'PrCklMXl < n w !F»r>ON NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1275 THE SOLUTION OF THE LAMINAR -BOUNDARY -LAYER EQUATION FOR THE FLAT PLATE FOR VELOCITY AND TEMPERATURE FIELDS FOR VARIABLE PHYSICAL PROPERTIES AND FOR THE DIFFUSION FIELD AT HIGH CONCENTRATION By H. Schuh Translation of ZWB Forschimgsbericht Nr. 1980, August 1944 Washington May 1950 DOCUMENTS DEPARTMENT RATIONAL ADVTSOEY COMMITTEE FOE AERONAUTICS TECHNICAL MEMOEANDUM 1275 THE SOLUTION OF THE LAMINAE -BOUNHAEY-LAYEE EQUATION FOE THE FLAT PLATE FOE VELOCITY AND TEMPER ATUEE FIELDS FOE VARIABLE PHYSICAL PEOPEETIES AND FOE THE DIFFUSION FIELD AT HIGH CONCENTRATION By H. Schuh SUMMAEY In connection with Pohlhausen's solution for the temperature field on the flat plate, a series of formulas vere Indicated "by means of which the velocity and temperatiire field for varia'ble physical charac- teristics can he computed hy an integral equation and an iteration method "based on it. With it, the following cases were solved: On the assumption that the viscosity simply varies with the temperature while the other fluid properties remain constant, the velocity and tempera- ture field on the heated and cooled plate, respectively, was computed at the Prandtl numbers 12 ."3 and 100 (viscous fluids). A closer study of these two cases resulted in general relations: The calculations for a gas of Pr numher 0.7 (air) were conducted on the assumption that all fluid properties vary with the temperature, and the velocities are low enough for the heat of friction to he discounted. The result was a thickening of the "boundary layers, "but no apprecla"ble modification in shearing stress or heat-transfer coefficient. The effects of density and viscosity or density and heat conductivity have opposite effect for velocity and temperature field and almost cancel one another. Formulas allowing for the heat produced hy the friction were indicated, "but no calculations were carried through in view of the already existing report "by Crocco . The methods of solution developed here were finally applied also to the case of diffusion of admixtures, where at higher concentration finite transverse velocities occur at the wall. ■* Uber die Losung der laminaren Grenzschichtgleichung an der e"benen Platte fur Geschwindigkelts- und Temperaturfeld "bei veranderlichen Stoffwerten und fur das Diffusionsfeld "bei hoheren Konzentrationen.''^ Zentrale fur wissenschaftliches Berichtswesen der Luftfahrtforschimg des Generalluftzeugmeisters (ZWB) Berlin-Adlershof , Forschungsbericht Nr. I980, August I8, 19I4.I1. NACA TM 1275 I. UTTRODUCTION The lamlnar-lioundary-layer equation for the flat plate lined up with the flow and with constant fluid properties was solved hy Prandtl (reference 1) and Blasius (reference 2) for the flow field and "by Pohlhausen (reference 3) for the temperature field. Since temperature and velocity field coincide when kinematic viscosity (v) and tempera- ture conductivity (a) are identically eq^ual ( Pr = - = 1 J , Pohlhausen's formula for the temperature field contains a solution for the velocity field also in the form of an integral equation. Piercy and Preston (reference k) , proceeding from a rough approximation, indicated that, with the aid of this integral equation and an iteration method, the well-known Blasius solution can he obtained in a few steps. This method of solution has the advantage of being simple and requiring relatively little time. It is - as is shown in the following - particularly suitable for boundary-layer calculations involving variable fluid properties, because a first, and usually fairly close, approximation, is already available in the solution for constant fluid properties . Crocco (reference 5) and von Kirman and Tsien (reference 6) (see also reference 9^ 10) computed velocity and temperature field for variable fluid properties. In both reports, the differential equations are put in a different form from the elsewhere conventional boundary- layer calculation by changing to new variables. Crocco obtains two simultaneous differential equations of the second order which he solves for a gas with the Prandtl number Pr = 0.725 (air). Von Karman and Tsien treat the case of Pr = 1 and have to solve only one differential equation, since then the temperature la related in a simple manner to the velocity. In the following, it is shown that a number of boundary- layer problems for the flat plate can be solved in a comparatively simple manner, involving merely quadrature, by means of the cited integral equation and an iteration method. II. SOLUTION OF BOUNDARY-LAYER EQUATION FOR VARIABLE PHYSICAL PROPERTIES The boundary-layer equations for velocity and temperature field at the flat plate at variable density read (reference 3) P^^+ pvr^= T^{ Mr^ 1 (1) ^Su ^ _a_ / Su \ NACA TM 1275 3 ^ + ^ = (la) with u and v the speed In flow direction and at right angle to It, T, the temperature, i the distance from the plate leading edge, y the distance from the vail, p the density, n the viscosity, Cp the specific heat, and X the heat conductivity. In the equation for the temperature field, the heat produced "by friction is, at first, not taken into account; as long as the speeds are not excessive and the temperature differences not too small, this is justified. For constant density, equations (l) and (2) can he reduced to an ordinary differential equation (reference 3) on the assumption that u and T are a function merely of the one (dimensionless) = 1-^ coordinate | = •^U — . Since the density depends only on the tempera- ture, the idea suggests Itself that the same simplification is possihle also for variahle density. We put where U is the velocity at the edge of the "boundary layer, Tq and T-]_ the wall temperature and the temperatiore at the edge of the houndary layer, respectively. The quantity "^-^ in the dimensionless | denotes the kinematic viscosity for the fixed temperatiore T, , for which in suitahle manner the wall temperature (k = O), or the tempera- ture at the edge of the houndary layer (k = 1), is chosen. The boundary conditions for flow and temperature field read y = i = (X) = 0=0 (M y ^ to 1-^ oo (D = 1 9=1 Putting p^-*CS ^-fd) >^ = x(u (5) NACA TM 1275 where the suhscript k denotes the density at temperature T, , gives "by (la) (6) hence, "by (l), after introduction of (5) and (6), the more suitahle form From (7), regarded as differential eq^uatlon for the quantity cp— and f temporarily as a known function of |, the following expression for OD is derived J(^) y ^ ' J Q ^> ^ .' This disposes of the integration constant from consideration of the houndary condition (k) . Likewise, there is afforded for the dimen- sionless temperature 9 the expression K(co) ' ' Jq X ^^^ where Pr, is the Prandtl numher with the density at temperature T . For constant density (cp = >jf = X = 1) , velocity and temperature field are independent of each other and (9) gives the Pohlhausen expression (reference 3) for "the temperature field, which represents the solution for the velocity field at Pr = l(v = a). When the velocity field is known, the solution for the temperature "by (9) is ohtainahle by simple quadrature. But the calculation of the velocity field runs into difficulties, at first, "because in (8) the still unknown velocity appears on the right-hand side in the expression for f . The methods of solution "by Plercy and Preston proceed from a random approximation for co with which f and J( I ) in (8) are WACA TM 1275 computed. The improved value oj obtained forme the starting point for the next step, etc. Figure 1 represents the several steps of this approximation method. The intentionally rough approximation cd = 1 over the entire "boundary layer was chosen as original solution; the corresponding first approximation-'- cd is given "by the error integral. After the third approximation, the shearing stress shows a mere difference of ^4- .5 percent from the exact value . Instead of continuing the process mechanically, the final solution to he expected was estimated from the variation of the previously computed approximations and utilized as hasis for the subsequent step; the solution cu contained hut a -^-percent error in shearing stress. With this method of solution, the improvement effected "by each step can "be estimated according to order of magnitude. The equations (8) and (9) are identical for constant density and Pr = 1 . Assuming that the approximate solution for en was such that for each individual value co the corresponding | coordinate differed by a constant factor ^ from the | coordinate of the exact solution, the effect of factor % Is then ohviously Just as great as that of quantity Pr for the temperature field. Pohlhausen found, on the hasls of his numerical calculations, that the heat- transfer coefficient is proportional to v Pr, thus the shearing stress at the wall Is afflicted at each new step hy an error of only ahout one-third of the error of the preceding step. For variable density, the discussed solution steps of mathe- matical nature can he combined with the steps of physical nature : Step 1: as starting point the known solutions for constant density are assumed : (a) The Blasius solution (reference 2) for the velocity profile (h) Pohlhausen 's method for the temperature field Step 2: (a) Calculation of velocity profile by (8), the temperature variation being based on the density of the tempera- ture profile according to step 1(b) (b) Calculation of temperature field by (9) with the velocity profile according to step 2(a); relation of density to temperatvire as in step 2(a) lit took a subsidiary worker 10 hours to reach the final solution of the velocity field in figure 1. NACA TM 1275 The process is repeated till the final solution ia sufficiently exact, usiially req^ulring three to four steps. A fev general remarks ahout the influence of the temperature variaTsility of the physical properties .- The flow with constant physical properties can he regarded as first approximation, and the prohlem is then to ascertain the differences which are produced hy variatle physical properties. The q^uality of this approximation depends, of course, on the temperature assumed for the physical properties at the isothermal flow. Choosing the wall temperature or the temperature at the edge of the houndary layer as reference tempera- ture for the isothennal flow so resiilts on the basis of physical point of view as well as on the hasis of the equations that an increase of the viscosity or density inside the "boundary layer is accompanied by an increase in the resistance; similarly, an increase in heat con- ductivity and density effects a greater heat transfer. But the magnitude of the effect of variability of the separate physical prop- erties is contingent upon the ratio of the boundary-layer thickness of the temperature and velocity field. (The ratio of both ia proportional according to Pohlhausen . ) This is Illustrated by the following case, which is, at the same time, of practical importance. The temperature boundary layer is assumed very small compared to the flow boundary layer; consequently, the variation of the physical properties within the thermal boundary layer can be disregarded for the shearing stress and the latter computed as if the temperature at the edge of the boundary layer reaches to the wall. The same holds for the velocity profile, with the exception of a small area within the thermal boundary layer, where the velocity profile by the viscosity variation ia deformed correspondingly. But for the temperature profile this area Is exactly decisive. From the equality for the shearing stresses the velocity gradients at the walls are: ^ ^o for Pr ' -^ CD ( 10 ) 11 the subscript 11 denotes the isothermal flow with the physical prop- erties at temperature T^. The variability of density is noneffective for the field of flow, in this instance. It can be mathematically derived from the formulas (8) and (9). The ratios for the temperature field are discussed in the next chapter by means of the two examples. WACA TM 1275 III. FLOW AND TEMPERATURE EIELD FOE VISCOUS FLUIDS In accordance with the physical properties of viscous fluids, the velocity and temperature field were computed on the assumption that only the viscosity should change with the temperature ty the following formula -H- J ^^ ^ M (11) where "b and T^, are constants, chosen so as to reproduce the tem- perature variation as closely as posBitle . The suhscript k is to te or 1, depending upon the choice of the physical properties in the dimensionless | . The choice was "b = 3 (viscous lubricating oil) and the two cases of a heated and cooled plate computed with -s and 8 and Pr^ = 12.5 and 100; It thus concerned Identically great temperature differences of the same fluid, since Pr is for the present formed with the physical properties at wall temperature. Choosing Tq as reference temperature gives by (ll) cp=^=/ ' \ (12) ^^o The result of the calculation ty the iteration method of the preceding section is seen in figures 2 and 3- In "both graphs, the dimensionless wall distances | and I-, , formed with \i and \j. , are plotted to the scale l:v8 and yo:!, respectively, so that the actual wall distance y Is the same for "both atscissas. Besides the solution a>, which took three steps to compute, the isothermal velocity profiles (cu) and {(d) at constant density at temperature T and T-, are shown plotted against the dimensionless coordinates | and L . In the subsequent compilation t and a denote the shearing stress at the wall and the heat-transfer coefficient I a. = Xi-^j 1; _ For the "isothermal" temperature profiles (0) and (G) , the o 1 Prandtl numbers at temperatures T and T must he inserted. For example, in figure 2: Pr =12.5 and Pr = 100. o 1 8 NACA TM 1275 n and (a) are the corresponding values In isothermal flow with the Tiscoslty at temperature T, and T . Tahle I •^o a P^o (^o)^ (a)^ Heated wall 0.81+1 1.20 12.5 1.58 1.811 Cooled wall l.o8 0.98 100 0.255 3.01 Although in hoth cases the thermal houndary-layer thickness is far from small compared to the flow "boundary layer, the shearing stress can still he computed satisfactorily "by the isothermal formula with the viscosity of the wall temperature. The assumptions to equation (10) are thus shown for Pr > 10 . The conditions are more complicated for the heat-transfer coefficient; from (9), It follows that the heat-transfer coefficient a is proportional to (Pr) y/T yvx , wherein, according to Pohlhausen, a is, with high accuracy, assumed as 0.661+ "V^Pr. Bearing in mind that Pr = '^, it follows that the heat- transfer coefficient is inversely a' "^ proportional to the sixth root of the viscosity. Since all physical properties except the viscosity have "been assumed constant, there results, when it is referred once to the wall temperature, the other time to the temperature at the edge of the boundary layer (13) A comparison with the foregoing tabulation indicates that (a) supplies a poorer approximation for the heat-transfer coefficient than (c(-)q> this is readily explained by the variation of the velocity profile (figures 2 and 3). It is to be expected that the conditions are similar at higher Prandtl numbers. NACA TM 1275 Another reference point for the heat-transfer coefficient is found in the velocity gradient at the wall; "by (10) and allowing for (3), there follows (lU) io the suhscripts io and il denoting isothermal flow at tempera- ture T and T-, . These relations are confirmed in figures 2 and 3- With these formulas, limits can he indicated for the heat-transfer coefficients (figures 2 and 3) • One is given according to (13) ^y ('^)x> hecause the velocity profile (o))-, yields at all points higher velocities at cooled and lower velocities at heated wall. The other limit is given hy a velocity profile of isothermal form, where the ahsclssa scale is so modified that its gradient at the wall agrees with the actual velocity dlstrihution . From the remark ahout the convergence of the method of solution in II, it follows then that the heat-transfer coefficient Is proportional to the third root of the velocity gradient at the wall; for this extreme value, the second equation of (Ih) gives: \ — (^) • Summed up, the limits of the heat-transfer coefficients, hy a change in viscosity, are the upper signs applying to heated, the lower to cooled wall. Hence, the following approximate rule for viscous fluids (Pr>10): For computing the resistance, the physical properties are referred to the temperature at the edge of the "boundary layer; for heat transfer, to the wall temperatiare . IV. FLOW AND TEMPEEATURE FIELD AT Pr = .? (AIR) FOE TEMPERATURE "VARIABILITY OF EVERY PHYSICAL PROPERTY In the -50° to 1^0° temperature range the physical properties of the air can he represented hy the following formulas 10 NACA TM 1275 H = K^T ,0.780 =Yil! T = temperature in a'bsolute degrees . With ^ = _ Tq - Ti \ = K T 3 0.821 there results iL = 9 = fl +^(1 - 9^ 0.78 and. similar expressions for \|r and X. The calculations for a heated plate and t3 = -j^ and - showed only moderate differences in the velocity and temperature field from the form for isothermal flow (Table 2). For the investigation of the conditions at higher temperature differences, the case T-j^ = 20 and Tq = 620° C was computed. The velocity and temperature fields already exhibit, according to figure k, appreciable differences from the form for constant physical properties; |q and |]_ are formed with the physical properties at temperatures T^ and T-|_, respectively. This results in a substantial thickening of the boundary layer for both fields; nevertheless, wall shearing stress and heat-transfer coefficient indicate only minor departures from the values for constant physical properties . Table 2 Heating l^^ljo al ■^o ■^o (^o)i a a (a)l (^o)o ^-l 0.575 0.U90 1.02 1.00 1.01 1.00 . = 1 .511^ .1+20 1.05 1.00 1.02 1.00 Tq = 620° C T;j_ = 20° C .286 .235 1.11 .93 1.03 .96 NACA TM 1275 11 The explanation for it is that in air the growth of the viscosity with the temperature acts in the sense of a resistance increase, the drop in density in the sense of a resistance decrease, and "both effects practically cancel one another at Pr = 0.7, where thermal and flow "boundary -layer thickness are ahout equally great. The conditions for the temperature field are almost identical, "because the heat con- ductivity and the viscosity are similarly affected hy the temperature. The frictional heat can he allowed for in similar manner; equation (2) contains then an additive term |j(-2H j on the right-hand side, and the solution reads i^l e = A(i) B(|) 1 + B(«.) A(<») A(l) - B(0 le-E(n 41 = 2. /^k ^e (Tl - To) r 1 P ^'^e^(^) d^ ^0 X Jo e-^^^) dl > (16) R(n =Pr^ f ^^^ e 2C3, The iteration method can he applied again, although a little more paper work is involved. For constant physical properties, equation (16) reduces to Eckert's solution (reference 8). The thermometer prohlem (vanishing temperature gradient at the wall) can also he solved hy suitahle variation of the Integration constant. In view of Crocco's calculations for a gas with Pr = O.725, it was decided not to cal- culate any model prohlems hy the new method . 12 NACA TM 1275 V. APFLICATIOW TO A DIFFUSION PROBLM The concentration field for the prohlem of diffusion at the flat plate can te calculated in the same manner as the temperature field. 3,^ The differential equation reads u^ + v|£ = k^ (17) dx dy ay2 where k is the diffusion factor and c the concentration which is defined as quantity of gas or vapor per unit volume . The physical properties are regarded as constant, hut it is also taken into account that for greater concentrations the velocity v at the wall no longer disappears, as already pointed out hy Wusselt (reference 7)- When fluid from a wall is vaporized, say "by a gas such as gas flowing along a wetted wall, sutstance passes continuously into the flow. Hence v(0)>0 at the wall. When, on the other hand, vapor condenses at the wall or when air containing ammonia, for example, passes over hlotting paper Impregnated with hydrochloric acid, it results in v(0)<0.^ The houndary conditions for v are according to the equations (lOO) and (101) of reference (7) 5 _ JC/BC\ 1 = y(0) (18) Po where c is the concentration of the gas or vapor, for which the wall is permeahle, Cq the concentration at the wall, p the corresponding partial pressure, and p the total pressure. Introduction of the flow velocity U and the dimensionless I results in ( l\ = 1 k ^1 " ^o (i.c] .^ \U4 2 Co Z:^ _ ^\j\Ad^Vvx c = c - c o_ (19) ^1 - Co ■^ckert reported a solution of this prohlem at the 19^3 meeting of the VDI committee for heat research in Bayreuth, where an approxima- tion method similar to Pohlhausen's method for the flow houndary layer was used . h " , Damkohler s estimate for the present problem was published in Z. fur Elektrochemie, 19^2, p. I78. WACA 1M 1275 13 where c. and C-, are the concentration at the wall and at the edge cdI dl + M = - - of the boundary layer. Gimilarly to (5), It gives 1 = \fl\a u Vux L and similarly to (9) r the solution for the concentration field if (f(l)-M)d| k ^1 - % /dC\ (20) ^ _ V = 1411 L(n = H f(l) = 2 [ f J Q U > (21) dl where ^ is a q^uantity analogous to the Prandtl number. To obtain the k velocity en, simply put Y = 1 in equation (21). k The concentration gradient at the wall is contained in M; hut (21) can, in the first instance, he solved for any M values and the quantity computed with the aid of the value obtained from The velocity and concentration fields for the the solution for calculated M and W values are represented in figures 5 and 6, the concentration gradient at the wall, in figure 7- M >0 denotes evapora- tion at the plate; M<0, condensation and absorption at the plate; the value 0.6 chosen for the quantity )L is applicable in good approxima- k 5 tlon for the diffusion of water and ammonia in air. Strictly speaking, for the specified higher concentrations, the density and viscosity of the two fluids are dependent on the concentration; and the diffusion factor, on the temperature. Cases of that kind can be calculated with the aid of the described method. If the diffusion is 5 It n According to Ten Bosch: Die Warmeubertragung, Berlin 193d> pp. 189 and 257- Ik NACA TM 1275 accompanied "by a heat transfer, the solution for the concentration field can equally "be applied to the temperature field with good approximation. In the same way, the heat transfer can he derived from the solution for the concentration field, when air is exhausted or hlown at the plate with transverse velocities at the wall corresponding to equation (20). CONCLUDING NOTE After completion of the calculations the writer received knowledge of a report hy Schlichting and Bussmann (reference 11) ahout the velocity profile at the flat plate for exhaustion where the transverse velocity at the wall was expressed "by Between the present value M and C the following relation exists (see also (I9) and (20)). C = -M The present velocity distrihutions agree to ahout 1 percent with those calculated hy Schlichting (hy a different method), with exception of M = 1, where the writer plainly chose too few approximation steps and the differences are therefore a little greater. For the present calculation three steps were usually sufficient. Translated "by J. Vanier National Advisory Committee for Aeronautics NACA TM 1275 15 REFERENCES II 1. Prandtl, L.: UelDer FlusslgkeltalDewegung "bel sehr klelner EeiTDung. Verhandl. d. III. Intern. Math. Kongr . Heidel^berg I90U, p. k&k . (Available as NACA TM 1^52.) It 2. Blasius, H.: Grenzschichten in Flusalgkelten mlt klelner Eeibung. Z. Math. Phys. Bd . 56, I908, p. 1^. (Available as NACA TM I256.) 3. Pohlhausen, E.: Der Warmeauetausch zwischen feate^ Korpem und Flussigkelten mlt klelner Eelbung und klelner Warmeleltung. ZAMM Heft 2, 1921, p. 115. k. Piercy, N. A. V., and Preston, J. H:: A Simple Solution of the Flat Plate Problem of Skin Friction and Heat Transfer. Phil. Mag. May I936, p. 995. 5. Crocco, L.: Sullo strato limlte laminare nei gas lungo una parete piana. Eend. Circ. Mathem. Palermo Bd . LXIII, 19^+0 /i+l. 6. Von Karman, Th., and Tsien, H. S.: Boundary Layer in Compressible Fluids. Jour. Aero. Sciences, Vol. 5, No. 6, I938, p. 227. 7- Nusselt, W.: Warmeubergang, Diffusion und Verdunstung. ZAMM Band 10, Heft 2, 1930, p. 105- 8. Eckert, E., and DrewJ||tz, 0.: Der Warmeubergang an elne mit grosser Geschwindigkeit langs angestromte Platte. Forsch . Ing.-Wes. Bd. 11, I9U0, p. 116. (Available as NACA TM lOi+5.) 9. Hantzsche, W., and Wendt, H.: Zum Kompresslbilltatseinfluss bei der laminaren Grenzachicht der ebenen Platte. Jahrb . 19^0 der Dtsch. Luftfahrtforachung, p. I 5I7 . 10. Hantzache, W., and Wendt, H.: Die laminare Grenzschicht der ebenen ti II II Platte mit und ohne Warmeubergang unter Berucksichtigung der KompresBibilitat . Jalirb. I9U2 der Dtsch. Luftfahrtforachung, p. I Uo. 11. Schlichting, H., and Busamann, K.: Eiakte Losungen fur die laminare Grenzachicht mit Abaaugung und Ausblasen. Schrlften der Dtsch. Akademie der Luftfahrtforschung, Heft 2, 19'<-3, p. 25. 16 MCA TM 1275 Figure 1,- The several approximations for computing the velocity distribution at the flat plate by the method of Preston and Piercy (constant physical quantities). 1.0 ^6)=^-^ Tj o (e), No e-_ y/ / II / ^^ --' ^-- 0.6 (e)A 1 / 1 1 . / / / --'^ ^ "^ h :> ^ ^ / ^ ^ ^h), fj, y, 8 0.2 c / / ^ / / ^ • Pro =12.5 r / y^ O.t 1 r 1.0 1 • e y u ' ^1 2\v,y. K 1. 2. 3 .0 ^ Figiire 2.- Velocity and temperature distribution at a heated plate for variable viscosity. Viscosity exponent b = 3. (oj)q, (9)o> and (oj) , (e) isothern^al velocity and temperature distributions, Vq and V, kinematic viscosity at wall temoerature T and temperature T]_ at the edge of the boundary layer. NACA TM 1275 17 2^ yoX Figure 3.- Velocity and temperature distribution at the cooled plate. Viscosity exponent b = 3. (w)^, (e)^, and (u)^, (e)^ isothermal velocity and temperature distributions, v and v., kinematic viscosity at wall temperature Tq and temperature T^^ at the edge of the boundary layer. Figure 4.- Velocity and temperature distribution at a heated plate for Pr = 0.7 (air); all physical quantities constant with temperature. 18 NACA TM 1275 C0 = 1.0 u_ U 0.6 0.2 1 M= -1.0 <^ ^^ — — ^ //o.2y //// /o.5 / k 1//// V / . . k Ci-Co idC r ^^.0 /.O 2.0 — 3.0. S 2 1 vx Figure 5.- Velocity field at diffusion with higher concentrations, where finite transverse velocities occur at the wall (see text for equations (18) to (20)). C = cr Co Co 1 M= - 1.0:. . /^ -^ 0.6 "^■^\y ^ yf Z^ /0.5 / 2^=0.6 k 0.2 A w "^ iO ,, . k c,-Co IdC] Po 1 /^ ^ \ V 1.0 2.0 — 3.0. e=.- y, u vx Figure 6.- Concentration distribution to figure 5. NACA TM 1275 19 2.0 M A 1.0 y-Z-O fdC\ 1 V k 0.6 N--'-'- ^-0 1 Co [Po 1 0.5 \ ., k Ci-Co (dC\ V \ ^^ ldC\ \d^)o • ■ 1 r 1 9 3 A i '. 5 t 3 7 — ^A/ L-n^. Figure 7.- Concentration gradient at the wall and quantity M plotted against N (see text for equations (19) and (20)). 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