed in the erfect gas and in their ttality are referred t as dust. It is assumed that the dust articles are highly disersed in the gas hase such that the dusty gas can be cnsidered as a cntinuus medium where the cnservatin Eqs. (1) (3) aly. All relaxatin rcesses are excluded such that n relative mtin and n temerature differences between erfect gas and slid articles ccur. The slid articles are als assumed t have n thermal mtin, and, hence they d nt cntribute t the ressure f the mixture. As a result, the ressure and the temerature T f the entire mixture satisfy the thermal equatin f state f the erfect gas artitin. The equatin f state f the mixture subject t the equilibrium cnditin, is given as 1 k = ρrit Z, (4) 1 where k = ms m, is the mass cncentratin f slid articles ( m s ) in the mixture ( m ) taken as a cnstant in the whle flw field, Z is the vlumetric fractin f slid articles in the mixture, R i is the gas cnstant and, T is the temerature f the mixture. The relatin between k and Z is given by Pai et al. [14] as fllws: Zρ s k =, (5) ρ where Z = ρ ρ, while ρ s is the secies density f the slid articles and a subscrit Z refers t the initial values f Z, and ρ. It is ntable that in equilibrium flw, the mass cncentratin f slid articles k is a cnstant in the whle flw-field. As a result, Z ρ = cnstant, in the whle flw-field. The mass cncentratin f slid articles k = 0 refers the case f a erfect gas, i.e. dust-free case. The initial vlume fractin Z f the slid articles is, in general, nt cnstant. But the vlume ccuied by the slid articles is very small because the density f the slid articles is much larger than that f the gas [15], hence, Z may be assumed as a small cnstant. The initial vlume fractin f the small slid articles is given by Pai [1] as Page 6 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 Vs k Z = =, (6) V + V G(1 k ) + k g s where the vlume f the mixture V is the sum f the vlume f the erfect gas at the reference state V g and the vlume f the articlesv s which remains cnstant. The vlumetric arameter G is defined as G = ρ ρ, which is equal t the rati f the s g density f the slid articles t the initial density f the gas. Hence, the fundamental arameters f the resent mdel are k and G which describe the effects f the dust lading. Fr the dust lading arameter G, we have a range f G = 1 t G, i.e. V s 0. Fig. 1 shws the variatin f Z the initial vlume fractin f small slid articles with k the mass cncentratin f slid articles fr the values f G frm1 t. Fig.1 Variatin f Z with k fr varius values f G. It is ntable that Z remains almst unaffected with increase in k fr large values f G, the dust lading arameter. It is wrth mentining that if G = 1 then Z = k. In this case, the dust and the erfect gas are virtually indiscernible cncerning their secific vlume, but the vlume fractin f the dust still behaves as an incmressible slid hase. The additin f slid articles des nt increase the inertia f mixture, and des nt Page 7 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 additinally slw dwn any wave ragatin. On the ther hand, the initial vlume fractin f small slid articles Z f the dust linearly increases with k, as the vlumetric extensin Vs f the dust increases. The dust is cnsidered as an incmressible slid hase. The ccurrence f an incmressible hase in the mixture basically lwers the cmressibility f the mixture. The internal energy f the mixture is related t the internal energies f the tw secies and may be written as e = [ k C + (1 k ) C ] T C T, (7) where s v = vm C s is the secific heat f the slid articles, C v is the secific heat f the gas at cnstant vlume and C vm is the secific heat f the mixture at cnstant vlume. Fr equilibrium cnditins, the secific heat f the mixture at cnstant ressure is C = k C + ( 1 k ) C, (8) m s where C is the secific heat f the gas at cnstant ressure. The rati f the secific heats f the mixture is then given as C Γ = C m vm γ + δβ = 1+ δβ s s, (9) where γ = c cv is the secific heat rati f the gas, β s = Cs Cv is the secific heat rati f slid articles and δ = k 1 k ). It is ntable that Γ decreases with increase in ( k, hwever, it increases with increase in values fγ. Eliminating the temerature frm Eqs. (4), (6) and (7), we may nw write the secific internal energy f mixture as fllws: 1 Z e =, (10) Γ 1 ρ Fr isentric change f state f the gas-slid article mixture and thermdynamic equilibrium cnditin, we can calculate the s-called equilibrium seed f sund f the mixture fr a given k by using the effective rati f secific heats and effective gas cnstant R = ( 1 k ) R as fllws: M i Page 8 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 a = ρ d d 1/ S Γ = (1 Z) ρ 1/ Γ(1 k ) RiT = (1 Z) 1/, (11) where subscrit s refers t the rcess f cnstant entry. The initial seed f sund a f the mixture is defined as a Γ = (1 Z )ρ. (1) The deviatin f the behavir f a dusty gas frm that f a erfect gas is indicated by the adiabatic cmressibility f the mixture and is defined [3] as ( 1 Z ) 1 ρ τ = =, (13) ρ S Γ where ( ρ ) S dentes the derivative f ρ with resect t at the cnstant entry. The vlume f slid articles lwers the cmressibility f mixture, while the mass f slid articles increases the ttal mass, and therefre may add t the inertia f mixture. This can be shwn in tw limiting cases f mixture at the initial state. Fr G = 1, it fllws frm the Eqs. (6) and (4) that Z = k, ρ = RiT and τ = (1 k ) Γ, i.e. the resence f slid articles linearly lwers the cmressibility f mixture in the initial state. In the ther limiting case, i.e. fr G t zer. Accrding t Eq. (6), the vlume fractin cmressibility τ = 1 Γ, the vlume f the slid articles Vs tends Z is equal t zer. In this case, the is nt affected by the dust lading. The slid articles cntribute nly t increasing the mass and inertia f mixture. Further, the exressin fr the change-in-entry acrss the shck frnt is given by Anand [0] as fllws: s = C vm ln( ) C m ln( ρ ρ ) + C ln[(1 Z) (1 Z )], (14) where C ( 1 k ) R ( Γ 1), and C Γ( 1 k ) R ( Γ 1). t + u vm = i Using Eq. (10), Eq. (3) transfrms int r m m = i u u + ρ a + D = 0. (15) r r Nw, let us cnsider a shck wave ragating int a hmgeneus mixture f a erfect gas and sherically small slid articles. In a frame f reference mving with the shck Page 9 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 frnt, the jum cnditins at the shck are given by the rinciles f cnservatin f mass, mmentum and energy acrss the shck, namely, ρ ( U u) = ρ U, (16) + ρ ( U u) = + ρu, (17) ( U u) e + + ρ = e + ρ + U, (18) where U and u are, resectively, the shck frnt ragatin velcity and the velcity f mixture. The flw quantities with the suffix and withut suffix dente, resectively, the values f flw quantities in ustream regin, i.e. ahead f shck frnt and in dwnstream regin, i.e. behind f shck frnt. Als, effects due t viscsity and thermal cnductivity are mitted and it is assumed that the dusty gas has an infinite electrical cnductivity. The ustream Mach number M, which characterizes strength f shck, is defined as M = U, (19) a The shck jum relatins fr ressure, temerature, density and velcity f mixture in terms f M are given by Anand [0] as T T ΓM ( Γ 1) =, (0) ( Γ + 1) [M Γ ( Γ 1)][(1 Z ) + M ( Γ 1+ Z )] =, (1) ] M ( Γ + 1) [ M ( Γ 1) + ρ = ρ M u a ( Γ + 1) M ( Γ 1+ Z ) + (1 Z, () ) (1 Z )( M 1) =. (3) ( Γ + 1 M ) 3. The gemetrical shck dynamics thery and analytical slutins In this sectin, we develed the gemetrical shck dynamics mdel t rvide a simlified, arximate treatment fr the ragatin f imlding shck waves in a tw-hase mixture f a erfect gas and small slid articles. The gemetrical shck Page 10 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 dynamics [] rvides ractically accurate results esecially fr cntinuusly accelerating imlding shck waves. Cnsequently, the resent mdel is well suited fr studying shcks in the self-ragating limit, in which the frnt is mving steadily r accelerating. Accrding t the gemetrical shck dynamics arach, the characteristic frm f the gverning Eqs. (1), () and (15), is easily btained by frming a linear cmbinatin f Eqs. (1) and (15) in nly ne directin in ( r, t) -lane. The linear cmbinatin f these tw equatins can be written as t u u u + ( u + λ ) ± λ ρ + ρ ( a + u λ) + Dρ a = 0, (4) r t r r The cnditins that this cmbinatin invlves the derivatives in nly ne directin, are given by t = r r t ( u + λ ) r = ( u + λ) u and λ ρ = ρ( a + λ u) t Eqs. (5) and (6) give u r r, (5) r λ = a + λ u, (6) t r λ = ± a i.e., = u ± a, (7) t It shws the fact that the characteristic curves in ( r, t) -lane reresent the mtin f ssible disturbances whse velcity differs frm the velcity f mixture u by the value ± a (seed f sund), resectively, fr diverging and cnverging shck waves. Nw, Eq. (4) can be written as t + ( u ± a) r u u u ± a ρ ± ρ a ( u ± a) + Dρ a = 0, (8) t r r The abve Eq. (8) is exact and hlds thrughut the flw since it is just a cmbinatin f the basic Eqs. (1), () and (15). By using abve Eq. (8) we may write the characteristic frm f the gverning Eqs. (1), () and (15), i.e. the frm in which equatin cntains derivatives in nly ne directin in the ( r, t) -lane, as u dr dr d + ρ a du + D ρ a = 0 alng C + i. e. = u + a, (9) u + a r dt Page 11 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 and u dr dr d ρ a du + D ρ a = 0 alng C i. e. = u a, (30) u a r dt Eqs. (9) and (30) reresent the characteristic equatins fr exlding and imlding shck waves, resectively. The gemetrical shck dynamics arach states that when relevant equatins are written first in the characteristics frm, the differential relatin which must be satisfied alng a characteristic can be alied t the flw quantities just behind the shck frnt. Tgether with the shck jum relatins, this rule determines the ragatin f the shck waves. We assume here that the shck jum relatins t hld, f curse, within the rder f arximatin determine by a cnstant value f U. We aly here the differential relatin (30) alng the negative characteristic C behind the shck wave. Tgether with the shck jum relatins, we are able t describe the shck velcity r the related quantities in terms f the quantities just ahead f the shck frnt. Eq. (30) is valid nly alng the negative characteristic curve C in the ( t) r, -lane, behind the imlding shck frnt. The idea f the characteristic rule f Whitham [] is t aly, n the negative characteristic curve C alng the imlding shck frnt. We thus neglect the difference in the cnstants f integratin btained when Eq. (30) is slved n different characteristics that intersect the shck frnt. These differences arise frm the nn-unifrmity f the flw behind the shck, s the characteristic rule effectively ignres the influence f the flw behind the shck wave n the shck ragatin. Because the effect f the flw behind the shck n the shck dynamics is ignred, the methd is very gd fr situatins where the shck wave accelerates with time, s that features f the flw behind d nt catch u with the shck. The excellent examles f flws with this characteristic are cnverging shck waves, which are the subject f this aer. Nw, assuming that the negative characteristic curve C alies n the shck frnt, we can use the shck jum relatins given by Eqs. (0) (3) t write the quantities in it, which are thse immediately behind the shck frnt, in terms f thse ahead f shck frnt and Mach number. The shck jum relatins we use here are the shck cnditins fr the dusty gas [0] rather than the shck cnditins fr an ideal gas used by Whitham [1]. Page 1 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 Nw, substituting the shck jum relatins given by Eqs. (0) (3) int the negative characteristic curve C we get a first rder rdinary differential equatin in M as dm dr λ ( M ) + D = 0, (31) M r where φ1( M ) λ ( M ) =, φ ( M ) φ ( M ) 3 φ ( M ) = ( M 1){ΓM ( Γ 1)} { M ( Γ 1+ Z ) + (1 Z 1 φ ( M ) = (1 Z )( M + 1)[{( Γ 1) M + } { M ( Γ 1+ Z ) + (1 Z )} ψ ], φ ( M ) = M ( M 1)ψ, and 3 + ψ ( M ) = [Γ M ( Γ 1)]/[( Γ 1) M + ]. On integratin, Eq. (31) disclses a relatin between the ragatin distance and Mach number as 1 dm r( M ) K = ex λ( M ) D M, (3) where K is a cnstant f integratin. The variatin f the functin λ ( M ) with the ustream Mach number M fr β = 1, γ = 7 / 5 and varius values f k and G are shwn in Fig.. It is bvius that s λ( M ) varies little fr small values f M and remains unaffected fr large values f M. Thus, the ragatin distance-mach number relatin fr a dusty (3) may be easily written as M Kr λ ( M ) D =, where K is a cnstant. Thus, the analytical exressin fr the nn-dimensinal ragatin velcity f shck may be written as U a λ ( M ) )}, D = Kr. (33) This equatin is valid fr the imlding shck waves in the mixture f a erfect gas and small slid articles and is the main result f the resent study. Thus, the characteristic rule fr the ragatin velcity is λ ( M ) λ ( M ) 1/ U r fr cylindrical shcks and is / U r fr sherical shcks, where r is the radius f shck frnt. It is wrth Page 13 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 mentining that fr strng diverging shck waves in a erfect gas, Whitham [ age73] btained the characteristic rule as U r 1/ n fr cylindrical shck waves and U r fr sherical shck waves, where n = ( 1+ (1 ) ( γ + 1) µ )( 1+ µ + 1/ M ) µ = ( γ 1) M + ) ( γ M ( γ 1) ) / n µ, and. The characteristic rule may be used fr investigating the nature and behavir f the flw variables behind the imlding shck waves in the tw-hase gas-article flws. Fig. The variatin f λ (M ) with M fr varius values f k and G. Nw, the crresnding analytical exressins fr the distributin f ressure, temeraturet, density ρ, velcity f mixture u, and seed f sund a just behind the imlding shck frnt can be easily written as K = D λ Γr ( Γ 1) ( Γ + 1), (34) T T [K Γr ( Γ 1)][(1 Z ) + K ( Γ 1+ Z ) r D λ K ( Γ + 1) [( Γ 1) K r + ] D λ = D ] λ r D λ, (35) Page 14 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 ρ ( Γ + 1) =, (36) D λ ρ ( Γ 1+ Z ) + K (1 Z ) r u a (1 Z = D )( K r K( Γ + 1) λ 1) r D λ, (37) a a (1 Z = D )( K r K( Γ + 1) λ 1) r D λ. (38) Further, the analytical exressin fr the adiabatic cmressibility f mixture just behind the imlding shck frnt is btained by using Eqs. (13) and (34) as ( ) ( 1 Z )( Γ + 1) τ =. Γ λ D ( K Γr ( Γ 1) ) Finally, the analytical exressin fr the change-in-entry acrss the imlding shck frnt in a dusty gas flw is easily btained by using Eqs. (14), (34) and (36) as: D λ s (1 k ) K Γr ( Γ 1) = ln R i ( Γ 1) ( Γ + 1) Γ(1 k ) ( Γ + 1) ln D (40) λ ( Γ 1) ( Γ 1+ Z ) + K (1 Z ) r Γ(1 k ) 1 Z + ln ( Γ 1) 1 Z Thus, the influence f the mass cncentratin f slid articles and dust lading arameter n the shck velcity, the flw field quantities and the change-in-entry can be exlred frm the abve analytical exressins (33) (40). (39) 4. Results and discussin In the resent aer the general analytical slutin fr imlding shck waves in a tw-hase mixture f a erfect gas and small slid articles was btained by adting the gemetrical shck dynamics, due t Whitham [] and further the general slutin was examined and exlred fr cylindrical and sherical shck waves. The gal f the resent investigatin was t examine the influence due t dust articles n the imlding shck waves as they fcus at the centre f cnvergence and the regin f flw field immediately behind the imlding shck frnt. The general characteristic rule fr the ragatin velcity U a f imlding shck waves in a dusty gas is given by Eq. (33). The nn- Page 15 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 dimensinal analytical exressins fr the distributin f ressure, temeraturet T, density ρ ρ, velcity f mixtureu a, seed f sund a a, and adiabatic cmressibility τ ) f mixture immediately behind the imlding shck ( frnt are given by Eqs. (34) (39), resectively. Finally, the analytical exressin fr the change-in-entry s Ri acrss the imlding shck frnt in the dusty gas is given by Eq. (40). These analytical exressins were derived by assuming that the disturbances due t the reflectins, wave interactins in the wake, etc. d nt vertake the imlding shck waves. It is wrth mentining that the analytical exressins fr the shck velcity, ressure, temerature, density, velcity f mixture, seed f sund, adiabatic cmressibility f medium and change-in-entry acrss the shck frnt are functins f the ragatin distance r frm the rigin O, i.e. the centre f cnvergence, the ustream Mach number M, the mass fractin (cncentratin) f slid articles k in the mixture, the dust lading arameter, i.e. the rati f the density f the slid articles t the initial density f the gas G, the secific heat rati f the slid articles β and the secific heat rati f the gasγ. It is ntewrthy that the effects due t the small slid articles enter thrugh the arameters such as the mass fractin f slid articles in the mixture, the rati f the density f the slid articles t the initial density f the gas and the secific heat rati f the slid articles. It is ntable that the analytical slutins fr the imlding cylindrical and sherical shck waves in a dusty gas are rincially identical and nly a dimensinality index D differs the tw cases. The numerical cmutatins f the flw variables behind the imlding shck frnt are carried ut using MATHEMATICA and MATLAB cdes. Fr the urse f numerical calculatins, the tyical values f the secific heat rati f the slid articles and the rati f secific heats f the gas are taken t be 1 and 7/5, resectively. The value f β = 1andγ = 7 / 5 crresnds t the mixture f air and glass articles [15]. In ur analysis, we have assumed the initial vlume fractin f slid articles Z t be a small cnstant. Frm Fig. 1 it is bvius that the values f k frm 0 t 0.4 withg frm 1 t give small values f Z, in general. Therefre, the values f the cnstant arameters are taken t be γ =7/5, β s = 1, k =0, 0., 0.4, G =1, 10, 100,, and M =5 fr the general urse f numerical cmutatins. s s Page 16 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 The value f k = 0 crresnds t the case f a erfect gas r dust-free gas. The value G = 1 crresnds t Z = k, i.e. the case when initial vlume fractin f slid articles in the mixture is equal t the mass fractin f slid articles. It is very useful t mentin that the resent analysis serves an analytical descritin fr the ragatin f imlding shck waves thrugh an in-viscid, nn-heat cnducting and electrically infinitely cnducting mixture f a erfect gas and small slid articles. In the resent investigatin, the tw cases are cnsidered-cylindrical and sherical cnverging shcks t highlight the differences between the D and 3D cnvergence. Page 17 f 31
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Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 Fig. 3 The variatins f U a,, T T, ρ ρ, u a, a a, τ ( ) and s Ri distributin behind the imlding cylindrical shck wave with r fr varius values f k and G. The nn-dimensinal analytical exressins fr the shck velcity, ressure, temerature, density, velcity f mixture, seed f sund, adiabatic cmressibility and change-in-entry behind the imlding cylindrical and sherical shck frnt are, resectively, btained by taking the dimensinality index D = 1 and D = in the Eqs. (33) (40). In numerical cmutatins, the value f cnstant K =.30014659 fr cylindrical shck waves and K = 1.05813487 btained by assuming that fr sherical shck waves, which is U = 5a at r = 0. 5 fr β = 1, γ = 7 / 5, M = 5, k = 0. G = 10. The variatins f the shck velcity U a, ressure, temeraturet T, density ρ ρ, velcity f mixture u a, seed f sund a a, adiabatic cmressibility f mixture τ ( ) and change-in-entry distributin s Ri behind the imlding cylindrical shck frnt and sherical shck frnt with the ragatin distance r fr β = 1, γ = 7 / 5, M = 5 and varius values f k and G are, resectively, shwn in s s and Page 1 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 Figs. 3 and 4. It is bserved frm Figs. 3 and 4 that the shck velcity, ressure, temerature, velcity f mixture, seed f sund, adiabatic cmressibility and changein-entry increase raidly and tend t infinity as the cylindrical r sherical shck wave araches at the axis r centre f cnvergence. It is ntable that the distributin f density immediately behind the cylindrical (r sherical) shck frnt increases as the shck wave mves twards the axis (r centre) f cnvergence excet fr G = 1, esecially with k = 0. 4. It is ntable that the velcity f cylindrical and sherical shck decrease with increase in the mass cncentratin f slid articles in the mixture. The velcity f cylindrical and sherical shcks increases with the dust lading arameter G fr k = 0., hwever, fr k = 0. 4 the velcity f cylindrical and sherical shcks first decreases u t G = 10 and then it increases. This behavir f the velcity f cylindrical and sherical shcks, esecially fr the case f = 0.4, G = 10 differs greatly frm the case f a erfect (dust-free) gas. Similarly the ressure behind the cylindrical and sherical shck frnt decreases as the mass cncentratin f slid articles increases in the mixture. The ressure behind the cylindrical r sherical shck frnt increases with the dust lading arameter G fr k = 0.. Hwever, fr k = 0. 4, the ressure first decreases with increase in the value f dust lading arameter G u t G = 10 and then the ressure increases. This behavir f the ressure, esecially fr the case f k = 0.4, G = 10 differs greatly frm the dust-free case. It is imrtant t nte that the temerature behind the cylindrical and sherical shck frnts decreases with the mass cncentratin f slid articles in the mixture. It is als ntable that nly fr value f G = 1, the temerature behind the cylindrical shck frnt increases with the mass cncentratin f slid articles in the mixture. An increase in the dust lading arameter G leads t a decrease in the temerature behind the cylindrical and sherical shck frnts. This behavir f the temerature fr bth dimensinalities, esecially fr the case f k = 0.4, G = differs greatly frm the dust-free case. The density behind the cylindrical r sherical shck frnt decreases with increase in the mass cncentratin f slid articles fr the values f G 10, hwever, the density increases fr the values f G 100. The density behind the cylindrical and sherical shck frnt increases with the k Page f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 dust lading arameter G. This behavir f the density, esecially fr the case f k = 0.4, G = 1differs greatly frm the dust-free case. The velcity f mixture, the seed f sund and the adiabatic cmressibility just behind the cylindrical r sherical shck frnt decrease with the mass cncentratin f slid articles in the tw-hase flw f a erfect gas and small slid articles. It is ntable that the velcity f mixture, the seed f sund and the adiabatic cmressibility immediately behind the cylindrical shck frnt increase with increase in the value f dust lading arameter G. The velcity f mixture, seed f sund and adiabatic cmressibility f mixture just behind the sherical shck frnt increase with the dust lading arameter G fr k = 0.. Hwever, fr k = 0. 4 the velcity f mixture, seed f sund and adiabatic cmressibility first decrease with increase in the value f dust lading arameter G u t G = 10 and then they increase. This behavir f the velcity f mixture, seed f sund and adiabatic cmressibility, esecially fr the cases f k = 0.4, G = 1 (cylindrical shcks) and k = 0.4, G = 10 (sherical shcks) differs greatly frm the dust-free case. The change-inentry acrss the cylindrical r sherical frnt decreases with increase in the mass cncentratin f slid articles in the mixture. It is bvius frm Figs. 3 and 4 that an increase in the dust lading arameter G leads t an increase in the change-in-entry fr k = 0.. Hwever, fr k = 0. 4 the change-in-entry first decreases with increase in the value f dust lading arameter G u t G = 10 and then it increases. This behavir f the change-in-entry, esecially fr the case f = 0.4, G = 10 differs greatly frm the dust-free case. It is ntable that large changes are fund in the values f flw variables fr small values f the dust lading arameter G and vice versa. The variatins in the flw variables with the dust lading arameter G are mre rnunced at higher values f k. This may be hysically interreted as fllws. At cnstant decrease in k k, there is a substantial Z with increase in G (see Fig. 1), i.e. the vlume fractin f slid articles in the undisturbed medium becmes, cmaratively, very small. It is wrth mentining that in the case f G = 1, small slid articles with the density equal t that f the erfect gas in the mixture ccuy a significant rtin f the vlume which decreases the Page 3 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 cmressibility f the medium remarkably. Hwever, fr examle in the case f G = 100, small slid articles with the density equal t hundred times that f the erfect gas in the mixture ccuy a very small rtin f the vlume and, therefre, cmressibility is nt reduced much; but, the inertia f the mixture is increased significantly due t the article lad. An increase in k frm 0. t 0.4 fr G = 100 means that the erfect gas in the mixture cnstituting 80% f the ttal mass and ccuying 99.75% f the ttal vlume nw cnstitutes 60% f the ttal mass and ccuies 99.34% f the ttal vlume. Due t this reasn, the density f the erfect gas in the mixture is highly decreased, which vercmes the effect f incmressibility f the mixture and ultimately causes a decrease in the shck velcity, and the abve-mentined behavir f the flw variables. Figs. 3 and 4 illustrate similar effects fr an increase in k frm 0. t 0.4 fr ther values f the dust lading arameter G. Thus, the vlumetric fractin f the dust lwers the cmressibility f mixture, hwever, the mass f the dust lad may increase the ttal mass, and hence it may add t the inertia f mixture. Bth effects due t the additin f dust, the decrease f mixture s cmressibility and the increase f mixture s inertia may bviusly influence the ragatin f shck waves. Page 4 f 31
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Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 Fig. 4 The variatins fu a,, T T, ρ ρ, u a, a a, τ ( ) and s Ri distributin behind the imlding sherical shck wave with r fr varius values f k and G. It is wrth mentining that the effects f dust articles mdify the numerical values f the shck velcity, ressure, temerature, density, velcity f mixture, seed f sund, adiabatic cmressibility and change-in-entry frm their values fr the erfect gas. Hwever, the trends f variatins f the shck velcity, ressure, temerature, density, velcity f mixture, seed f sund, adiabatic cmressibility and change-inentry remain arximately unaffected, in general, fr the cylindrical and sherical cnverging shck waves in a dusty gas. Thus, the resent analysis rvides a fairly accurate and cmlete descritin f the influence f dust articles n the shck velcity and flw variables behind the imlding cylindrical and sherical shck waves in a mixture f erfect gas and small slid articles. 5. Cnclusins The resent wrk investigates the effects f dust lading arameters n the shck Page 8 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 velcity and the flw field behind the imlding cylindrical and sherical shck waves ragating in a mixture f erfect gas and small slid articles. The fllwing cnclusins may be drawn frm the findings f the current analysis: 1. The effects due t the mass cncentratin f slid articles and the dust lading arameter G, generally, d nt change the trends f variatins f the shck velcity and flw variables behind the cylindrical and sherical shck waves but they mdify the numerical values f the shck velcity and flw variables frm their values fr the erfect (dust-free) gas case.. The shck velcity, ressure, temerature, velcity f mixture, seed f sund, adiabatic cmressibility f mixture and change-in-entry behind the cylindrical (r sherical) shck frnt increase as the shck wave fcuses at the axis (r centre) f cnvergence. The density f mixture behind the cylindrical (r sherical) shck frnt increases as the shck wave fcuses at the axis (r centre) f cnvergence excet fr G = 1. 3. The shck velcity, ressure, temerature, density, velcity f mixture, seed f sund, adiabatic cmressibility and change-in-entry behind the cylindrical and sherical shck frnts decrease with increase in the mass cncentratin f small slid articles in the mixture. 4. The shck velcity, ressure, density, velcity f mixture, seed f sund, adiabatic cmressibility and change-in-entry increase, hwever, the temerature decreases with an increase in the value f dust lading arameter G. 5. The trends f variatins f the shck velcity and flw variables just behind the cylindrical r sherical shck waves in tw-hase gas-article mixture are similar t that f behind the cylindrical r sherical shck waves in a erfect gas. The article cncerns with the imlding rblem, hwever, the methdlgy and analysis resented here may be used t describe many ther hysical systems invlving nn-linear hyerblic artial differential equatins. The tential alicatins f this study include simulatin f rblems in shck wave lithtrisy, exlsive detnatin and cmressible flw in tw-hase gas-article fluids. This mdel may be used t describe Page 9 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 sme f the verall features f the shck waves frm undergrund nuclear exlsin [4] and in the dusty envirns f stars, stellar medium, duble-detnatin suernvae, etc. The mdel s advantages lie in the fact that it is caable f describing the flw field behind the imlding shck waves in an ideal gas as well as actual dusty envirnments. The mdel develed is f interest t the astrhysicists, fluid dynamicists, gehysicists and scientists wrking n imlding shck waves. References [1] G. F. Carrier, Shck waves in a dusty gas, J. Fluid Mech. 4 (1958) 376-38. [] G. Rudinger, Sme rerties f shck relaxatin in gas flw carrying small articles, Phys. Fluids 7 (1964) 658-663. [3] A. R. Kribel, Analysis f nrmal shck waves in article laden gas, J. Basic. Eng. 86 (1964) 655-664. [4] E. Outa, K. Tajima, H. Mrii, Exeriments and analysis f shck waves ragating thrugh a gas-article mixture, Bull. Jn. Sc. Mech. Eng. 19 (130) (1976) 384-394. [5] F. E. Marble, Dynamics f dusty gas, Annu. Rev. Fluid Mech. (1970) 397-446. [6] A. N. Kraik, R. I. Nigmatulin, V. K. Starkv, L. E. Sternin, Mechanics f multihase media. In: Advances in Science and Technlgy. Hydrmechanics, VINITI, Mscw, 197, vl.6,. 93 174. [7] F. Higashin, T. Suzuki, The effect f articles n blast wave in a dusty gas, Z. Naturfrsch. 35a (1980) 1330-1336. [8] A. I. Ivandaev, A. G. Kutushev, R. I. Nigmatulin, Gasdynamics f multihase media. Shck and detnatin waves in gas-article mixtures. In: Advances in Science and Technlgy. Mechanics f Liquids and Gases, VINITI, Mscw, 1981, vl.16,. 09 87. [9] O. Igra, G. Ben-Dr, Dusty shck waves, Al. Mech. Rev. 41 (1988) 379-437. [10] S. I. Pai, S. Lu, Theretical and Cmutatinal Dynamics f a Cmressible Flw, Van Nstrand-Reinhld and Science Press, New Yrk, 1991, 699 [11] G. Ben-Dr, Dusty shck waves - an udate, Al. Mech. Rev. 49 (1996) 141-146. [1] S. I. Pai, Tw Phase Flw. Vieweg Tracts in Pure and Alied Physics,vl. 3, Vieweg, Braunschweig, 1977 (Chater V). Page 30 f 31
Acceted fr ublicatin in Internatinal Jurnal f Nn-Linear Mechanics (014) htt://dx.di.rg/10.1016/j.ijnnlinmec.014.05.001 [13] G. Rudinger, Fundamentals f Gas-Particle Flw, Elsevier, Amsterdam,1980. [14] S. I. Pai, S. Menn, Z. Q. Fan, Similarity slutin f a strng shck wave ragatin in a mixture f a gas and dust articles, Int. J. Eng. Sci. 18 (1980) 1365-1373. [15] H. Miura, I. I. Glass, Develment f the flw induced by a istn mving imulsively in a dusty gas, Prc. R. Sc. A 397 (1985) 95-309. [16] H. Steiner, T. Hirschler, A self-similar slutin f a shck ragatin in a dusty gas, Eur. J. Mech. B Fluids 1 (00) 371-380. [17] T. Sait, M. Marumt, K. Takayama, Numerical investigatins f shck waves in gas-article mixtures, Shck Waves 13 (003) 99-3. [18] G. N. Naidu, K. Venkatanandam, M. P. Ranga Ra, Arximate analytical slutins fr self-similar flws f a dusty gas with variable energy, Int. J. Eng. Sci. 3 (1985) 39-49. [19] R. K. Anand, The ragatin f shck waves in a channel f variable crss sectin cntaining a dusty gas, Phys. Scr. 86 (01) 05401(7). [0] R. K. Anand, Shck jum relatins fr a dusty gas atmshere, Astrhys. Sace Sci. 349 (014) 181 195. [1] G. B. Whitham, On the ragatin f shck waves thrugh regins f nn-unifrm area r flw, J. Fluid Mech. 4 (1958) 337-360. [] G. B. Whitham, Linear and Nnlinear Waves, Wiley, New Yrk, 1974. [3] E. A. Melwyn-Hughes, Physical Chemistry, Pergamn, Lndn, 1961. [4] F.K. Lamb, B.W. Cllen, J.D. Sullivan, An arximate analytical mdel f shck waves frm undergrund nuclear exlsin, J. Gehys. Res. 97 (199) 515-535. Page 31 f 31