Analytical and numerical study of the diffusion of chemically reactive species in an Eyring-Powell fluid over an oscillatory stretching surface

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1 Bulgarian Cheical Counications Volue 49 Nuber (pp. 7 Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an oscillatory stretching surace S.U. Khan * N. Ali T. Hayat 4 Departent o Matheatics COMSATS Institute o Inoration Technology Sahiwal 57 Pakistan Departent o Matheatics and Statistics International Islaic University Islaabad 44 Pakistan Departent o Matheatics Quaid-i-Aza University 45 Islaabad 44 Pakistan 4 Nonlinear Analysis and Applied Matheatics (NAAM Research Group Departent o Matheatics Faculty o Science King Abdulaziz University Jeddah 598 Saudi Arabia Received October 5; Revised July 4 6 This study deals with the unsteady low o an Eyring-Powell luid induced by an oscillatory stretching surace in presence o cheical reaction. The elastic sheet is stretched periodically back and orth in its own plane. The equations governing the low are derived eploying undaental law o ass oentu and diusion. The independent variables in the governing equations are reduced by using diensionless variables which are solved by using two dierent techniques naely hootopy analysis ethod and an iplicit inite dierence schee. Solutions obtained by both ethods are copared and ound in excellent agreeent. The physical variables such as longitudinal velocity coponent and ass concentration are exained in detail or various values o the paraeters o interest. Keywords: Eyring-Powell luid cheical eects oscillatory stretching sheet Hootopy analysis ethod inite dierence schee. INTRODUCTION The study o convective low under the inluence o agnetic ield and cheical reaction has practical applications in any areas o science and engineering. This phenoenon plays a vital role in cheical industry petroleu industry cooling o nuclear reactors packed-bed catalytic reactors etc. In view o all these applications any researchers studied the eects o cheical reaction on the low o dierent luids. The specialized literature on this topic is discussed in the ollowing paragraphs. Chabre and Young [] discussed the diusion o a cheically reactive species in a lainar boundary layer low over a lat plate. Andersson et al. [] studied the lainar boundary layer low induced by a stretching sheet in the presence o cheical reaction eects. Takhar et al. [] discussed the diusion o cheically reactive species in a second-order luid over a stretching sheet. Akyildiz et al. [4] studied the diusion o cheically reactive species in a second-grade luid over a porous stretching surace. Hayat and Abbas [5] used hootopy analysis to analyze the eects o cheical reaction in a Maxwell luid. In another paper Hayat et al. [6] exained the eects o ass transer in a unsteady low o Maxwell luid over a stretching sheet. The eect o cheical reaction and variable viscosity with heat and ass transer or a Hieenz low through a Darcian porous * To who all correspondence should be sent: E-ail: sk_iiu@yahoo.co ediu was investigated by Seddeek et al. [7]. Aziz [8] used a nuerical technique to discuss the eects o cheical reaction and heat ass transer in a viscous luid. Ferdows and Qase [9] investigated the eects o the order o cheical reaction on a boundary layer low with heat ass transer over a linearly stretching surace. Krishnendu [] discussed the eects o ass transer in presence o cheical reaction over a porous lat plate. Mukhopadhyay and Bhattacharyya [] used a shooting ethod to analyze the irst-order constructive/destructive cheical reaction in a low o Maxwell luid over a stretching sheet. The cheically reactive hydroagnetic low o a second-grade luid in a sei-porous channel was discussed by Abbas et al. []. Apart o these soe recent attepts regarding lows o dierent luids in presence o cheically reactive species can be ound in res. [-7]. Motivated by the studies entioned above the ai o this paper is to analyze the unsteady low and ass transer o cheically reactive species. The rheological behavior o the luid is captured by the constitutive equation o the Eyring-Powell odel. This odel has already been used by several authors to discuss non-newtonian lows [8-]. Unlike typical studies the stretching sheet is assued to be oscillatory. The idea o low over an oscillatory stretching sheet was introduced by Wang []. The work o Wang [] was extended by ew authors including Siddapa et al. [] Abbas et al. [4 5] Zheng et al. [6] and Ali et al. [7]. 7 Bulgarian Acadey o Sciences Union o Cheists in Bulgaria

2 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an The proposed study extends the analysis o Wang [] by considering an Eyring-Powell luid odel in the presence o cheically reactive species. The solution o the governing proble is obtained by a hootopy analysis ethod (HAM and a inite dierence schee. A coparison o both solutions is ade. Based on the nuerical solution a paraetric study is carried out to quantiy the eects o various eerging paraeters on the low and concentration characteristics inside the boundary layer. FLOW ANALYSIS Let us consider an unsteady two-diensional and agnetohydrodynaic (MHD low o an incopressible Eyring-Powell luid past over an oscillatory stretching sheet coinciding with plane y = (see Fig.. Fig.. Geoetry o the proble. The elastic sheet is periodically stretched back and orth with a velocity uw = bx sinωt ( x is the coordinate along the sheet b is the axiu stretching rate and ω represents the requency. A agnetic ield o agnitude B is applied in the direction perpendicular to the sheet. Let c w denotes the concentration at the surace while the concentration ar away ro the surace is c. The continuity oentu and concentration equations or an Eyring-Powell luid can be expressed as [] u v + = x ( u u u u + u + v = ν + t x ρβ C u u σ B u ρβ C ρ ( c c c + u + v = t x c D k ( cc ( where u and v are velocity coponents along x and y directions respectively ν represents the kineatic viscosity ρ is the density β and C denote the aterial paraeters o the Eyring- Powell odel c is the concentration ield D is the concentration expansion coeicient k is the cheical reaction rate. Eqs. (-( are subjected to the conditions u = uω = bx sin ωt v = c = cw at y = t > (4 u c c at y (5 Let us introduce appropriate variables [ 4] y = b y τ = tω ν (6 u = bx y ( y τ v = νb ( y τ c c ( y =. cw c (7 Utlizing Eqs. (6 and (7 Eq.( is identically satisied and Eqs. ( and ( becoe ( + K yyy S yτ y + yy M y λk yy yyy = (8 yy + Sc( y Sτ Scβ = (9 with boundary conditions y ( τ = sin τ ( τ = ( τ = ( y ( τ = ( τ =. ( In the above equations K = / µβ C and λ = xb/νc are diensionless aterial luid paraeters S ω / b is the ratio o the oscillation requency o the sheet to its stretching rate M = σb is the Hartann nuber Sc = υ / D is the Schidt nuber and β = k/ b denotes the cheical reaction paraeter. According to Javed et al. [8] Eq. ( is subject to the constraint λ K <<.

3 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an The skin-riction coeicient C w C is deined as τ = w ρu ( where τ w denotes the shear stress at the wall. In view o (6 and (7 Eq. ( takes the ollowing or [9] K C = + K ( / Rex yy β yy y= ( where Re x = uwx/ ν is the local Reynold nuber. HOMOTOPY ANALYSIS METHOD Hootopy analysis ethod is one o the powerull analytic approaches to solve nonlinear partial and ordinary dierential equations. This ethod was proposed by Liao [8] and then used by any authors or solution o dierent nonlinear probles [9-4]. Now we briely describe the application o this ethod to the boundary value proble developed in the previous section. The boundary conditions lead to the ollowing initial guesses or ( y τ and ( y ( ( τ = τ ( ( y = exp ( y. y sin exp y (4 Introducing linear operators ( = ( = (5 satisying C+ Cexp( y + Cexp ( y = (6 ( ( C4exp y + C5exp y = (7 where C i ( i =...5 are constants. The zero th -order deoration probles deined ( p ( y τ; p ( y τ = p N ( y τ; p (8 ( p ( y τ; p ( y τ = p N ( y ; p τ ( y τ; p (9 ( τ ( τ y ; p ( τ ; p = = y ; p sin τ =. y ( p ( p y= ( τ; = τ; = ( where p [] is an ebedding paraeter. The associated nonlinear operators N and N are ( y τ; p ( y τ; p N ( y τ ; p ( K S = + y τ ( y τ; p ( y τ; p ( y τ; p ( y τ; p ( y τ; p M + ( y τ ; p ( λk ( ( y τ; p N y τ; p ( y τ; p = + ( ( ; y τ p ( y τ; p Sc ( y τ; p S βsc τ The zero th -order deoration probles deined above have the ollowing solutions corresponding to p = and p = ( y τ; = ( y τ ( y τ; = ( y τ (4 ( y τ; = ( y τ ( y τ; = ( y τ. (5 Using Taylor's series expansion we can write ( τ = ( τ + ( τ y ; p y y p ( y τ = ( τ y ; p =! p p= ( = ( + ( y τ; p y τ y τ p ( y τ = ( y τ p ; =.! p p= (6 (7 The convergence o the above series solution depends upon and. We assue that and are selected so that Eqs. (5 to (6 converges at p =. Thereore y τ = y τ + y τ (8 ( ( ( = ( y ( y ( y τ = τ + τ (9 =

4 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an The th -order o the deoration proble is ( y τ χ ( y τ = hr ( y τ ( ( ( = ( ( τ ( τ ( τ = = = y τ χ y τ hr y τ ( ( ( ( τ = τ = ( R y K S M y τ ( τ = ( + + k k k k k k = k k` ` λk l= ( ( (4 R y τ = S Sc βsc + τ (5 k Sc k. k = χ = (6 >. The general solution at th -order can be expressed as ( y τ = ( y τ + C+ Cexp( y + Cexp ( y (7 y τ = y τ + C exp y + C exp y. (8 ( ( ( ( 4 5 where ( y τ and ( y τ indicate the particular solutions. The constants C i (i = 5 using conditions ( and ( get values ( τ C = C5 = C = (9 C =C ( τ C = ( τ. 4 DIRECT NUMERICAL SOLUTION OF THE PROBLEM The syste o nonlinear partial dierential equations (8 (9 with the boundary conditions ( and ( are solved nuerically using a inite dierence schee with Fortran sotware. We use coordinate transoration η = /( y + to transor the sei-ininite physical doain y [ to inite calculation doain η [] i.e.: y = =η η 4 = η + η =η y τ τ =η 6η 6 η. Using the above transorations in Eqs. (8 and (9 S = η + 6 ( + K η η + τ η 4 6 ( + K η η + ( + K η η η M λkη λkη 6λKη 6 4λKη η 9 8 λkη 4λKη 4 4λKη 9 (4 4λKη 4λKη 4 η + η Sc η S + τ (4 βsc = η = = at η = (4 = η = sin τ = at η = (4 In the second step we discretize Eqs. (8 and (9 or L equally spaced points η = ( η η η... η + [ ] with a step size o L η = / ( L + at tie instants τ = ( τ τ... i where τ = τ + i τ. The nuerical values o n n... n n n... n are ( L and ( L sought at these points at each tie level provided that the boundary conditions at η = η = and η = η L + = are known. The initial conditions or velocity ield are: ( ητ = = and ( ητ = =. (44 We construct the sei-iplicit tie dierence schee or and as ollows:

5 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an ( K ( n ( n S = η + τ 6 η ( n ( n 4 ( K η ( n ( n ( + K η + ( n ( n η 6+ η η M + + ( n ( n λkη 4λKη ( n ( n ( n λkη 6λKη 6 ( n ( n ( n ( n ( n 9 8 λkη 4λKη 4 ( n ( n ( n 4λKη 4λKη ( n ( n ( n ( n 9 4 λkη (45 The advantage o the schee is that only linear equations or each new tie step ( n + are to be solved. Two systes o linear equations or i ( n and + i at the tie step ( n + can be solved by using Guassian eliination. CONVERGENCE OF HAM SOLUTION AND ITS COMPARISON WITH A NUMERICAL SOLUTION The convergence o HAM solution depends upon the suitable choice o auxiliary paraeters and. The h-curves are plotted in Fig. (a-b to show the convergence region o the HAM solution or a particular set o involved paraeters. Fro these igures it is clear that or this choice o paraeter value a convergent solution can be obtained when.4 < and <. To check the accuracy o HAM solution the nuerical values o ''( τ at dierent order o approxiations are shown in Table. The convergent values o '' ( τ can be obtained by increasing order o approxiation. Figs. and 4 show the coparison o HAM solution with a nuerical solution at two dierent orders o approxiation. An excellent agreeent between HAM solution and nuerical solution can be achieved at 5 th order o approxiation. ( S Sc Sc ( n ( n ( 4 η η = + τ η Scβ. (46 Table. The convergence o the HAM solution o ''( τ or dierent order o approxiation with S =. M =. K =.5 λ =.4 and τ = τ =.5π and τ =.5 π respectively. Order o approxiation τ = τ =.5π τ =.5π

6 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an RESULTS AND DISCUSSION In this section graphical results obtained via a inite dierence ethod are displayed in order to exaine the eects o involved paraeter on velocity coponent ' and concentration ield. Fig. 5(a-c deonstrates the eects o the relative aplitude o requency to the stretching rate S luid paraeters λ and Hartann nuber M on the evolution series o the velocity coponent ' at a ixed distance y =.5 ro the sheet respectively. Fig. 5(a shows that the aplitude o the low otion at this location decreases by increasing S. However this decrease is arginal and it is anticipated that such a trend is prevalent even or larger values o S. Fig. 5(b depicts that the aplitude o the low increases by increasing luid paraeter λ. This increase in the aplitude o low otion is attributable to the increased eective viscosity induced by larger values o λ. The inluence o Hartann nuber M on tie evolution o the velocity coponent ' is shown in Fig. 5(c. The igure reveals that the aplitude o the velocity coponent ' decreases by increasing Hartann nuber. The velocity proile ' or various values o S at our dierent distances τ = 8.5 π τ = 9 π τ = 9.5π and τ = π are shown in Fig. 6(a-d. Fig. 6(a shows that at tie instant τ = 8.5π the velocity decreases by increasing S. The back low occurs near the surace where ' gets negative. The eects o S at tie interval τ = 9π are depicted in Fig. 6(b. The velocity ' at this tie instant oscillates near the surace and inally approaches zero. The aplitude o oscillation is ound to increase with an increase in S. Fig. 6(c elucidates that at tie instant τ = 9.5 π the velocity ' gets the value - at the wall and becoes zero ar away ro the surace without peroring oscillation. A decrease in the aplitude o velocity is observed at this tie instant. Fig. 6(d reveals that at tie instant τ = π the velocity ' is zero both at the surace and ar away ro the surace. It is also observed that at this tie the aplitude o back low increases by increasing S. Fig. 7(a shows the variation o ' or dierent values o λ at τ = 8.5 π. Here it is observed that the velocity approaches ro at the surace to zero ar away ro the surace. The occurrence o back low near the surace is also observed at this tie instant that is ound to increase by increasing λ. Fig. 7(b indicates that at this tie instant τ = 9 π the velocity oscillates near the surace beore approaching zero ar away ro the surace. Moreover the aplitude o the back low is ound to increase with λ. The variation o ' with λ at tie instant τ = 9.5π is shown in Fig. 7(c. Here the agnitude o velocity decreases by increasing λ. Occurrence o back low at τ = π with its strengthening or larger values o λ is observed in Fig. 7(d. Fig. 8(a-d illustrates the eects o Sc β λ and S on the concentration proile. Fig. 8(a is plotted to observe the eects o Sith nuber Sc on the concentration proile. Fro this igure we observe that ass concentration decreases or large values o Sc. Moreover it is also seen ro this igure that the concentration boundary layer thickness decreases by increasing Schidt nuber Sc. These eects ay be attributable to the increase in the rate o solute transer ro the surace by increasing the Schidt nuber. The eect o cheical reaction paraeter β on concentration is shown in Fig. 8(b. Here we again observe that the ass concentration decreases by increasing β. Fig. 8(c illustrates the eects o luid paraeter λ on the concentration proile. Fro this igure it is clear that the concentration increases by increasing luid paraeter λ. Fig. 8(d shows opposite eects i.e. ass concentration decreases by increasing the relative aplitude o requency to the stretching rate paraeter S. Fig. 9(a-d describes the eects o relative aplitude o requency to the stretching rate S Hartann nuber M and luid paraeters K and λ on the tie-series o shear stress at the wall or the irst ive periods τ [ π ]. Fig. 9(a shows the inluence o the relative aplitude o requency to the stretching rate S on the skin-riction coeicient / Re x C by keeping other paraeters ixed. It is clear ro this igure that the aplitude o oscillation o the skin-riction coeicient increases by increasing the relative aplitude o requency to the stretching rate S. Fro Figure 9(b it is clear that the skin riction coeicient oscillates with tie and the aplitude o oscillation increases or large values o Hartann M. The eects o luid paraeter λ and K are illustrated in Figs. 9(c and (d respectively. These igures show an opposite trend i.e. the skin riction coeicient / Re x C decreases onotonically by increasing these luid paraeters. 5

7 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an CONCLUSIONS In the present study we have investigated the ass transer in a unsteady low o an Eyring- Powell luid odel over an oscillatory stretching sheet. The non-siilar solution o the governing nonlinear partial dierential equations is obtained analytically by a hootopy analysis ethod and nuerically by a inite dierence schee. The low and ass transer characteristics are explained graphically or the several values o the involved paraeters. The ain indings can be suarized as: The convergence o the HAM solution is largely dependent on the choice o auxiliary paraeters and the order o approxiation. The aplitude o the low velocity at a ixed distance ro the sheet decreases with increasing ratio o oscillating requency to stretching rate S and Hartann nuber M while a converse Eyring-Powell luid paraeter λ trend is coputed with increasing λ. The concentration proile increases or large values o Eyring-Powell luid paraeter λ while it decreases or large values o Schidt nuber Sc ratio o oscillating requency to stretching rate S and cheical reaction paraeter β. The concentration boundary layer thickness decreases with increasing Schidt nuber Sc ratio o oscillating requency to stretching rate S and cheical reaction paraeter β. In contrast an increasing Eyring-Powell luid paraeter λ increases the concentration boundary layer thickness. The aplitude o the skin riction coeicient increases with increasing the Hartann nuber M and the ratio o oscillation requency o the sheet to its stretching rate S while it is suppressed with increasing the Eyring-Powell luid paraeters λ and K. In the liiting case when λ K our results reduce to the corresponding results o Wang []. Fig.. The curves at 6 th order o approxiation: (a or velocity; (b or concentration proile. Fig.. Coparison o '( y τ obtained ro HAM solution (solid lines and the nuerical solution (open circles. 6

8 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an Fig. 4. Coparison o concentration ield ( y solution (open circles. τ obtained ro HAM solution (solid lines and the nuerical Fig. 5. Velocity proile as a unction o tie: (a eects o S ; (b eects o λ ; (c eects o M. 7

9 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an Fig. 6. Velocity ield or dierent values o S. Fig. 7. Velocity ield or dierent values o λ. 8

10 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an Fig. 8. Concentration ield (a eects o ; Sc (b eects o β ; (c eects o λ ; (d eects o S. Fig. 9. The skin riction coeicient o λ ; (d eects o K. / Re x C as a unction o tie: (a eects o S ; (b eects o M ; (c eects 9

11 S.U. Khan et al.: Analytical and nuerical study o the diusion o cheically reactive species in an Eyring-Powell luid over an Acknowledegents: We are grateul to the reviewer or his/her coents. The irst author is grateul to the Higher Education Coission o Pakistan or inancial assistance. REFERENCES. P. L. Chabre J. D. Young J Phys. Fluids 48 (958.. H. I. Andersson O. R. Hansen B. Holedal Int. J. Heat Mass Tran (994.. H.S. Takhar A.J. Chakha G. Nath Int. J. Eng. Sci. 8 (. 4. F.T. Akyildiz H. Bellout K. Vajravelu J. Math. Anal. Appl. (6. 5. T. Hayat and Z. Abbas Z. angew. Math. Phys (8. 6. T. Hayat M. Awais M. Sajid Int J. Mod. Phys. B 5( 86 (. 7. M. A. Seddeek A. A. Darwish M. S. Abdeleguid Co. Nonlinear Sci. Nu. Siulation (7. 8. M.A. El-Aziz Che. Eng. Coun (. 9. M. Ferdows M. Qase A. J. Fluid Dynaics (6 89 (.. K. Bhattacharyya Che. Eng. Bulletin. 5 6 (.. S. Mukhopadhyay K. Bhattacharyya J. Egyp. Math. Society 9 (.. Z. Abbas B. Ahad S. Ali J. Appl. Mech. Tech. Physics 56(5 878 (5.. T. Hayat M. Mustaa S. Asghar Nonlinear Analysis: Real World Applications (4 86 (. 4. Z. Ziabakhsh G. Doairry H. Bararnia H. Babazadeh J. Taiwan Inst. Che. Engineers 4 (. 5. S. A. Kechil J. Porous Media ( 5 (9. 6. T. Hayat M. Awais Abreen Sadar Awati A. Hendi Nonlinear Analysis: Modelling and Control 7 ( 47 (. 7. S. A. Shehzad T. Hayat M. Qasi S. Asghar Braz. J. Che. Eng. 87 (. 8. V. Sirohi M. G. Tiol N.L. Kalathia Reg. J. Energy Heat Mass Transer 6 9 ( T. Javed N. AliZ. Abbas M. Sajid Che. Eng. Coun. 7 (.. T. Hayat Z. Iqbal M. Qasi S. Obaidat Int. J. Heat Mass Tran (.. T. Hayat S. Asad M. Mustaa A. Alsaedi Plos One 9(7 e4 (4... C. Y. Wang Acta Mech (988.. B. Siddappa S. Abel V. Hongunti. ll Nuovo Ciento D 7 5 ( Z. Abbas Y. Wang T. Hayat M. Oberlack Int. J. Nonlinear Mech. 478 (8. 5. Z. Abbas Y. Wang T. Hayat M. Oberlack Int. J. Nuer. Meth. Fl (9. 6. L.C. Zheng X.Jin X. X. Zhang and J. H. Zhang Acta Mech. Sinica 9(5 667 (. 7. N. Ali S. U. Khan Z. Abbas Z. Naturorsch. 7(7a 567 (5. 8. S. J. Liao J. Ship Res. 6 ( S. J. Liao Coun. Non-linear Sci. Nuer. Siulation 6 (6.. S. J. Liao Int. J. Non-Linear Mechanics 4(4 759 (999.. S. J. Liao J. Fluid Mech (.. M. Turkyilazoglu Int. J. Theral Sciences 5(5 8 (.. M. Turkyilazoglu Math. Cop. Modelling 5 99 (. 4. S. Abbasbandy A. Shirzadi Stud. Nonlinear Sci. (4 7 (. АНАЛИТИЧНО И ЧИСЛЕНО ИЗСЛЕДВАНЕ НА ДИФУЗИЯТА НА ХИМИЧЕСКИ АКТИВНИ ВЕЩЕСТВА ВЪВ ФЛУИД НА EYRING-POWELL НАД НАДЛЪЖНО ОСЦИЛИРАЩА ПОВЪРХНОСТ С.У. Хан * Н. Али Т. Хаят 4 Департамент по математика Институт по информационни технологии COMSATS Сахиуал Пакистан Департамент по математика и статистика Международен ислямски университет Исламабад Пакистан Департамент по математика Университет Куаид-и-Азам Исламабад Пакистан 4 Изследователска група по нелинеен анализ и приложна математика Департамент по математика Научен факултет Университет Крал Абдулазиз Джеда Саудитска Арабия Постъпила на октомври 5 г.; коригирана на 4 юли 6 г. (Резюме Това изследване засяга нестационарния поток на флуид на Eyring-Powell породен от надлъжно осцилираща повърхност в присъствие на химична реакция. Еластичният лист се разтяга периодично напред-назад в собствената си равнина. Изведени са уравненията на движението прилагайки основните закони за запазване на масата на количеството движение и на дифузията. Намалени са независимите променливи чрез въвеждането на безизмерни променливи а уравненията се решават по два метода: хомотопен анализ и неявна диференчна схема. Сравнението на резултатите по двата метода показва отлично съгласие. Физичните променливи (надлъжната скорост и масовата концентрация са подробно изследвани за различни параметри.