The Mathematics of Thermal Diffusion Shocks

Transcription

1 The Mathematics of Thermal Diffusion Shocks Vitalyi Gusev,WalterCraig 2, Roberto LiVoti 3, Sorasak Danworahong 4, and Gerald J. Diebold 5 Université du Maine, av. Messiaen, 7285 LeMans, Cedex 9 France, 2 Deartment of Mathematics McMaster University, Hamilton, Ontario, Canada L8S 4K, 3 Universita di Roma La Saienza, Diartimento di Energetica, Via Scara, 4 6 Roma,Italy 4 School of Science, Walailak University, 222 Thaiburi, Thasala District, Nakorn Si Thammarat,86, Thailand, 5 Deartment Chemistry, Brown University, Providence, RI, USA, 292 Thermal diffusion, also known as the Ludwig-Soret effect, refers to the searation of mixtures in a temerature gradient. For a binary mixture the time deendence of the change in concentration of each secies is governed by a nonlinear artial differential equation in sace and time. Here, an exact solution of the Ludwig-Soret equation without mass diffusion for a sinusoidal temerature field is given. The solution shows that counter-roagating shock waves are roduced which slow and eventually come to a halt. Exressions are found for the shock time for two limiting values of the starting density fraction. The effects of diffusion are shown on the develoment of the concentration rofile in time and sace are found by numerical integration of the artial integration of he nonlinear differential equation. I. Introduction The searation of liquid mixtures in a thermal gradient was discovered by Ludwig[] in 856, and was described theoretically somewhat later by Soret[2]. The searation of mixtures by a thermal gradient takes lace not only in liquids, but also in gases and solids as is described by nonequilibrium thermodynamics[3 6]. Recently, a new method of roducing the Ludwig-Soret effect has been introduced where crossed laser beams roduce thermal gradients[7 ]. The method relies on interference in the electric fields of two hase coherent light beams to write an otical grating in a weakly absorbing liquid mixture. At the nodes of the electric field, there is no absortion, while at the antinodes heat is deosited resulting in a temerature field that is sinusoidal in sace[]. A robe laser is directed at the grating at the Bragg angle, and roduces diffracted beams in resonse to either aeriodic change in absortion or by a eriodic change in index of refraction that forms in the mixture as a result of thermal diffusion. A recording of the intensity of the diffracted beam gives the time develoment of searation of the comonents of the mixture. Here we discuss solution of the Ludwig-Soret equation in a one-dimensional geometry for a sinusoidal temerature distribution. A derivation of an exact solution of the Ludwig-Soret roblem with negligible mass diffusion is given in Section II. The time develoment of the shock fronts[2] are given in Section III. Section IV gives an exression for the shock velocity and exressions for the shock formation time for two limiting cases of the starting dentsity fraction. The effects of diffusion are shown through numerical solutions to the Ludwig-Soret equation. Derivation of the concentration rofile in sace at longtimesisgiven. II. Solution to the Thermal Diffusion Equation Thermal diffusion in a binary mixture is governed by the couled differential equations [4] T ρc P = (λ T + κ c ) () t c = (c c 2 D T + D c ), (2) t where t is the time, ρ is the solution density, c P is the secific heat caacity, λ is the thermal conductivity, T is the temerature above ambient, κ is a constant, D is the thermal diffusion coefficient, D is the mass diffusion coefficient, and c and c 2 are density fractions of the secies that make u the solution. Since the total mass of the solution is the sum of the masses of the two comonents of the mixture, the densities of the two secies satisfy the relation ρ + ρ 2 = ρ, whereρ and ρ 2 are the densities of the two comonents and ρ is the density of the mixture. If the exression relating the densities of the two secies is divided by ρ, the result is an exression for the density fractions c and c 2 of each of the comonents which obey the relation c + c 2 =. The first term on the right hand side of Eq. is the Fourier law for heat conduction; the second describes the Dufour effect, which refers to roduction of a temerature change as a result of imosition of a concentration gradient. The first term on the right hand side of Eq. 2 describes thermal diffusion, while the seconddescribesmassdiffusion. Consider a temerature field of the form T = T [ + sin(kx)], where 2T is the eak temerature, K is a wavenumber determined by the otical fringe sacing in the grating, and x is the coordinate. If the Dufour effect is taken as negligible [4], then Eq.2 becomes c(z,t) τ = α z {c(z, t)[ c(z,t)] cos z} + 2 c(z,t) z 2, (3) Tyeset by REVTEX

2 2 where c 2 has been written in terms of c, and the subscrit droed from the density fraction c, where the thermal diffusion factor α, isdefined as α = D T /D, and the dimensionless quantities τ and z are defined by τ = K 2 Dt, and z = Kx. If the last term in Eq. 3 describing diffusion is ignored, then the differential equation of motion for c(z,t) can be written c τ = f z, (4) where a flux f(c, z) is defined as f(c, z) = αc( c)cosz and τ is given by τ = K 2 D. Equation 4 is the differential form of a conservation equation that exresses the buildu of c in a volume as a consequence of a flux change in sace. Since for a eriodic temerature field the density fraction must also be eriodic in z, it follows that c(2 π, τ) = c(,τ). Integration of Eq. 4 over one eriod of the temerature field, i.e. from z =to z =2π, gives the integral form of Eq. 4 as 2π c(z, t) dz =2πc, (5) where c is the density fraction at time t =,assumed to be a constant throughout the cell. The integral for the density fraction over z exresses simle mass conservation for the Ludwig-Soret effect. The Eulerian descrition of the rofile c = c(z, τ) by Eq. 3 can be transformed[3] into a Lagrangian descrition yielding the couled air of ordinary differential equations, dz f(c, β) = = α(2c ) cos z (6) dτ c β) = f(c, = αc( c)sinz, (7) dτ β that determine the motion of oints with coordinates z = z(τ,c, z ) and c = c(τ,c, z ) on the zc lane for a oint initially at (c, z ) at time τ =. As discussed in Ref. [3], Eqs. 6 and 7 form a Hamiltonian system of equations as found in classical mechanics, with the flux function in the resent roblem taking on the role of the Hamiltonian function in classical mechanics. Numerical integration of Eqs. 6 and 7 can be used to give the z coordinate and the value of c as a function of time for any oint initially at (z,c ), as shown in Fig.. Since the temerature field has a eriod of 2π, the solutions to Eq. 3 must also have the same eriodicity. As can be seen from Fig., the oints π/2 and the oint 3π/2 reresent cold oints of the grating so that the solutions to Eq. 3 are identical at these oints, and for any oints dislaced from these oints by a multile of 2π. From FIG. : Density fraction c versus dimensionless distance z for τ =,.5, 3.5, and 6., for a starting density fraction c =.3. The curves can be determined by numerical integration of Eqs. 6 and 7 or through solution of Eqs. 2 and 3. The vertical lines indicates the ositions of the shock fronts. The shock velocity is found by determining the line that makes the areas of the darkened regions adjacent to each of the vertical lines equal. Eqs. 6 and 7 the sloe of the curves on a hase ortrait is found to be c( c)sinz = dz (2c ) cos z, which can be written as d [c( c)cosz] =. (8) dz It follows then that the quantity in brackets in Eq. 8 is a constant, taken here to be k, so that c( c)cosz = k. (9) The hase ortrait for the Ludwig-Soret equation, as shown in Fig. 2, gives the trajectories of oints with secified starting values of c and z.by differentiation of Eq. 9 with resect to c, it can be seen that the maximum value of the left hand side occurs when cos z =,an =/2; hence the values of k are restricted to k /4. By solving Eq. 7 and Eq. 9 for sin z and cos z, resectively, squaring and adding the resulting exressions, it is easily shown that k c( c) 2 which can be written + c( c) c2 ( c) 2 k 2 2 µ 2 dτ =, = ±dτ. ()

3 3 FIG. 2: Phase ortrait for the density fraction c versus dimensionless distance along the grating z from Eq. 9. Points on the zc lanemovealongcurvesofconstantk as time rogresses. It is convenient to define the dearture of the density fraction from a value of /2 through the variable c,and to exress the quantity in the denominator of Eq. in terms of two arameters a and b, defined by a = q c = c k b = q 4 k. Owing to the restriction on the magnitude of k, the inequality a b is automatically satisfied. Hence, Eq. then can be integrated from a starting density fraction c at time τ =along a curve of constant k to a density fraction c at time τ giving c 2 c 2 τ (a2 c 2 )(b 2 c 2 ) = ± dτ, () where the integration is taken over a trajectory in hase sace with a constant value of k. The left hand side of Eq. can be exressed as an ellitic integral[4] of the first kind F, so that the solution to the Ludwig-Soret roblem for a sinusoidal temerature field, neglecting diffusion is F arcsin c 2, 4 k 4 k 4 +k q (2) F arcsin c 2, 4 k 4 k = ±τ 4 + k. 4 +k It can be seen that Eq. 2 gives τ exlicitly for any starting oint (z,c ) that arrives at the newoint c; the new coordinate z is determined through c ( c )cosz = c( c)cosz = k. (3) FIG. 3: Density fraction c versus dimensionless distance z from Eq. 6. with α =for τ =,, 2,and 3.. The oints lotted as dots ( ) are those that originally lie in the region <z<π/2 at τ =. The oints lotted as squares ( ) are those that lie in the region π/2 <z< at time τ =. From the oint of view of calculating density fraction rofiles as shown in Fig., it is convenient to secify values of z, c and τ and to solve Eq. 2 for the new values of c through a numerical root search. Alternately, the oints (c, z) can be substituted directly into Eq. 2 throughuseofeq. 3todeterminek. Note that for ositive values of α, theminussignisusedineq.2for oints moving in the hot regions of the grating <z<π, and the lus sign is used for motion in the cold regions π<z<2π, or π <z<. Inversion of Eq. 2 to give c exlicitly as a function of z for any given time τ, does not aear to be ossible; however it is straightforward to determine the rofiles of the density fraction versus coordinate through numerical methods. III. Motion of the Points in Phase Sace Consider the motion of a oint located between and π/2 with a value of c < /2. It can be seen from Fig. that oints in this region move downwards and towards the c axis. For such oints the motion is determined first by Eq. 2 with a negative sign is used, and after assing the c axis at z =, by Eq. 2 with the ositive sign. Any oint in this region starting at (c,z ) reaches its minimum value at z =, which from Eq. 3, is given by = 2 r 4 c ( c )cosz. (4)

4 4 The time required to arrive at is therefore τ min = c min c x2 ( x) 2 [c ( c )cosz ] 2, (5) and the total time τ, from the starting oint to the oint (c, z) is τ = τ min+ c x2 ( x) 2 [c ( c )cosz ] 2. (6) which can be written in terms of ellitic integrals as in Eq. 2. Since the shocks in the region π/2 <z< originate from oints in the region >z<π/2, Eq. 2is exression that determines the time of shock formation. Similar reasoning gives the maximum density fraction c max for any oint starting at (c,z ) located between and π/2 as c max = 2 + r 4 c ( c )cosz, (7) where τ max, thetimeittakestomovefromc to to c max, is given by τ max = τ min + c max x2 ( x) 2 [c ( c )cosz ] 2. (8) The curves in Fig. 3 show how oints originally in the region between and π/2 move downwards to their minimum values at z =, ass to the left of the c axis and then rise uwards as time rogresses. IV Shock Formation A. Determination of the shock velocity It is clear from examination of Fig. 2 that the solutions to Eq. 2 exhibit nonfunctional behavior after a certain time. Similar solutions for different values of c show that nonfunctional behavior is found for any value of c if τ is allowed to become large. When the sloe of any curve becomes unbounded, i.e.. immediately before the density fraction rofile becomes nonfunctional, shocks are formed, and the mathematics must be treated in a different manner. Consider the vertical line to the right of the density fraction maximum at 3π/2 indicating the osition of the shock front on the curve in Fig. for τ =3.5. For the shock front, the density fraction immediately to the left of the front is denoted c l and that to the right of the shock by c r. The area of the curve bisected by the vertical line, can be equated to the roduct of the velocity of the shock dz sh /dτ, the difference in c and the time interval dτ,that is, dz cl sh dτ (c l c r )dτ = dz, c r which, together with Eq. 6, gives dz cl sh dτ (c l c r )dτ = α cos zdτ (2c ). (9) c r After evaluation of the integral in Eq. 9 the shock velocity becomes dz sh = α(c l + c r ) cos z. (2) dτ It can be seen that two shocks, one on either side of the oint z =3π/2 are formed, each of which travels with the same seed towards the cold region of the grating, the different velocities being determined by the change in sign of the cosine function on either side of z =3π/2. The left-going shock comes to a halt when c l =and c r =, that is, where a comlete searation of the mixture has been attained, at which oint the rofile of c in sace is asquarewavecenteredatz =3π/2. The shock velocity can be obtained equally by writingeq. 6asdz sh /dτ = [f(c l,z) f(c r,z)]/(c l c r ), which, after substitution of the exression for the flux function, gives Eq. 2. As noted in Ref. [2], Eq. 2, which exresses the thermal diffusion shock velocity in terms of density fractions on either side of the shock, is an exact analog of the Rankine-Hugoniot relations for one-dimensional fluid shocks: Eq. 2 gives the thermal diffusion shock seed in terms of the "state variables" c l and c r, the Rankine-Hugoniot relations exress the shock seed in terms of ratios of the state variables of the fluid on either side of the shock. The time of formation of the shock can be found from numerical integration of the Hamiltonian system of equations and finding the oint in sace and time where the rofile reverses direction. Figure 4 shows a lot of the dimensionless time τ required for shock formation versus the initial density fraction c found from Eqs. 6 and 7. Determination of a general exression for the time of shock formation in closed form is difficult; however, solutions for two limiting cases, c. c,an = can be found. B. Shock formation time: c. /2 Consider the region between π/2 and π/2. When c is aroximately equal to /2, the shock must form at a oint to the left of the c axis, but near the oint z =. If the cosine function is aroximated as cos z = z 2 /2 and substituted into Eq. 9 then the rofiles of shown in

5 for oints in the region π/2 z. The modified density fraction c and coordinate z then become z = c cos(φ + τ ) cos 2 φ sin 2 τ sin φ c = c cos(φ + τ (26) ) cos 2 φ sin 2 cos φ. τ The condition for the aearance of a shock is that dz / should aroach zero, which from Eqs. 26 can be found as dz = 5 FIG. 4: Time for formation of a shock τ versus the starting density fraction c from numerical integration of the Hamiltonian system of equations. The oint (N) at c = /2, is the limining value of τ from Eq. 28. The oints ( ) are from Eq.3for c Fig. reduce to circles described by µ 2 z 2 + c 2 = k (2) on the zc lane for k 4. Substitution of z from Eq. 2 into Eq. 7 with the sine function aroximated as sin z = z gives, which can be integrated to give arcsin q c arcsin 4 + k q = ±dτ, (22) 4 + k c 2 c q 4 + k Further maniulation of Eq. 23 gives for z< = τ. (23) 2 c c 2 c 2 + z 2 + c z c 2 + z 2 =sinτ, (24) where z has been defined as z = z/2 2 and τ = τ / 2. It is convenient at this oint to exress the coordinates c and z in Eq. 24 in cylindrical coordinates, c = r cos φ and z = r sin φ, where φ is an angle measured clockwise with resect to the negative c axis Equation 24 gives, after some algebraic maniulation, r = c cos(φ + τ ) cos 2 φ sin 2 (25) τ cos(2φ + τ )[cos 2 φ sin 2 τ ]+sin(2φ)cos(φ + τ )sinφ sin(2φ)cos(φ + τ )sinφ sin(2φ + τ )[cos 2 φ sin 2 τ ] (27) The shock for c aroximately equal to zero is exected to aear at φ = π so that Eq. 27 reduces to dz / = cot τ =;hence, it follows that τ = π/2, for the time of shock generation, and the time for shock formation is τ ' π 2. (28) As can be seen in Fig. 4 the shock time given by Eq. 28 is in excellent agreement with the numerical results. C. Shock formation time: c = A second case where the time of shock formation time can be found in closed form is when the initial density fraction is small. Consider oints in the region π/2 < z<π/2. Starting from Eq. 6, the time for the density fraction to reach a value c starting from the oint (z,c ) canbeshowntobegivenby τ = c c x 2 ( x) 2 c 2 ( c)2 cos 2 z +2 c. x 2 ( x) 2 c 2 ( c ) 2 cos 2 z (29) It is clear from lots found from the solution to Eq. 6 or through numerical integration of Eqs. 6 and 7, that the oints first forming the shock lie originally in the region <z<π/2 and which move downward and to the left along the curves shown in Fig. 2. For small c,theshock formsatvaluesofc ' /2 near π/2 from oints z on the ositive z axis where z '. Equation 29 can then be aroximated as τ ' /2 c x2 c 2 min +2 c x2 c 2 min

6 6 giving τ as τ ' ln c +ln à c +! c 2 c2 min where ' c cos z. for small c is thus Thetimefortheshocktoform τ ' ln c (3) The lot in Fig. 4 gives two oints from Eq. 3. Evaluation of the shock time for values smaller than those given in Fig. 4 shows that the aroximation given by Eq. 3 gets successively better for small values of c. For examle, for c =5 5, numerical integration gives τ =.6 whereas Eq. 3 gives τ =9.9 FIG. 5: Density fraction c versus dimensionless distance z for α=4. The times for the lots are τ =,.2,.4, and.8 for lots with successively higher eak values of c V. Effects of Mass Diffusion A. Results from numerical integration The effects of mass diffusion were determined by numerical integration of Eq. 3 using the finite difference method, which gives a solution by considering small changes in the density fraction resulting from a change in time τ and sace z. A solution for c is obtained by selecting successively smaller values of τ and z until the solution converges. The first order derivative of the density fraction with resect to z at a time t k at the oint z k is given, for instance, by c z = c(t k,z i+ ) c(t k,z i ). (3) 2 z Since derivatives must be comuted at the end oints of the integration range in sace, the eriodicity of the solution is used to give values for the required derivatives. The modulus function, mod(a, b), isintroduced to reresents the remaining integer of division of a by b. For examle, mod(3, 2) is equal to one and mod(4, 2) is equal to zero. If a is less than b, the value of the modulus function is a, where a is a ositive integer, including zero. The exressions for the first and second sace derivatives become c z = c(t k,z mod(i+,imax)) c(t k,z mod(i,imax)) 2 z 2 c z 2 = c(t k,z mod(i+,imax)) 2c(t k,z i )+c(t k,z mod(i,imax)) z 2 To simlify the notation, z i and τ k are now relaced by their indices, giving Eq. 3 as c(k +,i)=c(k, i)+α T τ cos(2π i )( 2c(k, i)) i max (32) c(k, mod(i +,imax )) c(k, mod(i,i max )) 2 z α T τ sin(2π i µ τ )(c(k, i) c 2 (k, i)) + i max z 2 [c(k, mod(i +,i max )) 2c(k, i)+c(k, mod(i,i max ))]. The lots shown in Fig. 5 indicate that the effect of diffusion is to remove the high satial frequencies in the distribution. Since the satial rofile of c deends on α and τ only, it is ossible to calculate the satial Fourier comonents of the distribution after obtaining c from numerical integration, and hence to determine the intensity of light diffracted from an absortion grating, as was done in Ref. [2]. By minimizing the error in a fit to the intensities of the diffracted light beams, the thermal diffusion factor can be determined from exerimental data. B. Density fraction distribution for long times When τ becomes long, thermal diffusion is exactly balanced by mass diffusion and the satial rofile of the density fraction must aroach a limiting form. It can be seen that when /dτ =, Eq. 3 reduces to an ordinary differential equation in z, a first integral of which is given by ˆT + αc( c)d dz dz = C, (33)

7 7 where C is a constant, and a dimensionless satial derivative of the temerature field d ˆT/dz has been substituted for the cosine function. For temerature fields that are eriodic in z it is exected that in the region <z<2π, both /dz and d ˆT/dz will be zero at the some value of z. For the resent roblem, both of these quantities are zero at the oint z =3π/2; hence the constant C must be zero and Eq. 33 can be integrated as c(z) c(z ) c( c) = α The general solution to 34 is thus, z z d ˆT dz. (34) dz c(z) = +Fe αt (z) (35) where F is a constant given by [ c(z )] ex[ α ˆT (z )]/c(z ) For the resent roblem with a sinusoidal temerature field, the long time distribution becomes c(z) =. +Feαsin z (36) An equivalent exression for the linearized Eq. 3 has been given in Ref. [3]. Since the value of c(z ) is not known without solution to Eq. 3, F (α, c ) can be determined for a given value of α and c through use of the mass conservation law, Eq. 5. Deending on the value of α, curves of c versus α from Eq. 36 can resemble a sinusoidal function for small α, or a nearly square wave for large α. VI. Discussion The exressions in Eqs. 2 and 6 rovide an exact solution to the thermal diffusion roblem without mass diffusion. If Eq. 3 is divided by α so that the time derivative is exressed with resect to τ, it is clear that the effects of diffusion are small when α is large or when the gradient of the density fraction is small, such as at short times. The conditions for the time of aearance of the shocks is straightforward, however, the comlexity of an exlicit exression for the shock time in terms of α and c aears to be great. The formation of a shocks follows from the nonlinearity of the deendent variable c in Eq. 3. Perhas the unique feature of shocks roduced by thermal diffusion in sinusoidal temerature fields is that they come as counter-roagating airs of waves, the direction of which is governed by the sign of the cosine function in Eq. 2 to the right and left of z =3π/2.. The left-going wave near z =3π/2 has the roerty that it slows to a seed of zero when c l aroaches one, and c r aroaches zero. At this oint in time, a comlete searation of the mixtures, ossible only in the absence of diffusion, takes lace. The arallel between thermal diffusion shocks and fluid shocks has been given in Ref. [2]. In both cases, the mathematics are treated by ignoring the dissiative term in the equations of motion to find the roerties of the shock waves. In fluid shocks, viscosity causes a broadening of the shock front, just as mass diffusion broadens the thermal diffusion shock fronts as shown by the results of numerical integration given here. The extent to which thermal diffusion shocks are visible in the laboratory is clearly a function of how large the thermal diffusion factor can be made. ACKNOWLEDGEMENT SD and GD acknowledge the suort of the US Deartment of Energy, Office of Basic Energy Studies under grant ER WC acknowledges artial suort for his research from the NSF under grant DMS-728, the NSERC under grant # , and the Canada Research Chairs Program [] C. Ludwig, Sitzber. Akad. Wiss. Vien Math.-naturw. 2, 539 (856). [2]C.Soret,Arch.Sci.Phys.Nat.Geneve3, 48 (879). [3] D.D.Fitts,Nonequilibrium Thermodynamics, McGraw- Hill, New York, 962. [4] S. R. degroot and P. Mazur, Non-Equilibrium Thermodynamics, North Holland, Amsterdam, 962, As noted in this reference the Dufort effect is generally small and can be ignored. [5]E.A.Mason,R.J.Munn,andR.J.Smigh, Thermal diffusion in gases, in Advances in Atomic and Molecular Physics, edited by D. R. Bates, age 33, New York, 966, Academic Press. [6]R.D.Present, Kinetic Theory of Gases, McGraw Hill, New York, 958. [7] K. Thyagarajan and P. Lallemand, Ot. Commun 36, 54 (978). [8] F. Bloisi, Ot. Commun. 68, 87 (988). [9] W. Köhler, J. Chem. Phys. 98, 66 (993). [] W. Köhler and S. Wiegand, Measurement of transort coefficients by an otical grating technique, in Thermal Nonequilibrium Phenomena in Fluid Mixtures, editedby W. Köhler and S. Wiegand, age 9, Sringer Verlag, Berlin, 2. [] H. J. Eichler, P. Gunter, and K. W. Pohl, Laser-Induced Dynamic Gratings, Sringer, Berlin, 985. [2] S. Danworahong, W. Craig, V. Gusev, and G. J. Diebold, submitted for ublication. [3] W. L. Craig, S. Danworahong, and G. J. Diebold, Physical Review Letters 92, 259 (24). [4] L. M. Milne-Thomson, Handbook of Mathematical Functions with Formulas, Grahs, and Mathematical Tables, volume 55, age 596, National Bureau of Standards, Washington D.C., 964, Some authors (e.g. I. Gradshyteyn and Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 98) give the ellitic

8 8 integral with the second argument as a/b instead of a 2 /b 2, which we believe to be correct.