The Restoration Coefficient and Reynolds Analogy in a Boundary Layer with Injection and Suction over the Entire Prandtl Number Range

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1 ISSN , Fluid Dynamics, 11, Vol. 46, No. 4, pp Pleiades Publishing, Ltd., 11. Original Russian Text I.I. Vigdorovich, A.I. Leont ev, 11, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 11, Vol. 46, No. 4, pp The Restoration Coefficient and Reynolds Analogy in a Boundary Layer with Injection and Suction over the Entire Prandtl Number Range I. I. Vigdorovich and A. I. Leont ev Received November 5, 1 Abstract The restoration and Reynolds analogy coefficients are calculated for a laminar self-similar boundary layer on a permeable plate over the entire possible range of variation in the Prandtl number and the injection and suction parameter. Keywords: boundary layer, injection, suction, restoration coefficient, Reynolds analogy, Prandtl number. DOI: /S The restoration coefficient r can be introduced by the formula T r = T e 1 + r γ 1 M e ), where T r is the temperature of a heat-insulated rigid surface in a gas flow restoration temperature), T e and M e are the temperature and the Mach number at the outer boundary layer edge, and γ is the heat capacity ratio. The restoration coefficient characterizes the degree of deviation of the restoration temperature from the total gas temperature. The Reynolds analogy coefficient α is the double ratio of the Stanton number to the surface friction coefficient α = St/c f. In the generic case a knowledge of the two quantities makes it possible to solve the problem of heat transfer between the gas stream and the wall. In accordance with the classic approximation formulas [1], for a laminar boundary layer on an impermeable plate r Pr, α Pr /3.1) over the Prandtl number range.5 Pr. However, in recent years a series of new problems has appeared such that the above relations are not sufficient to solve them. This concerns calculations of the energy separation of a gas stream using the method proposed in [, 3]. The maximum gain in the energy separation may be expected either for very small or very large restoration coefficients, i.e., for such Prandtl numbers which are beyond the above-mentioned interval. Moreover, rather significant factors affecting heat transfer are gas injection and suction through the surface in the gas stream. In the present study the restoration and Reynolds analogy coefficients are determined over the entire possible range of variation of the Prandtl number and injected and sucked gas flow rates on the basis of the well-known exact self-similar solution for a laminar boundary layer on a permeable plate. 565

2 566 VIGDOROVICH, LEONT EV 1. FORMULATION OF THE PROBLEM We will consider a laminar boundary layer flow on a flat permeable plate located in a gas stream at zero angle of attack. The equation of motion and the continuity and energy equations have the form: ρu u x + ρv u y = μ u ), y y ρu) x + ρv) y =, ρu T x + ρv T y = 1 Pr y μ T ) + μ ) u. y c p y Here, Pr = μc p /λ is the Prandtl number, μ, λ,andc p are the dynamic viscosity and thermal conductivity coefficients and heat capacity of the gas, respectively. The mass flow rate ρ w v w proportional to the distance from the leading plate edge in the power of 1/ is given on the surface. The solution of the problem considered is self-similar and in the Crocco variables [1] has the form: μ u y = ρe μ e u 3 e Fη), T = T e θη), η = u. 1.1) 4x u e Here, the subscript e denotes the free-stream parameters. The functions F and θ must satisfy the ordinary differential equations FF + η =, 1.) [ θ + Prγ 1)M e] F +1 Pr)F θ =. 1.3) Equation 1.) is written for the density and the viscosity related by the condition ρμ = const. This viscosity law makes it possible to carry out calculations with sufficient accuracy [1] by solving the dynamic problem independently of the thermal problem. We introduce the injection and suction parameter b by means of the formula b = ρ wv w ρ e u e Rex, Re x = ρ eu e x μ e. 1.4) Then, the boundary conditions for Eq. 1.) on the wall and in the free-stream flow are F )=b, F1)=. 1.5) After determining the function Fη), the boundary layer velocity profile can be calculated using the relation [1] y θη)dη Rex = x Fη).. DYNAMIC PROBLEM Many investigations [1, 4, 5] are devoted to the boundary-value problem 1.), 1.5). The solution valid over the entire range of the parameter b up to the critical one when the boundary layer is pushed aside can be obtained by reducing the order of Eq. 1.), i.e., by going over to solution of a first-order equation. The change of variables F = η 3/ f ξ ), ξ = lnη.1) FLUID DYNAMICS Vol. 46 No. 4 11

3 THE RESTORATION COEFFICIENT AND REYNOLDS ANALOGY 567 Fig. 1. Integral curves of Eq..5). transforms Eq. 1.) to the form: Hence after substitution we obtain the first-order equation f f + f + 3 ) 4 f + =. df = z f ).) dξ zz + z = 3 4 f f..3) In accordance with 1.5) and.1), f = at the outer boundary layer edge. The function F specifies the tangential stress and, as can be seen from.1), may differ from zero on the wall if f =. After change of variables f = w 3/, z = w 3/ y.4) equation.3) takes the form: dy dw = 3y + y + 3/4 + w 3 )..5) wy In Fig. 1 we have plotted integral curves of this equation for w 1. Point, 3/) is the node in whose neighborhood the solution has asymptotics and depends on a single parameter a. From.),.4) we have FLUID DYNAMICS Vol. 46 No y = 3 + aw + Ow ), w.6) d lnη = 3dw wy..7)

4 568 VIGDOROVICH, LEONT EV Substituting expansion.6) in this equality, after integrating it with allowance for.1) we obtain η = Cw + Ow ), w, F = F)=C 3/, b = a C1/,.8) wherec is a constant of integration. As can be seen from.8), values a > correspond to injection, a = to an impermeable plate, and a < to suction. Increase in the parameter a which specifies the angle of inclination of integral curves at the point, 3/) corresponds to increase in the injected gas flow rate. However, the limiting case of critical injection when the wall friction vanishes corresponds to an integral curve of another family which departs from the saddle point, 1/) Fig. 1) and has the asymptotics In this case y = 1 w3 + Ow 6 ), w..9) η = C 1 w 3 + Ow 6 ), F = C 3/ 1 w 3 + Ow 6 ), w, b = 1 C1/ 1, where C 1 is a constant of integration. The algorithm of solving the boundary value problem 1.), 1.5) reduces to integration of Eq..5) with the initial condition.6) on the interval w 1 and Eq..3) on the interval 1 f. In the last case an initial condition following from relation.4) is used. The solution is specified by the formulas f η = exp df z ), F = η 3/ f, f 1, η = we, F = E 3/, w 1, [ Ew)=exp w 3 + y)dw wy + df z ]..1) As follows from.1) and.4), F = η 1/ z + 3 ) f = η1/ w 3/ y + 3 )..11) Substituting.6) and.1) in.11), we obtain the following relation between the boundary condition on the wall injection and suction parameter) with the parameter a b = a E1/ ). In the case of the critical injection the corresponding formulas have the form: η = w 3 E 1, F = w 3 E 3/ 1, w 1, [ E 1 w)=exp 3 w 1 + y)dw wy + df z ]. FLUID DYNAMICS Vol. 46 No. 4 11

5 THE RESTORATION COEFFICIENT AND REYNOLDS ANALOGY 569 Fig.. Profiles of the tangential stress in the boundary layer for various injection and suction parameters: curves 1 8 correspond to b =.35,.3,.93,,.8,.19,.36, and.6, respectively. Hence we can see that the boundary layer is pushed aside when the injection parameter has the critical value b = 1 E1/ 1 ). In Fig. we have plotted the graph of the function Fη); the critical value b =.6. From 1.1),.8) there follows the expression for the plate friction coefficient: c f = F Rex..1) The friction coefficient decreases monotonically with increase in the injection and suction parameter and vanishes for the critical value b Fig. 3). 3. THERMAL PROBLEM Using relations 1.1), we can represent the heat influx q = λ T / y in the form: q = c pt e Fη)θ η) ρe μ e u e. 3.1) Pr x Solving Eq. 1.3), we obtain the temperature and heat influx distributions over the boundary layer cross- FLUID DYNAMICS Vol. 46 No. 4 11

6 57 VIGDOROVICH, LEONT EV c f Rex Fig. 3. Plate friction coefficient as a function of the injection and suction parameter b. section θη)=1 C Fη) Pr 1 dη +γ 1)M e Pr ϕη, Pr)dη, η η Fη)θ η)=cfη) Pr γ 1)M e PrFη)Pr Fη) 1 Pr dη, η 1, 3.) ϕη, Pr)= Here, C is a constant. Eliminating C from relations 3.), we have [ ] Fη) Pr 1 dζ. 3.3) Fζ) θ)=1 +γ 1)M e PrBPr) θ )APr), BPr)= ϕη, Pr)dη, 3.4) [ ] Fη) Pr 1 APr)= dη. 3.5) Hence and from 3.1), for the heat flux on the wall we obtain [ q w = ρ e u e c p T e St 1 + r γ 1 ] M e θ), St = αf, Re x F r = PrBPr), α = 1 PrAPr). In accordance with formulas 3.6), calculation of the restoration and Reynolds analogy coefficients reduces to evaluation of integrals 3.4) and 3.5). FLUID DYNAMICS Vol. 46 No )

7 THE RESTORATION COEFFICIENT AND REYNOLDS ANALOGY 571 When Pr = 1wehaver = 1andα = 1. We will find the value of the derivative of the function BPr) when Pr = 1. We can represent integral 3.3) in the form: Differentiating, we obtain ϕη, Pr)= ϕη, 1) Pr Integration of this expression by parts gives ϕη, 1) Pr = [ exp Pr 1)ln Fη) ] dζ. Fζ) = ln Fη) Fζ) dζ. ηf η) Fη) dη 1 [F η)] dη. 3.7) 4 The second of integrals 3.7) is obtained with allowance for Eq. 1.). From 3.4), 3.7) we have B1) Pr = b 1 F η)dη. 4 The last integral can be evaluated integrating Eq. 1.) from zero to unity and using integration by part Thus, finally we obtain the expression F η)dη = 1 F b. B1) Pr = b + 1 F b 1 4 from which for the Prandtl number close to unity the restoration coefficient can be represented in the form: r = Pr + b + F b 1 ) PrPr 1). We note that by virtue of formulas 1.4) and.1) in Fig. 3 we have actually reproduced the dependence of the quantity F on the parameter b. 4. STRONG SUCTION We will consider the asymptotic behavior of the solution when the sucked-gas flow rate increases indefinitely. As can be seen in Fig. 3, as b, the quantity F increases. We will seek the function F in the form: Fη)=F 1 η) +F 1 ψη) +O F 3 ), F, 4.1) where the second term of the expansion must satisfy the boundary conditions FLUID DYNAMICS Vol. 46 No ψ)=ψ1)=. 4.)

8 57 VIGDOROVICH, LEONT EV Expressing the second derivative F from Eq. 1.) and integrating by parts two times with allowance for the boundary conditions, we obtain F = bη 1) +1 η) η dη F + η η1 η)dη. 4.3) F Substitution of expansion 4.1) in the right-hand side of Eq. 4.3) and evaluation of the integrals give Fη)=bη 1) +F 1 1 η)[1 η ln1 η)] + O F 3 ). Comparing this expansion with 4.1) and taking conditions 4.) into account, we have b = 1 F + 1 F 1 + OF 3 ), ψη)=η 1)[η + ln1 η)]. 4.4) In accordance with 4.1), in the first approximation the relation between the velocity and the tangential stress in the flow considered is linear. From 1.4),.1) and the first of expressions 4.4) it follows that the relation between the friction coefficient and the sucked gas flow rate has the form: ρ w v w ρ e u e + c f =. The well-known flow in the asymptotic boundary layer with suction [1] has the same properties. Using representation 4.1), we will now calculate the coefficients α and r. Substituting expansion 4.1) in integral 3.5), we have: { APr)= 1 1 } Pr 1)[η + ln1 η)] 1 η) Pr 1 dη + O b 4). 4b Here, on the basis of the first of expressions 4.4) we use the quantity b as a large parameter. In accordance with formula 3.6), evaluating the integrals, we obtain 1 Pr)Pr + ) α = 1 + 4b PrPr + 1) Analogous evaluation of integral 3.4) gives + O b 4). 4.5) Pr 1)Pr + 3) r = 1 + 6b + Ob 4 ). 4.6) PrPr+ 1) Thus, in the case of strong suction in passage to the limit as b the restoration and Reynolds analogy coefficients are equal to unity for any Prandtl number. 5. ASYMPTOTICS FOR LARGE PRANDTL NUMBERS As Pr, the asymptotic representations of integrals 3.4) and 3.5) can be obtained using the Laplace method. Taking into account qualitatively different behavior of the function Fη) for positive and negative values of the parameter b Fig. ), we will individually consider the cases of suction, impermeable plate, and injection. Suction. In the case of suction and for the impermeable plate the function Fη) decreases monotonically on the interval [, 1]. Owing to this fact, as Pr, the neighborhood of zero makes the main contribution to asymptotics of integral 3.5). For suction the expansion at zero has the form: Fη)=F + bη + Oη 3 ). FLUID DYNAMICS Vol. 46 No. 4 11

9 THE RESTORATION COEFFICIENT AND REYNOLDS ANALOGY 573 Hence there follows the estimate Consequently, for suction APr) [ exp Pr 1) bη ] F [ exp Pr 1)ln 1 + bη )] dη F F dη bpr 1), Pr α b, Pr. 5.1) F The asymptotics of integral 3.3) can also simply be calculated. The quantity Fη)/Fζ) reaches a maximum when ζ = η. From the expansion Fη) Fζ) = 1 + F η) η ζ) +..., Fη) ζ η for the function ϕ we have [ ϕη, Pr) exp Pr 1) F ] η) η ζ) dζ Fη) Fη) PrF η). 5.) Hence, in accordance with formula 3.6), after integration we obtain r Fη) dη F, Pr. 5.3) η) Thus, both quantities tends to finite limits during suction. Relations 5.1) and 5.3) are in agreement with 4.5) and 4.6). In passage to the limit as Pr,wefind α = 1 1 4b + O b 4). The same result follows from 5.1) if we take into account expansion 4.4). Substituting expansion 4.1) in integral 5.3), with allowance for 4.4) we have r = [ 1 η + 1 ] 4b 1 η ) dη + O b 4) = b + O b 4). The same expression we can obtain from 4.6) in passage to the limit as Pr. Impermeable plate. As follows from Eq. 1.), when b = the expansion at zero has another form: Fη)=F η3 + Oη 6 ) 5.4) 3F and evaluation of asymptotics of the integral gives APr) [ ] exp Pr 1) η3 3F dη FLUID DYNAMICS Vol. 46 No ) 1/3 F e Prx x /3 dx = Γ ) ) 1/3 F. 9Pr

10 574 VIGDOROVICH, LEONT EV Here, Γ is the gamma function. Hence on the base of 3.6) there follows the expression for the Reynolds analogy coefficient ) 9 1/3 Pr /3 α, Pr. 5.5) F Γ1/3) With allowance for the numerical value of the coefficient the formula has the form α.89pr /3 so that representation.1) remains actually valid also as Pr. In the case of the impermeable plate integral 5.3) diverges at zero by virtue of expansion 5.4). This means that the neighborhood of the point η = makes the main contribution to the asymptotic of integral 5.4). From 5.4) we have Fη) Fζ) 1 + ζ 3 η 3 3F, η, ζ and, making the change of variable ζ = ηx in integral 3.3), for small values of the argument the function ϕη, Pr) can be represented in the form: ϕη, Pr) η [ 1 + η3 x 3 ] 1) Pr 1 3F dx, η As Pr, the neighborhood of the point x = 1 makes the main contribution to the integral. Making the change of variable of integration x = 1 y and retaining only the term of the first order in y, we obtain ϕη, Pr) η ) 1 η3 y Pr 1 [ F dy = F Prη 1 η, Pr. 1 η3 F ) Pr ], The expression on the right-hand side has no singularity at zero and can be integrated, the upper limit of integration having no value from the point of view of determining the principal term of asymptotics and it can be taken from considerations of convenience of calculations BPr) F Pr F /3 [1 η3 F Evaluation of the last integral gives ) Pr 1] d 1 η ) = F4/3 Pr + 3 F /3 ) Pr 1 1 η3 η dη. F BPr) F4/3 + F4/3 ΓPr)Γ/3). Pr ΓPr + /3) Hence, using the asymptotics of gamma function [6] lnγx) x 1 ) lnx x + O1), x, finally we have BPr) F4/3 Γ/3) F4/3 Pr + /3) /3 Pr, r F4/3 Γ/3)Pr 4/3 F Pr + /3) /3, Pr. 5.6) Thus, as Pr, the quantities α and r have power-law asymptotics. For the restoration coefficient relation.1) ceases to be valid since, in accordance with formula 5.6), the quantity r as Pr 1/3. FLUID DYNAMICS Vol. 46 No. 4 11

11 THE RESTORATION COEFFICIENT AND REYNOLDS ANALOGY 575 Injection. We will now consider the case of injection when the fact that the function Fη) is nonmonotonic and reaches a maximum at a certain point η on the integration interval plays a significant role. It is precisely the neighborhood of the maximum makes the main contribution to asymptotics of integral 3.5). From the expansion Fη)=F m η m η η m ) +..., η η m F m for function 3.5), we obtain APr) Fm F ) Pr 1 [ exp Pr 1) η mη η m ) ] dη Fm Pr F Pr 1 e x dx = ηm Pr FPr m F Pr 1 F m π η m Pr. 5.7) Hence on the basis 3.6) we have α 1 ) Pr F ηm, Pr. 5.8) F F m πpr In order to evaluate asymptotics of integral 3.4) we will represent it in the form of the sum of two terms: BPr)= η ϕη, Pr)dη + η ϕη, Pr)dη, 5.9) where the quantity η is taken from the condition Fη )=F, η >. We will consider the first of integrals 5.9) and the behavior of the integrand of integral 3.3). When η < η the function 1/Fζ) reaches a maximum at the point ζ = on the interval ζ η. From the expansion at zero for the function ϕ we have [ ] Fη) Pr 1 η ϕη, Pr) F Fζ)=F + bζ + Oζ 3 ) [ exp Pr 1) bζ ] dζ F [ Fη) F ] Pr 1 F bpr. Now, in evaluating asymptotics of the integral of the function ϕ we can use the result 5.7). This gives η ϕη, Pr)dη F b Fm F ) Pr π η m Pr 3/. We will now consider the second of integrals 5.9). When η > η the function 1/Fζ) reaches a maximum at the point ζ = η on the interval ζ η. The situation is completely analogous to the aboveconsidered case of suction and asymptotics of the function ϕ can be given by formula 5.). After integration we obtain FLUID DYNAMICS Vol. 46 No η ϕη, Pr)dη Pr 1 η Fη) dη F η).

12 576 VIGDOROVICH, LEONT EV Thus, the first of integrals 5.9) makes the main contribution to asymptotics of the function BPr) and the asymptotic representation of the restoration coefficient based on formula 3.6) have the form: r F b Fm F ) Pr π η m Pr 1/, Pr. 5.1) From formulas 5.8), 5.1) it follows that for injection the Reynolds analogy coefficient decreases exponentially, while the restoration coefficient increases exponentially with the Prandtl number Pr. 6. ASYMPTOTICS FOR SMALL PRANDTL NUMBERS AsPr, the neighborhood of the point η = 1 makes the main contribution to integral 3.5). As follows from Eq. 3.5), Fη) gη), η 1, g = 1 η) ln1 η). We will now represent integral 3.5) in the form: APr)=F 1 Pr ] [Fη) Pr 1 gη) Pr 1 dη + F 1 Pr gη) Pr 1 dη. The first integral converges when Pr = and after the change of variable η = 1 e x the second integral takes the form: ) Pr + 1 gη) Pr 1 dη = Pr 1 e Prx x Pr 1)/ dx = Pr 1 Γ Pr Pr+1)/. 6.1) Hence for the Reynolds analogy coefficient we obtain ) 1 Pr Pr Pr 1)/ α = F Γ.5Pr + 1)) We will now represent function 3.4) as follows: + O1), Pr. 6.) BPr)= hη, Pr)dhη, Pr), hη, Pr)= Fη) Pr 1 dη. Hence after integration by parts we have BPr) +B Pr)=h1, Pr)h1, Pr). With allowance for relation 6.1), as Pr, passage to the limit in the equality gives ) Pr + 1 BPr) Pr 1 Γ Pr Pr+1)/ Fη) 1 Pr dη + O1). Hence for the restoration coefficient we obtain ) Pr + 1 r = Pr Γ Pr 1 Pr)/ Fη) 1 Pr dη + OPr), Pr. 6.3) FLUID DYNAMICS Vol. 46 No. 4 11

13 THE RESTORATION COEFFICIENT AND REYNOLDS ANALOGY 577 Fig. 4. Reynolds analogy coefficient as a function of the Prandtl number: curves 1 1 correspond to b = 1.47,.59,.35,.3,.93,,.8,.19,.36, and.6, respectively When Pr =, with allowance for the numerical value of the coefficient formula 6.3) has the form r.96 Pr. In practice, this coincides with representation.1). For the Reynolds analogy coefficient the approximation formula is invalid since, in accordance with asymptotics 6.), α as Pr 1/. We note that the asymptotic representations 4.5), 4.6) obtained as b and for an arbitrary Prandtl number are not in agreement with the representations 6.), 6.3) valid as Pr and for an arbitrary b. This means that the expansions 4.5), 4.6) are not uniformly suitable as Pr. Integral 3.4) has one more singularity, namely, in the case of critical injection when F = b η + Oη ), η, it converges when Pr =. We will represent the function 3.4) in the form: BPr)= { Fη) Pr 1 ] [Fζ) 1 Pr b ζ) 1 Pr + b ) 1 Pr Pr η Pr Fη) Pr 1 dη. } dζ dη 6.4) The first integral in 6.4) converges and the second term tends to infinity as Pr. Consequently, in the case of critical injection, as Pr the restoration coefficient tends to infinity as Pr) 1. In Figs. 4 and 5 we have plotted the coefficients α and r as functions of the Prandtl number for various values of the injection and suction parameter. In accordance with the asymptotic representation 6.), as Pr the Reynolds analogy coefficient tends to infinity Fig. 4a). The larger the parameter B, the faster increase in α. To the contrary, in Fig. 5a the curve corresponding to critical injection lies below all other curves, in this case r Pr to a high accuracy everywhere, except for a very small neighborhood of zero. In Fig. 4b curve 6 corresponds to the impermeable plate; the curves corresponding to suction lie above this curve. As mentioned above, as Pr they tend to a finite limit. The curves corresponding to injection tend exponentially to zero and lie below curve 6. The parameter 6 affects even stronger the behavior of the restoration coefficient when Pr > 1 Fig. 5b). For suction, as the Prandtl number increases, the quantity r tends to a finite limit. The smaller the parameter b, the lower the limit. For injection the restoration coefficient tends exponentially to infinity; in this case, in accordance with 5.1), the exponent increases FLUID DYNAMICS Vol. 46 No. 4 11

14 578 VIGDOROVICH, LEONT EV Fig. 5. Restoration coefficient as a function of the Prandtl number: curves 1 14 correspond to b =.6,.46,.36,.4,.19,.1,.8,.4,,.93,.3,.58, 1.47, and 7.9, respectively. indefinitely with the injection parameter so that for critical injection the function rpr) exists only on the finite interval < Pr <. Summary. The classical formulas have a bounded interval of applicability and are invalid for the restoration coefficient r at large Prandtl numbers and for the Reynolds analogy coefficient α at small Prandtl numbers. In these cases the quantities α and r have power-law asymptotics with other exponents. Injection and suction affect significantly α and r and modify the behavior of these coefficients when Pr > 1. As Pr, for suction the functions αpr) and rpr) tend to a finite limit and for injection rpr) increases exponentially, while αpr) decreases exponentially. In the case of critical injection r Pr on the interval < Pr 1, and, as Pr, the function rpr) tends to infinity so that for critical injection the restoration coefficient exists only on the interval < Pr<. The work was carried out with support from the Russian Foundation for Basic Research projects No ). REFERENCES 1. L. Howarth et al. Eds.), Modern Developments in Fluid Dynamics High Speed Flow, Vol. 1 Oxford, 1953; Izd-vo Inostr. Lit., Moscow, 1955).. A.I. Leont ev, Gasdynamic Method of Energy Separation of Gas Streams, Teplofizika Vysokikh Temperatur 35, No. 1, ). 3. A.I. Leont ev, Temperature Stratification of a Supersonic Gas Stream, Dokl. Ross. Akad. Nauk 354, No. 4, ). 4. G.G. Chernyi, Laminar Gas and Liquid Flows in a Boundary Layer with a Discontinuity Surface, Izv. Akad Nauk SSSR, OTN, No. 1, ). 5. G.G. Chernyi, Boundary Layer with a Discontinuity Surface. Flow past a Plate with Fluid Seepage through its Surface, Dokl. Akad Nauk SSSR 1, No. 5, ). 6. E. Janke, F. Emde, and F. Lösch, Tafeln Hönerer Funktionen Teubner Verlag, Stuttgart, 196; Nauka, Moscow, 1977). FLUID DYNAMICS Vol. 46 No. 4 11

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