Key words Shock waves. Dusty gas. Solid particles. Shock jump relations. Mach number

Transcription

1 Shck jum relatins fr a dusty gas atmshere Shck jum relatins fr a dusty gas atmshere R. K. Anand Deartment f Physics, University f Allahabad, Allahabad-00, India anand.rajkumar@rediffmail.cm Abstract This aer resents generalized frms f jum relatins fr ne dimensinal shck waves ragating in a dusty gas. The dusty gas is assumed t be a mixture f a erfect gas and sherically small slid articles, in which slid article are cntinuusly distributed. The generalized jum relatins reduce t the Rankine-Hugnit cnditins fr shcks in an idea gas when the mass fractin (cncentratin) f slid articles in the mixture becmes zer. The jum relatins fr ressure, density, temerature, article velcity, and change-in-entry acrss the shck frnt are derived in terms f ustream ach number. Finally, the useful frms f the shck jum relatins fr weak and strng shcks, resectively, are btained in terms f the initial vlume fractin f the slid articles. The cmutatins have been erfrmed fr varius values f mass cncentratin f the slid articles and fr the rati f density f slid articles t the cnstant initial density f gas. Tables and grahs f numerical results are resented and discussed. Key wrds Shck waves. Dusty gas. Slid articles. Shck jum relatins. ach number. Intrductin Understanding the influence f slid articles n the ragatin henmena f shck waves and n the resulting flw field is f imrtance fr slving many engineering rblems in the field f astrhysics and sace science research. When a shck wave is ragated thrugh a gas which cntains an areciable amunt f dust, the ressure, the temerature and the entry change acrss the shck, and the ther features f the flw differ greatly frm thse which arise when the shck asses thrugh a dust-free gas. The flw field, that devels when a mving shck wave hits a tw-hase medium f gas and articles, has a clse ractical relatin t industrial alicatins (e.g. slid rcket engine in which aluminum articles are used t reduce the vibratin due t instability) as well as industrial accidents such as exlsins in calmines and grain elevatrs. Therefre, a successful redictin f the beha vir f shck waves in a twhase medium f gas and slid articles is very crucial and imerative fr the successful design and eratin f rcket nzzles and energy cnversin systems. Underwater shck-wave fcusing techniques such as lithtrisy fr treatment f kidney stnes have been widely used in hsitals. Presently, these techniques have been alied nt nly t urlgy but als musculskeletal disrders, brain-neur surgery, cancer treatments and the theray f cerebral emblisms. This aer describes an interactin henmenn when anand.rajkumar@rediffmail.cm /6 Physics Deartment, University f Allahabad, Allahabad 00

2 Shck jum relatins fr a dusty gas atmshere a shck wave ragates in a tw-hase medium f a erfect gas and sherically small slid articles. The bjective f resent study is t btain jum relatins acrss a shck frnt in a twhase mixture f erfect gas and small slid articles. T authr s best knwledge, s far there is n aer rerting the shck jum relatins fr a tw-hase mixture f a erfect gas and small slid articles, btained using the Pai mdel (977). This mdel assumes the dusty gas t be a mixture f sherically small slid articles and a erfect gas. The dust hase cmrises the ttal amunt f slid articles which are cntinuusly distributed in the erfect gas. On the ne hand, the vlumetric fractin f the dust lwers the cmressibility f the mixture. On the ther hand, the mass f the dust lad may increase the ttal mass, and hence it may add t the inertia f the mixture. Bth effects due t the additin f dust, the decrease f the mixture s cmressibility and the increase f the mixture s inertia may markedly influence the shck wave sread. In the resent research aer, the authr has derived the generalized frms f jum relatins fr ne-dimensinal shck wave ragating in a dusty gas. These jum relatins reduce t the well knwn Rankine-Hugnit cnditins fr shck waves in an ideal gas. The dusty gas mdel given by Pai (977) is used here. The jum relatins fr the ressure, temerature, density, and article velcity are btained, resectively in terms f the ustream ach number, characterizing the shck strength. Besides these jum relatins the generalized exressin fr the adiabatic cmressibility f the mixture and change-in-entry acrss the shck frnt are als btained in terms f the ustream ach number. Further, the useful frms f the shck jum relatins fr ressure, density and mixture velcity and adiabatic cmressibility f mixture alng with the exressin fr the change-in-entry, in terms f the initial vlume fractin Z f the slid articles are btained fr the tw cases: viz., (i) when the shck is weak and (ii) when it is strng, simultaneusly. The Z is a functin fk, the mass cncentratin f the slid articles and G, the rati f the density f slid articles t the cnstant initial density f gas. If the arameter Z is taken zer, the generalized shck jum relatins reduce t the crresnding jum relatins fr shck waves in an ideal gas. These generalized jum relatins fr varius flw variables are very useful in the theretical and exerimental investigatins f strng as well as weak shck waves in the dusty envirnments. The backgrund infrmatin is rvided in Sect. as an intrductin. Sect. cntains the general assumtins and ntatins. In Sectin 3 the generalized frms f shck jum cnditins are resented. Sect. 4 mainly describes results with a brief discussin. The shck cnditins crresnding t an ideal gas are summarized in an aendix fr the cnvenience f reference. anand.rajkumar@rediffmail.cm /6 Physics Deartment, University f Allahabad, Allahabad 00

3 Shck jum relatins fr a dusty gas atmshere. Basic equatins The cnservatin equatins fr an unsteady, lane, cylindrically r sherically symmetric flw field between a shck and a istn mving behind it in a dusty gas under an equilibrium cnditin can be exressed cnveniently in Eularian c rdinates (see Anand 0b) as fllws: u u u 0 () t r r u u t r r u j r 0 e e u 0 u t r t r where u ( r, t) is the velcity f the mixture, ( r, t) the density f the mixture, ( r, t) the ressure f the mixture, e ( r, t) the internal energy f the mixture er unit mass, r is the distance frm the rigin, O and t is the time c-rdinate. The gemetry factr j is defined j by j d In A d Inr, where A( r) ( j ) r is the flw crss-sectin area. Then the ne-dimensinal flw in lane, cylindrical and sherical symmetry is characterized by j = 0,, and, resectively. 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 equatins () and (3). These terms are, f curse, imrtant fr evaluating the extent f the relaxatin zne behind the shck frnt, which is hwever, beynd the sce f this aer. 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. () - (3) aly. All relaxatin rcesses are excluded such that n relative mtin and n temerature differences between erfect gas and 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 ( k ) RiT (4) Z () (3) anand.rajkumar@rediffmail.cm 3/6 Physics Deartment, University f Allahabad, Allahabad 00

4 Shck jum relatins fr a dusty gas atmshere 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. The relatin between Z and the mass cncentratin f the slid articles in the mixture taken as a cnstant in the whle flw field is given by Pai et al (980) as fllws: Z k s (5) where Z, while s is the secies density f the slid articles and a Z subscrit refers t the initial values f Z, and. The initial vlume fractin f the slid articles Z 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 (see iura and Glass 985), hence, Z may be assumed as a small cnstant. The initial vlume fractin f small slid articles is (see Pai977) Vs k Z Z Z, and Vg Vs G( k ) k (6) 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 articles V s which remains cnstant. The vlumetric arameter G is defined as G s g which is equal t the rati f the density f the slid articles t the initial density f the gas. Hence, the fundamental arameters f the Pai 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 t G, i.e., V 0. The variatin f initial vlume fractin Z f small slid articles with mass cncentratin k f slid articles in the mixture fr varius values f G is shwn in Fig.. It is wrth mentining that Z increases arximately linearly with increasing fr G 0, while it increases very slwly fr values fg 50. Fig. shws the variatin f the vlumetric fractin Z f slid articles in the mixture with / fr different values f k and G. Nte that Z is rrtinal t the rati f /, and it increases linearly with increasing G 0. k fr G, while it increases very slwly fr the values f s k anand.rajkumar@rediffmail.cm 4/6 Physics Deartment, University f Allahabad, Allahabad 00

5 Shck jum relatins fr a dusty gas atmshere Fig. Variatin f Z with k fr varius values f G Fig. Variatin f Z with / fr varius values f k and G anand.rajkumar@rediffmail.cm 5/6 Physics Deartment, University f Allahabad, Allahabad 00

6 Shck jum relatins fr a dusty gas atmshere The internal energy f the mixture is related t the internal energies f the tw secies and may be written as e [ k C ( k ) C ] T C T (7) where s v v m C s is the secific heat f the slid articles, C v is the secific heat f the gas at cnstant vlume and C v m 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 ( k ) C (8) m where s 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 m C v m where c cv is the secific heat rati f the gas, C s / Cv is the secific heat rati f the slid articles and k k ). The variatin f the rati f secific heat f ( the mixture with the mass cncentratin (9) k f slid articles in the mixture fr, 7 / 5and varius values f is shwn in Fig.3. It is ntable that decreases with increasing k hwever it increases with increasing values f. Fig.3 Variatin f Г with k fr and varius values f anand.rajkumar@rediffmail.cm 6/6 Physics Deartment, University f Allahabad, Allahabad 00

7 Shck jum relatins fr a dusty gas atmshere Eliminating the temerature frm equatins (4), (6) and (7), we may nw write the secific internal energy f the mixture as fllws: Z e (0) 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 ( k ) R as fllws: i d ( k ) RiT a () d ( ) ( ) S where subscrit s refers t the rcess f cnstant entry. The initial sund seed a f the mixture is defined as a () ( Z ) Fig.4 shws the variatin f nn-dimensinal initial sund seed a ( ) f the mixture with k fr, 7 / 5and varius values fg. It is ntewrthy that the initial sund seed increases with increasing arximately linearly fr values f G 50. k fr G, while it decreases Fig.4 Variatin f a ( ) with k fr, 7 / 5and varius values f G anand.rajkumar@rediffmail.cm 7/6 Physics Deartment, University f Allahabad, Allahabad 00

8 Shck jum relatins fr a dusty gas atmshere Thus, the rati f the equilibrium sund seed f the mixture t that f a erfect gas a i is a s ( k ) ( ) k (3) ai Z Z s Fig. 5 shws the variatin f a ai with the rati f density / f the mixture fr, 7 / 5and varius values f k and G. It is ntewrthy that the rati f exnentially with increasing values fg 0. a ai increases k fr G, while it decreases arximately linearly fr Fig. 5 Variatin f a ai with / fr,. 4 and varius values f k and G The deviatin f the behavir f a dusty gas frm that f a erfect gas is indicated by the cmressibility defined as (see elwyn-hughes 96) where S Z S (4) dentes the derivative f with resect t at the cnstant entry. The vlume f the articles lwers the cmressibility f the mixture, while the mass f the slid articles increases the ttal mass, and therefre may add t the inertia f the mixture. This can be shwn in tw limiting cases f the mixture at the initial state. Fr G, it fllws frm the equatins (6) and (5) that Z = k, RiT and anand.rajkumar@rediffmail.cm 8/6 Physics Deartment, University f Allahabad, Allahabad 00

9 Shck jum relatins fr a dusty gas atmshere ( k ), i.e. the resence f the slid articles linearly lwers the cmressibility f the mixture in the initial state. In the ther limiting case, i.e. fr G, the vlume f the slid articles V s tends t zer. Accrding t equatin (6), the vlume fractin Z is equal t zer. In this case, the cmressibility is nt affected by the dust lading. The slid articles cntribute nly t increasing the mass and inertia f the mixture. Further, using the arriate relatins, we can write the exressin fr the change in entry acrss the shck frnt as s C ln( / ) C ln( / ) C ln[( Z) /( Z )] (5) vm m where C ( k ) R /( ), and C ( k ) R /( ) vm i m m i 3. Frmulatin f jum relatins in terms f ach number The jum cnditins acrss shck frnt relate the fluid rerties behind the shck, which is referred t as the dwnstream regin, t the fluid rerties ahead f the shck, which is referred t as the ustream regin. The hysical nature f the flw field imses tw sets f bundary cnditins, and in additin, it rvides als an integral relatin exressing the rincile f glbal cnservatin f energy inside the field. Nw, let us cnsider a shck wave ragating int a hmgeneus mixture f a erfect gas and small sherical slid articles f cnstant initial density. In a frame f reference mving with the shck frnt, the jum cnditins at the shck are given by the rinciles f cnservatin f mass, mmentum and energy acrss the shck, namely, ( U u) U (6) ( U u) U (7) ( U u) e e U where U and u are, resectively, the shck frnt ragatin velcity and the velcity f the mixture. The quantities with the suffix and withut suffix dente the values f the quantities in ustream regin i.e., immediately ahead f shck, and in dwnstream regin i.e., immediately behind the shck, resectively. Als, the effects f viscsity and thermal cnductivity are mitted and it is assumed that the dusty gas has an infinite electrical cnductivity. The ustream ach number as (say), which characterizes the strength f a shck, is defined as U (9) a In general, the ustream ach number in a given rblem is knwn and it is desired t determine the dwnstream ach number. The exressin fr dwnstream ach bs number can be easily btained in terms f ustream ach number bs, and written as (8) anand.rajkumar@rediffmail.cm 9/6 Physics Deartment, University f Allahabad, Allahabad 00

10 Shck jum relatins fr a dusty gas atmshere 4 4( Z ) ( Z) ( Z ) ( Z) bs (0) [( ) ( Z ) ( ) ] ( Z ) The rati f the different flw variables such as ressure, density, temerature, etc. acrss a shck wave in an ideal gas are exressed as functins f ustream ach number. They are generally referred t as the Rankine-Hugnit jum relatins r cnditins. Using Eqs. (0), (), (6)-(8), the ressure f the mixture, the temerature f the mixture, the density f the mixture and the velcity f the mixture immediately behind the shck frnt can be written as T T U ( ) a [ a (U a () a ) ][a ( ) U ( U a ) z ] () ( ) [ ( ) U ] U U ( ) U ( Z ) ( Z ) a u ( Z )( U a ) (4) U ( ) U The ressure f the mixture, the temerature f the mixture, the density f the mixture and the velcity f the mixture just behind the shck frnt in terms f the ustream ach number can be exressed as T T ( ) ( ) (3) (5) [ ( ) ][ ( ) ( ) z ] (6) ( ) [ ( ) ] ( ) ( Z ) ( Z ) u ( Z )( ) (8) a ( ) The exressin fr the cmressibility f mixture just behind the shck frnt is easily btained by substituting Eq. (5) int Eq. (4) and it can be written as Z ( ) ( ) (9) [ ( )] Further using Eqs. (5) and (7) in Eq. (5) the exressin fr the change in entry acrss the shck frnt can easily be written as (7) anand.rajkumar@rediffmail.cm 0/6 Physics Deartment, University f Allahabad, Allahabad 00

11 Shck jum relatins fr a dusty gas atmshere s ( k ) ( ) ln R i (30) ( k ) ( ) ( k ) Z ln ln ( Z Z Z ) ( ) The abve frms f the jum relatins acrss a shck frnt in a dusty gas flw are similar t the well-knwn Rankine-Hugnit cnditins acrss a shck frnt in an ideal gas flw [see Aendix] and the shck jum relatins are exlicitly written in terms f the ustream arameters nly. 3. Jum relatins fr weak shcks: Fr weak shck, is taken as, where is a arameter which is negligible in cmarisn with unity i.e., <<. Thus, the ressure, the density, the velcity f the mixture, and the sund seed just behind the weak shck can be, resectively, written as 4 (3) 4( Z ) u a 4( Z ) (3) (33) a a (34) 3. Jum relatins fr strng shcks: Fr strng shck, U a, thus the ressure, the density, the velcity f the mixture, and the sund seed just behind the strng shck can be, resectively, written as ( Z ) U (35) ( ) ( Z ) (36) ( Z ) U u (37) ( ) a ( Z ( ) ) U (38) 3.4 Strength f a shck wave: In shck wave analysis, the quantity, reresents the strength f the shck, and dentes the verressure. The shck seed increases with increasing verressure. anand.rajkumar@rediffmail.cm /6 Physics Deartment, University f Allahabad, Allahabad 00

12 Shck jum relatins fr a dusty gas atmshere Substituting fr frm equatin (3), we get Thus, fr shcks f any strength, we can write i.e., cnst. Case I. Fr Shcks f vanishing strength: Shck waves fr which ξ is almst zer, are referred t as shcks f vanishing strength. Fr such shcks,. Thus fr shcks f vanishing strength we can write,,, T, and s 0. Ri T Case II. Fr Strng s hcks: Since shck strength is rrtinal t ( cnst.), strng shcks are a result f very high values f ustream ach number. The maximum values f quantities in equatins ()-(4), are given belw lim lim lim T T 4 Results and discussin z The simlified frms f the shck jum relatins fr ne-dimensinal shck waves ragating in a dusty gas are derived which reduce t the well knwn Rankine-Hugnit shck cnditins fr ideal gas when mass cncentratin f slid articles in the mixture becmes zer. The jum relatins fr ressure, temerature, density, and velcity f mixture are btained, resectively in terms f the ustream ach number. The exressins fr the cmressibility f the mixture which shws the deviatin f the behavir f a dusty gas frm that f a erfect gas and the change in entry acrss the shck frnt are als btained, resectively in terms f the ustream ach number. Further, the useful frms f shck jum relatins fr ressure, density and article velcity in terms f the initial vlume fractin Z f small slid articles and the rati f secific heats f the mixture are btained fr the tw cases: viz., (i) when the shck is weak and (ii) when it is strng, simultaneusly. Fr the urse f numerical calculatins, the values f the and the rati f secific heats f the gas are taken t be and.4, resectively. The values and. 4 crresnds t the mixture f air and glass articles (iura and Glass, 985). In ur analysis, we have assumed the initial anand.rajkumar@rediffmail.cm /6 Physics Deartment, University f Allahabad, Allahabad 00

13 Shck jum relatins fr a dusty gas atmshere vlume fractin Z f slid articles t be a small cnstant. The values f 0., 0. 4 with G, 0, 00 give small values f Z. The exressin fr the dwnstream ach number is given by equatin (0). Fig. 6 deicts the variatin f dwnstream ach bs number with ustream ach number fr,. 4 and varius values f k bs and G. It is ntable that the dwnstream ach number bsdecreases with increasing and k, resectively while it increases with increasing G. This behavir, esecially fr the case f k 0. 4 and G, differs greatly frm the dust-free case. k Fig. 6 Variatin f bs with fr varius values f k and G The simlified shck jum relatins fr the ressure /, the temerature T / T, the density /, the velcity u / a f mixture, the cmressibility ( ) f mixture just behind the shck frnt are given by Eqs. (5) - (9), resectively. T see the effect f the mass cncentratin k and the mass-lading G f the dust n the flw field behind the shck frnt, the lts f dimensinless ressure, temerature, density, article velcity, sund seed and cmressibility with ustream ach number are resented in Fig. 7 fr,. 4, k 0, k 0., G, k 0., G 0, k 0., G 00, k k 0.4, G, k 0.4, G 0, 0.4, G 00, allwing a cmarisn with the dustfree case fr k 0. It is imrtant t nte that the ressure, temerature, density, anand.rajkumar@rediffmail.cm 3/6 Physics Deartment, University f Allahabad, Allahabad 00

14 Shck jum relatins fr a dusty gas atmshere velcity f mixture and sund seed increases with increasing while the cmressibility decreases. The ressure decreases slwly with increasing k while it is indeendent f G. This behavir f ressure, esecially fr the case f k 0. 4 differs greatly frm the dust-free case k 0. The temerature increases with increasing k fr values fg 0, while it decreases fr G 00 and it decreases with increasing G. This behavir f temerature, esecially fr the case f k 0. 4 and G differs greatly frm the dust-free case. The density behaves inversely; it decreases with increasing k fr G 0, while it increases fr G 00 and it increases with increasing G. This behavir f density, esecially fr the case f k 0. 4 and G 00 differs greatly frm the dustfree case. The velcity f mixture decreases with increasing k while it increases with increasing G. This behavir f velcity, esecially fr the case f k 0. 4 and G differs greatly frm the dust- free case. The seed f sund increases with increasing fr values f G 0, while it decreases fr G 00 and it decreases with increasing G. This behavir f sund seed, esecially fr the case f k 0. 4 and G differs greatly frm the dust-free case. The cmressibility decreases with increasing k k while it increases with increasing G. This behavir f cmressibility, esecially fr the case f k 0.4 and G differs greatly frm the dust-free case. It is wrth mentining that the cmressibility decreases raidly fr values f 4and then it becmes cnstant fr values f 4. It is interesting t nte that the increase in ressure and temerature can be infinitely large fr sufficiently large shck strengths (r ach number) but the density increase is limited t the value z. anand.rajkumar@rediffmail.cm 4/6 Physics Deartment, University f Allahabad, Allahabad 00

15 Shck jum relatins fr a dusty gas atmshere anand.rajkumar@rediffmail.cm 5/6 Physics Deartment, University f Allahabad, Allahabad 00

16 Shck jum relatins fr a dusty gas atmshere anand.rajkumar@rediffmail.cm 6/6 Physics Deartment, University f Allahabad, Allahabad 00

17 Shck jum relatins fr a dusty gas atmshere Fig.7 Variatins f /, /, T / T, u / a, a / a and ( ) with fr varius values f k and G anand.rajkumar@rediffmail.cm 7/6 Physics Deartment, University f Allahabad, Allahabad 00

18 Shck jum relatins fr a dusty gas atmshere The exressin fr the change in entry s given by Eq. (30). The variatin f the change in entry and varius values f Ri acrss the shck frnt in a dusty gas is s R with fr,. 4 k and G are shwn in Fig. 8. The change in entry increases with increasing. It is ntable that with increasing i k the change in entry increases raidly fr G, mderately fr G 0 and very slightly fr G 00 and it decreases with increasing G. In ther wrds, with increasing k the shck becmes strnger swiftly fr lwer values f G and gradually fr higher values fg. Accrding t the secnd law f thermdynamics, the entry f a substance cannt be decreased by internal rcesses alne, thus the dwnstream entry in a shck must equal r exceed its ustream value, s s. This entry increase, redicted by the mass, mmentum, and energy cnservatin relatins alne, imlies an irreversible dissiatin f energy, even fr an ideal fluid, entirely indeendently f the existence f a dissiatin mechanism. The exressin fr the change in entry s R acrss the shck frnt in a dusty gas is given by Eq. (30). Fr the sake f justificatin the Eq. (30), i.e. i s Ri is ltted with fr,. 4and varius values f k and G and shwn in Fig. 8. The change in entry increases with increasing. It increases with increasing k fr values fg 0, while it remains the same fr G 00 and it decreases with increasing G. This behavir, esecially fr the case f k 0. 4 and G differs greatly frm the dust-free case. Obviusly, the change in entry fr ideal gas ( k 0 ) is sitive when ustream ach number is greater than unity; hence, nly a shck frm suersnic t subsnic seed is ssible, with a crresnding rise in ressure acrss the nrmal discntinuity. It is imrtant t nte that fr curves excet fr k = 0 (dust free gas) s R has always sitive values fr values f greater than unity. Decrease in entry is imssible by secnd law f thermdynamics, thus shck waves cannt devel in a flw where ustream ach number is less than ne in case f a dusty gas as well as in an ideal gas. It cnfirms that the shck waves will arise in a flw f dust-laden fluids where ustream ach number is equal t r greater than unity as in the flw f ideal gas. It is remarkable and astnishing that in nn-ideal gas flws the shck waves may arise where ach number equal t r greater than 0.5 (Anand, 0a). i anand.rajkumar@rediffmail.cm 8/6 Physics Deartment, University f Allahabad, Allahabad 00

19 Shck jum relatins fr a dusty gas atmshere Fig. 8 Variatins f s R with fr varius values f k and G i 4. Weak shck waves The handy frms f jum relatins fr weak shck waves are given by Eqs. (3)-(34).These relatins are deendent f a arameter which is negligible in cmarisn with the unity, the rati f secific heat f the mixture, the mass cncentratin k f slid articles and the dust laden arameter G. The variatins f the ressure /, density /, velcity u / a f mixture and change in entry with fr,. 4and varius values f k and G are shwn in Fig. 9. It is imrtant t nte that the ressure, density, velcity f the mixture and change in entry increase with increasing. The ressure decreases very small in amunt with increasing k and it is indeendent f G. This behavir f ressure, esecially fr the case f k 0. 4 differs greatly frm the dust-free case. The density decreases with increasing r k fr values f G 0, while it increases fr G 00 and it increases with increasing G. This behavir f density, esecially fr the case f k 0. 4 and G differs greatly frm the dust-free case. The velcity f mixture als decreases with increasing k fr values fg 0, while it increases fr G 00 and it increases with increasing G. This behavir f velcity, esecially fr the case f k 0. 4 and G differs greatly frm the dust-free case. The change in entry slightly increases with increasing k and it als slightly decreases with increasing G. This behavir f change in entry, esecially fr the case f k 0. 4and G differs greatly frm the dust-free case. anand.rajkumar@rediffmail.cm 9/6 Physics Deartment, University f Allahabad, Allahabad 00

20 Shck jum relatins fr a dusty gas atmshere anand.rajkumar@rediffmail.cm 0/6 Physics Deartment, University f Allahabad, Allahabad 00

21 Shck jum relatins fr a dusty gas atmshere Fig.9 Variatins f /, /, u / a and s Ri with fr varius values f k and G anand.rajkumar@rediffmail.cm /6 Physics Deartment, University f Allahabad, Allahabad 00

22 Shck jum relatins fr a dusty gas atmshere 4. Strng shck waves The handy frms f jum relatins fr strng shck waves are given by Eqs. (35)-(38). These jum relatins are deendent f the shck strength U / a, the rati f secific heat f the mixture, the mass cncentratin the dust laden arameter G. Fig. 0 shws the variatins f the ressure /, velcity a k f slid articles and /, density u / f mixture, seed f sund a / a and change in entry with nndimensinal shck velcity fr,. 4 and varius values f k and G. It is imrtant t nte that the ressure, velcity f mixture, seed f sund and change in entry increase with increasingu / a but the density remains unchanged. The ressure decreases very small in amunt with increasing k and it is indeendent fg. This behavir f ressure, esecially fr the case f k 0. 4 differs greatly frm the dust-free case. The density decreases with increasing k fr values fg 0, while it increases fr G 00 and it increases with increasing G. This behavir f density, esecially fr the case f k 0. 4and G 00 differs greatly frm the dust-free case. The velcity f mixture remains unchanged with increasing k fr G, while it increases slightly fr values f G 0 and it increases with increasing G. This behavir f velcity, esecially fr the case f k 0. 4 and G 00 differs greatly frm the dust-free case. The seed f sund increases with increasing k fr G, remains unchanged fr G 0 decreases fr G 00 and it increases with increasing G. This behavir f seed, esecially fr the case f k 0. 4 and G 00 differs greatly frm the dust-free case. The change in entry increases with increasing k fr values fg 0, while it slightly decreases fr G 00 and it decreases with increasing G. This behavir f change in entry, esecially fr the case f k 0. 4 and G differs greatly frm the dust-free case. anand.rajkumar@rediffmail.cm /6 Physics Deartment, University f Allahabad, Allahabad 00

23 Shck jum relatins fr a dusty gas atmshere anand.rajkumar@rediffmail.cm 3/6 Physics Deartment, University f Allahabad, Allahabad 00

24 Shck jum relatins fr a dusty gas atmshere anand.rajkumar@rediffmail.cm 4/6 Physics Deartment, University f Allahabad, Allahabad 00

25 Shck jum relatins fr a dusty gas atmshere anand.rajkumar@rediffmail.cm 5/6 Physics Deartment, University f Allahabad, Allahabad 00 Fig.0 Variatins f /, /, a u /, a a / and i R s with a U fr varius values f k and G Acknwledgement: I acknwledge the encuragement f my wife, Nidhi during rearatin f the aer. Aendix Shck jum relatins fr ideal gas (Anand 000) a T T a u ln ln R s where a U and a

26 Shck jum relatins fr a dusty gas atmshere Jum relatins fr weak shcks in ideal gas 4 4,, 4a u and U a Jum relatins fr strng shcks in ideal gas U, u U and References, T T a U 8 T T, U, a Anand, R. K.: Effects f vertaking disturbances n the ragatin f shck waves thrugh unifrm and nn-unifrm media. Ph.D. Thesis, Dr. B. R. Ambedkar University, Agra. Cha. (000) Anand, R. K.: Jum relatins acrss a shck in nn-ideal gas flw. Astrhys Sace Sci 34, (0a) Anand, R. K.: The ragatin f shck waves in a channel f variable crss sectin cntaining a dusty gas. Phys. Scr. 86, 0540 (0b) iura, H., Glass, I. I.: Develment f the flw induced by a istn mving imulsively in a dusty gas. Prc. R. Sc. A 397, (985) elwyn-hughes, E.A.: Physical Chemistry. Pergamn, Lndn (96) Pai, S. I., enn, S., Fan, Z. Q.: Similarity slutin f a strng shck wave ragatin in a mixture f a gas and dust articles. Int. J. Eng. Sci. 8, (980) Pai, S. I.: Tw Phase Flw (Vieweg Tracts in Pure and Alied Physics 3), Cha. V. Vieweg, Braunschweig (977) anand.rajkumar@rediffmail.cm 6/6 Physics Deartment, University f Allahabad, Allahabad 00