A NON-LINEAR DIFFUSION EQUATION DESCRIBING THE SPREAD OF SMOKE

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1 Fizikos ir matematikos fakulteto Seminaro darbai, iauliu universitetas, 6, 2003, 758 A NON-LINEAR DIFFUSION EQUATION DESCRIBING THE SPREAD OF SMOKE Arvydas Juozapas JANAVIƒIUS, Gintaras L UšA iauliai University, P. Vi²inkio 9, 5400 iauliai, Lithuania; AYanavi@takas.lt iauliai University, Vytauto 84, 5400 iauliai, Lithuania; gintarasl@takas.lt Abstract. A non-linear diusion equation is derived by taking into account the local variations in the solvent density, within a mechanism of diusion driven by particle collisions. Analytical solutions in the case of sphericalsymmetric non-linear diusion equation for the spread of smoke in gas are obtained and discussed. In this case the solutions of the non-linear diusion equation are similar to solutions of the Bernoulli equation. We note that the obtained solutions can be used to describe the shape of smoke clouds. Key words and phrases: diusion in gas, non-linear diusion. Physics and Astronomy Subject Classication: 5.20+d, h.. Introduction We suppose that the smoke spreading process is similar to other diusion processes and, in the non-linear case, may be described by the non-linear diusion equation [][3]. In the case of a thin layer of uid (two-dimensional case), assuming that the frequency of the jumps depends on the particle coordinates, the time variation of the uid concentration n(x, y, t) is presented, see [][3], by = I x + I y, t I x = ν(x + L, y)n(x + L, y) + ν(x L, y)n(x L, y) 2ν n, I y = ν(x, y + L)n(x, y + L) + ν(x, y L)n(x, y L) 2ν n. Here L is the average free path of the particles in gases or the length of the jump from one equilibrium position to another in uids and solids, ν is the frequency of these jumps or the molecule collisions. Expanding the above

2 76 A non-linear diusion equation functions in power series, including three main terms, we obtain the following non-linear equation t = L2 ν n + 2L 2 (gradν)(gradn). () The average frequency ν(x, y, z, t) of collisions in gas is expressed [3] by ν(x, y, z, t) = σ N(x, y, z, t) u 2. (2) Here σ is the collision cross-section of the diusing particle, N is the density of the solvent particles, u 2 is the relative velocity of impurity or the diusing atoms and solvent particles. Assuming that the coordinate z is xed, we may use the last formula in the two-dimensional case. As usual, assume (n/n), but we must evaluate the fact that the density during the diusion process becomes equal to N +n and N n for a non-equilibrium and equilibrium process, respectively. It is important to evaluate the second term density on the right hand side of (). In our case only the collisions between the impurity atoms or molecules and vacancies in the solid states or uids are important. Then for the area of diusion front we have ν f (x, y, z, t) = 2 σ n(x, y, z, t) u 2. (3) We suppose that this quickly decreasing impurity concentration area in the front of diusing particles lls a half of the vacancies. Near the source of impurities all vacancies are lled with the atoms, and then we have ν(x, y, z, t) = σ n(x, y, z, t) u 2. (4) After substituting (3) and (4) in equation (), we obtain t = L2 σu 2(n n + (grad n) 2 ). (5) If we describe the diusion coecient D by D = 2L 2 σun, then the non-linear diusion equation becomes similar to equation (4.) from [2], and the diusion coecient is proportional to the impurity concentration in [2].

3 A. J. Janavi ius, G. L uºa 77 In the case of the diusion of the gases are expedient to change N to N + n in (2). After substituting the obtained expression in equation (), we have t = 2L 2 σnu n + 2L 2 σu(grad n) 2. Earlier we proposed that n N. In the case of constant N and if the diusion coecient is given by D = 2L 2 σnu, we have the diusion equation in gases t = D n + D N (grad n)2. Without any doubt, this equation is also valid in one- or three-dimensional cases. In the three-dimensional spherical case the last equation is of the form t = D ( d 2 n dr r dn dr ) + D N ( dn dr ) 2. (6) 2. The analytical solution of the non-linear diusion equation Using the similarity variable ξ = r Dt, we transform equation (6) into the following form d 2 ( n dξ ξ + 2 ) dn ξ dξ + ( ) dn 2 = 0. (7) N dξ Taking f =, (8) dn dξ we obtain that our equation is similar to the Bernoulli equation [4] ( df dξ 2 ξ + 2 ) f = 0. (9) ξ N Then, taking into account [4] and (8), we get the following solution df dξ = ξ 2 N exp [ 4 ξ2] ξ 2 exp [ 4 ξ2] dξ + C.

4 78 A non-linear diusion equation Here C is a constant established from the evaluation of the impurities gradient describing the moving into surroundings of the impurity particles quantity M(R, t) per time t through the spherical area 4πR 2. Hence, M = Dt4πR 2 dn dr r=r = 4π(Dt) 3/2 N exp [ 4 ξ2] ξ 2 exp [ 4 ξ2]. (0) dξ + CN r=r For the impurity particles dissipating in the diusion way, the quantity of the particles thrown out to surroundings is limited by the diusion coecient D and the diusion process M, because the maximum penetrating length [6] of diusing particles is.66 Dt. Since the real source of the particles has the nite dimensions r = R, the singular integral in the denominator of (0) is nite. Then, for r R, we have t R 2 /(.66 2 D). If the speed of diusion is large and exceeds the right side of equation (0), i.e., the particles of impurity spread into surroundings by the diusion, then a convection ow and dynamic waves arise. From (0) we may nd the integrating constant [ C = 4π(Dt) 3/2 exp ] 4 r=r ξ2 / M ξ 2 exp [ 4 ] N ξ2 dξ. r=r Integrating this expression once more, from the general solution of the equation, we can nd the radial density of smoke n(ξ) = ξ 2 exp [ 4 ξ2] dξ ξ 2 exp [ 4 ξ2] dξ + C + C 0. () N Demanding that the quantity of spreading particles in surroundings must be nite and setting the integrating constant C 0 = 0, we obtain M t = 4π R nr 2 dr. This equality can be used for computing the constant C from the nite quantity of the diusing particles which was introduced in the surroundings by the time interval t. Using a more convenient variable we may rewrite () as n(z) = N z 2 z = ξ 2, exp [ z 2] dz exp [ z z 2 ] dz + CN. 2

5 A. J. Janavi ius, G. L uºa 79 After integration we nd that n(z) = N ln z 2 exp [ z 2] dz + CN. (2) In view of (5), we can suppose that ξ ξ 0, ξ 0 =.66. In this case z <, and the exponential function in (2) may be expanded into the power series with nite number of terms exp( z 2 ) = z z4 6 z z8.... Since formula () cannot be integrated exactly, it is purposeful to nd a more handy solution. For this aim we can obtain an asymptotic solution f 0 of the equation (9) as ξ 0, namely, f 0 = C ξ 2. We look for the general solution in the form f 0 = C(ξ) ξ 2. Substituting the last expression into (9), we get the equation ξ 2 dc(ξ) dξ 2 ξ3 C(ξ) N = 0 which can be solved like the Bernoulli equation [4] by ξ 2 dn [ dξ = N exp [ 8 ξ4] exp 8 ξ4] dξ + CN. (3) From the last expression and the measured quantity of the inserted particles M at the origin of the coordinates through time t, we obtain M(0, t) = lim r 0 Dt4πr 2 dn C = 4π(Dt) 3 2 M(0, t). dr = 4π(Dt) C Integrating (3), we nd that the distribution of the density n of the inserted particles of impurities through time t is given by ξ 2 exp [ 8 n = N ξ4] dξ [. (4) exp 8 ξ4] dξ 4π(Dt) 2 3 N M(0,t) 3 2,

6 80 A non-linear diusion equation According to the Fick's law and (3), we obtain for the quantity of impurities M(r, t) which spread per time t through the spherical area 4πr 2 by the diusion process M(r, t) = 4π(Dt) 3 2 M(0, t)n exp [ 8 ξ4] ξ M(0, t) exp [, (5) 8 ] ξ4 dξ 4π(Dt) 3 2 N where ξ = r Dt. We obtained [ that the concentration of smoke in the surrounding is decreasing like exp r 4 ], depends on the diusion coecient, and on the introduced 8D 2 t 2 amount of impurities M(0, t) in the time t from the pollution source. We have the positive impurities ux (5) only in the case M(0, t) exp [ 8 ] ξ4 dξ 4π(Dt) 3 2 N < 0. (6) Hence, after replacement in (5), as in [5], for ξ, exp [ 8 ] ξ ξ4 dξ = 0 [ exp ] 8 ξ4 dξ, (7) the following conditions for physical spreading of impurities by diusion process can be obtained M(0, t) < 2.625π(Dt) 3 2 N. (8) Using formulae for dierentiation of integrals, see [5], and substituting (8) into (4), we can obtain a handy expression for the radial density of diusing impurities n(r, t) = N ξ ξ 2 ξ2 2 exp [ 8 2] ξ4 dξ2 exp [ ] 8 ξ 4 dξ 4π(Dt) 3 2 N/M(0, t). (9)

7 A. J. Janavi ius, G. L uºa 8 3. Conclusions The obtained solutions (5), (9) can be appropriate to describe the clouds or wreaths of smoke. The obtained solutions of non-linear diusion equation (7) dene the impurities ux and the radial density in the spherical case. The physical conditions for spreading of impurities by diusion processes is dened by (7). References [] B.-F. Apostol, On a non-linear diusion equation describing clouds and wreaths of smoke, Physics Letters A 235, (997). [2] A. J. Janavi ius, Method for solving the nonlinear diusion equation, Physics Letters A 224, 5962 (997). [3] A. J. Janavi ius, Random processes and transfer phenomenon, iauliai university (2002) (in Lithuanian). [4] P. F. Fil akov, Handbook of high mathematics, Kiev, Naukova Dumka (973) (in Russian). [5] G. Korn and T. Korn, Mathematical handbook for scientists and engineers, McGraw-Hill Book Company, New York (96). [6] D. Shaw, Atomic diusion in semiconductors, London, New York, Plenum Press (973). Difuzijos lygtis, apra²anti d umu sklidim A. J. Janavi ius, G. L uºa Kai difuzija vyksta plok²tumoje ir daleliu ²uoliu tankis priklauso nuo koordina iu, gaunama netiesine d umu daleliu difuzijos lygtis. J apibendrinus trima iam atvejui su sferine simetrija, surasti analiziniai sprendiniai. Bendrieji netiesines difuzijos lygties sprendiniai sutampa su Bernulio lygties sprendiniais. Gautieji sprendiniai gali b uti panaudoti d umu kamuoliams ir debesims apra²yti. Rankra²tis gautas