Transp Porous Med (2013) 99:413–425 DOI 10.1007/s11242-013-0193-3
Mixed Convection Boundary-Layer Flow on a Vertical Surface in a Porous Medium with a Constant Convective Boundary Condition J. H. Merkin · Y. Y. Lok · I. Pop
Received: 1 May 2013 / Accepted: 27 May 2013 / Published online: 13 June 2013 © Springer Science+Business Media Dordrecht 2013
Abstract The mixed convection boundary-layer flow on a vertical surface heated convectively is considered when a constant surface heat transfer parameter is assumed. The problem is seen to be chararterized by a mixed convection parameter γ . The flow and heat transfer near the leading edge correspond to forced convection solution and numerical solutions are obtained to determine how the solution then develops. The solution at large distances is obtained and this identifies a critical value γc of the parameter γ . For γ > γc a solution at large distances is possible and this is approached in the numerical integrations. For γ < γc the numerical solution breaks down at a finite distance along the surface with a singularity, the nature of which is discussed. Keywords Porous media · Boundary-layer flow · Mixed convection · Convective boundary condition
1 Introduction Convective boundary-layer flows in porous materials have many practical applications and, as a consequence, have received much attention through both theoretical and experimental studies, see the recent books by Vadász (2008), Vafai (2010), Nield and Bejan (2006) and Ingham and Pop (2005) for example. Mixed convection is an important aspect of this general area in which the flows generated by the buoyancy forces (natural convection) are comparable
J. H. Merkin (B) Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK e-mail:
[email protected] Y. Y. Lok Mathematics Division, School of Distance Education, Universiti Sains Malaysia, 11800 Pulau Pinang, Malaysia I. Pop Department of Mathematics, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania
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with the imposed outer flows (forced convection). Cheng (1977) was the first to treat the mixed convection flow on a vertical surface held at a constant temperature. Merkin (1980, 1985) later extended Cheng’s results for aiding flows to opposing flows. This situation is characterized by a mixed convection parameter and in this latter case a critical value of this parameter was identified, limiting the range of existence of solutions for opposing flow and giving dual solutions where these solutions existed. This basic problem has been extended in many different ways including different flow geometries, stagnation points and horizontal surfaces for example, and to a variable surface temperature and a variable surface heat flux. Much of this previous work on mixed convection in porous media has assumed either a prescribed surface temperature or a prescribed surface heat flux. More recently the idea of a convective surface condition has been suggested. Here the surface is assumed to be heated through one side of the surface being maintained at (usually) a constant temperature T f (say) with the other side of the surface, the one in contact with the flow, at a (variable) temperature Ts (say) providing a surface heat flux at a rate proportional to (T f − Ts ). This is a specific case of more general temperature-dependent boundary conditions treated by Merkin and Pop (2010). This concept of a convectively heated surface has been applied to several flow configurations where it has been seen that by applying a convective surface condition, significant differences can arise from the results reported previously when a prescribed surface temperature or heat flux had been applied. Here we re-consider the basic problem of a uniform flow over a vertical surface embedded in a porous material. Now we assume that there is a convective surface boundary condition. In a previous paper (Lok et al. 2013) we considered this set up but assumed that the heat transfer coefficient in the surface condition had a spatial variation to allow the system to be reduced to similarity form. Here we build on this by assuming that the heat transfer coefficient is constant, thus precluding a similarity solution. We find that our problem involves the single dimensionless parameter γ , which can be regarded as a mixed convection parameter, and for large γ the problem reduces to the one treated previously (Merkin 1985). The flow starts at the leading edge with the solution developing from the forced convection limit and, when we consider the flow at large distances, a critical value of γc of γ is found, previously identified in Merkin (1980) and Klemp and Acrivos (1976). The flow can proceed to large distance along the surface only if γ > γc . Otherwise for opposing flow of sufficient strength the solution breaks down through a singularity at a finite distance along the plate. We start by describing our model.
2 Equations We consider the steady mixed convection boundary-layer flow along a vertical surface embedded in a saturated porous material with a constant outer flow U0 and ambient temperature T0 . We assume that the surface in contact with the porous medium is heated convectively through the other side of the surface being maintained at constant temperature T f . We further assume that the flow in the porous material is described by Darcy’s law and we make the standard Boussinesq approximation. The governing equations are then, see Vadász (2008), Nield and Bejan (2006), Lok et al. (2013) for example, ∂u ∂v + =0 ∂x ∂y gβ K (T − T0 ) u = U0 + ν
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(1) (2)
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u
415
∂T ∂2T ∂T +v =α 2 ∂x ∂y ∂y
(3)
subject to the boundary conditions v = 0,
∂T = −h s (T f − T ) on y = 0, u → U0 , T → T0 as y → ∞ ∂y
(4)
Here x and y are the Cartesian coordinates measured along the surface and normal to it, u and v are, respectively, the velocity components in the x and y directions, T is the fluid temperature, g is the acceleration due to gravity, K is the permeability of the porous medium, β is the coefficient of thermal expansion, ν is the kinematic viscosity of the fluid and α is the effective thermal diffusivity of the porous medium. h s is the constant surface heat transfer coefficient, where the thermal conductivity of the surface has been subsumed into h s . We make Eqs. (1)–(4) dimensionless using the outer flow U0 to characterize the fluid temperature, namely we put u = U0 u, v = h s α v, x =
h 2s α x, U0
y = h s y, T − T0 =
U0 ν θ gβ K
(5)
The equations become, on dropping the bars for convenience, ∂ψ = 1+θ ∂y ∂ψ ∂θ ∂ψ ∂θ ∂ 2θ − = ∂y ∂x ∂x ∂y ∂ y2
(6) (7)
now subject to the boundary conditions ψ = 0,
∂θ = θ − γ on y = 0, ∂y
∂ψ → 1, θ → 0 as y → ∞ ∂y
(8)
where ψ is the (dimensionless) stream function (defined in the usual way). The condition in (8) that ψ = 0 on y = 0 assumes that the surface is impermeable and is derived from the condition that v = 0, or ∂ψ/∂ x = 0, on y = 0 for all x ≥ 0. The dimensionless parameter γ is given by γ =
gβ K T νU0
where T = (T f − T0 )
(9)
We note that γ can be positive (when T f > T0 ) or negative (when T f < T0 ). We can express (7) and (8) as, on using (6), ∂ψ ∂ 2 ψ ∂ψ ∂ 2 ψ ∂ 3ψ − = ∂ y ∂ x∂ y ∂ x ∂ y2 ∂ y3
(10)
subject to ψ = 0,
∂ 2ψ ∂y
2
=
∂ψ − (γ + 1) on y = 0, ∂y
∂ψ → 1 as y → ∞ ∂y
(11)
When γ = 0, Eqs. (10) and (11) have the simple solution ψ = y, θ = 0 and hence we exclude this possibility from our discussion. Before considering the general problem we note that the behaviour for γ large is determined through the transformation ˜ ψ = γ −1 ψ,
y˜ = γ y, x˜ = γ 2 x
(12)
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This leaves Eq. (10) essentially unaltered with boundary conditions (11) now being ˜ ∂ 2 ψ˜ ∂ ψ˜ −1 ∂ ψ ˜ = −1 + γ ψ = 0, −1 on y˜ = 0, → 1 as y˜ → ∞ 2 ∂ y ˜ ∂y ∂ y˜
(13)
The leading-order problem, obtained by letting γ → ∞ in (13), has been treated by Merkin (1980). It is the problem given by (10) and (11) for general values of γ that we now consider, starting with a solution valid for small x. 3 Solution for Small x To obtain a solution to (10) and (11) valid for small x we make the transformation ψ = (2x)1/2 f (x, η), η =
y (2x)1/2
Equations (10) and (11) then become ∂ f ∂2 f ∂3 f ∂2 f ∂ f ∂2 f + f = 2x − ∂η ∂ x∂η ∂ x ∂η2 ∂η3 ∂η2 ∂f ∂2 f 1/2 − (γ + 1) on η = 0, = (2x) f = 0, ∂η ∂η2
(14)
(15) ∂f → 1 as η → ∞ ∂η
(16)
Expression (16) suggests an expansion of the form f (x, η) = η + (2x)1/2 f 1 (η) + · · ·
(17)
where f 1 satisfies f 1 + η f 1 − f 1 = 0,
f 1 (0) = 0, f 1 (0) = −γ , f 1 → 0 as η → ∞
(18)
and where primes denote differentiation with respect to η. The required solution to (18) is ⎞ √ ⎛ ∞ 2 γ 2 2 (19) f 1 = √ ⎝e−η /2 − η e−s /2 ds ⎠ π η
Hence 2γ 2γ u w ∼ 1 + √ x 1/2 + · · · , θw ∼ √ x 1/2 + · · · , τw ∼ −γ + · · · (20) π π ∂ψ for x small, where u w = and θw = θ y=0 , with θw = u w − 1, are, respectively, the ∂ y y=0 ∂ 2ψ wall velocity and temperature and where τw = is the skin friction. ∂ y 2 y=0 We can extend these results noting that, by perturbing about the forced convection solution ψ = y, Eq. (7) becomes approximately ∂θ ∂ 2θ = 2 ∂x ∂y
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(21)
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subject to the boundary condition on θ given in (8) with the corresponding solution √ √ √ θ = γ erfc(η/ 2) − ex+y erfc(η/ 2 + x
(22)
where η is given in (14). Expression (22) gives
√ √ θw = γ 1 − ex erfc( x) , τw = −γ ex erfc( x) By expanding the expressions in (22) for x small we can extend (20) to √ 1/2 πx 2γ x 1/2 2x 1/2 1− + · · · , τw ∼ −γ 1 − √ + · · · θw ∼ √ 2 π π
(23)
To obtain a solution valid for larger values of x we need to solve (10) and (11) numerically which is what we consider next.
4 Numerical Solution To obtain our numerical solution we first make transformation (14) and it is then that we solve Eqs. (15) and (16) numerically. We use the same numerical technique that we have used previously, see Mealey and Merkin (2008), Merkin (2009, 2012) for example. For this we use u = ∂ f /∂η as our dependent variable with the numerical grid including points on η = 0. Thus both u w and θw are calculated directly as part of the numerical scheme. We used ξ = (2x)1/2 as our streamwise variable to smooth out the x 1/2 singularity near x = 0 seen in expansion (17). Hence we numerically integrate ∂ f ∂2 f ∂3 f ∂2 f ∂ f ∂2 f + f =ξ − (24) ∂η ∂ξ ∂η ∂ξ ∂η2 ∂η3 ∂η2 subject to f = 0,
∂2 f ∂η
2
=ξ
∂f − (γ + 1) on η = 0, ∂η
∂f → 1 as η → ∞ ∂η
(25)
In Fig. 1 we plot τw and θw against ξ for representative values of γ (labelled on the figure), the plots of τw for γ = −1.0 and −1.3 in Fig. 1a are almost identical. We see that both skin friction τw and surface temperature θw start with the values given in expansion (20) and approach constant values for large values of ξ (or x), with τw approaching zero in each case. We comment that the numerical integrations were continued to very much larger values of ξ than that used to plot Fig. 1. This approach to a constant value becomes slower as the value of γ is increased. We further note that our numerical results indicate that θw appears to approach the value of γ for ξ large. This leads us to consider the nature of the solution for x large. 5 Solution for Large x Our numerical solutions indicate that transformation (14) is also the appropriate one to use when looking for a solution for x large. We then look for a solution of Eqs. (24) and (25) valid for x, or ξ , large by expanding
f (η; ξ ) = f 0 (η) + ξ −1 f 1 (η) + ξ −2 φ2 (η) log ξ + f 2 (η) + · · · (26)
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(a)
(b)
Fig. 1 Plots of a τw and b θw against ξ = (2x)1/2 obtained from the numerical solution to (24) and (25) for γ = 4.0, 1.0, −1.0, −1.3
The leading-order problem is f 0 + f 0 f 0 = 0,
f 0 (0) = 0, f 0 (0) = (1 + γ ), f 0 → 1 as η → ∞
(27)
The problem given in (27) has arisen previously Klemp and Acrivos (1976), Merkin (1980, 1985), where it was seen that there was a critical value γc of γ , where γc −1.3541, with solutions to (27) possible only for γ ≥ γc . Thus we expect the solution to develop to large x only when γ ≥ γc . When γ = −1 we recover the Blasius solution (Rosenhead 1963) with f 0 (0) = 0.46960. When γc < γ < −1 there are dual solutions to (27) and our numerical integrations for γ in this range approach the corresponding upper branch solution as ξ increases, where the upper solution branch is that one which continues the Blasius solution for γ = −1. This is, perhaps, to be expected as it was shown in Merkin (2012) that this upper solution branch was the temporally stable branch.
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At O(ξ −1 ) we have f 1 + f 0 f 1 + f 0 f 1 = 0,
f 1 (0) = 0, f 1 (0) = f 0 (0), f 1 → 0 as η → ∞
(28)
Equation (28) can be integrated to get, for γ ≥ γc , f 1 = f 0 − (γ + 1)
(29)
from which it follows that f 1 (0) = f 0 (0) = 0. At O(ξ −2 ) we find that the equation has a complementary function (η f 0 − f 0 ) arising from the leading edge shift effect (Stewartson 1964) that satisfies all the required boundary conditions. Thus we need to modify expansion (26) to include the term in log ξ at this order Stewartson (1955). Then φ2 = a2 (η f 0 − f 0 ) for some constant a2 , which we expect to depend on γ , with f 2 then satisfying, on applying (29), f 2 + f 0 f 2 + 2 f 0 f 2 − f 0 f 2 = − f 0 2 + a2 f 0 f 0
f 2 (0) = 0, f 2 (0) = f 1 (0) = 0, f 2 → 0 as η → ∞
(30)
Following from Stewartson (1955) Eq. (30) has a solution which satisfies all the required boundary conditions only if the constraint ∞ a2 0
f 02 f 0 dη =
∞
f 0 f 0 2 dη
(31)
0
holds. It is expression (31) that then determines a2 . We note that the solution at this stage is not unique as any arbitrary multiple of φ2 can be added. We can evaluate the integrals in (31) to give, on using (27),
a2 1 − (1 + γ )2 = f 0 (0)2 (32) When γ = −1 and we have the Blasius solution, expression (32) gives a2 = 0.2205. From (26), (28) we have 1 −1 −1/2 −1 τw ∼ f 0 (0) + O(x log x) , θw ∼ γ + f 0 (0) (2x) + O(x log x) (2x)1/2 (33) as x → ∞, consistent with the results shown in Fig. 1. 6 Solution for γ <γ c As noted above we expect the numerical solution to not continue to large x when γ < γc and we find that this is the case. We now find that the numerical integration breaks down at a finite value xs of x, where xs depends on γ , and could not be continued past this point. The numerical procedure involved a procedure for halving the step size ξ in an attempt to maintain overall accuracy. As the numerical integration approached x s , ξ had to be successively halved reaching a value of 1.25 × 10−4 before terminating. The numerical results suggest that the solution is approaching a singularity as x approaches x s . We illustrate this in Figs. 2a and 2b with plots of τw and θw against x for representative values of γ (labelled on the figure). These plots show that xs depends on γ and decreases as the value of |γ | is increased.
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(a)
(b)
(c)
Fig. 2 Plots of a τw , b θw and c δ = y − ψ∞ (x) against x obtained from the numerical solution to (24) and (25) when γ < γc for γ = −2.0, −2.5, −3.0, −4.0
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6.1 Solution Near the Singularity To discuss the nature of the solution for x close to xs we put X = xs − x and look for a solution valid for X small. We then put ψ = X 2/3 F(X, ζ ), ζ =
y
(34)
X 1/3
Equation (10) then becomes ∂3 F ∂ζ 3
2 ∂2 F 1 − F + 3 ∂ζ 2 3
∂F ∂ζ
2 =X
∂ F ∂2 F ∂ F ∂2 F − ∂ X ∂ζ 2 ∂ζ ∂ X ∂ζ
(35)
subject to F = 0,
∂2 F ∂ζ 2
= β + X 1/3
∂F ∂ζ
on ζ = 0,
where β = −(1 + γ ) > 0
(36)
The outer boundary conditions are relaxed at this stage. Boundary conditions (36) suggest an expansion of the form F(ζ ; X ) = F0 (ζ ) + X 1/3 F1 (ζ ) + X 2/3 F2 (ζ ) + X F3 (ζ ) + · · ·
(37)
We find that the leading-order solution is F0 =
β 2 ζ 2
(38)
At O(X 1/3 ) we then obtain F1 −
βζ 2 F + βζ F1 − β F1 = 0, 3 1
F1 (0) = 0, F1 (0) = F0 (0) = 0
(39)
on using (38). Primes now denote differentiation with respect to ζ . Equation (39) has the complementary functions F1 = ζ and βζ 3 + 6 together with a solution that is exponentially large for ζ large. Hence to satisfy the boundary condition on ζ = 0 we can only have F1 = b1 ζ
for some constant b1 .
(40)
At O(X 2/3 ) we then have F2 −
2b2 βζ 2 4βζ 4β F2 + F2 − F2 = − 1 , 3 3 3 3
F2 (0) = 0, F2 (0) = F1 (0) = b1
(41)
b12 and a complementary function ζ together with 2β a further complementary function (not exponentially large at infinity) that can be expressed in terms of a confluent hypergeometric function (Slater 1960). To determine this we put dφ F2 = ζ φ and find that = satisfies the equation dζ βζ 3 5 2 s + − s + = 0 where s = (42) 3 3 9 Equation (41) has a particular integral
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and here primes denote differentiation with respect to s. Equation (42) has the solution (as defined in Slater 1960) = U (− 23 ; 53 ; s) not exponentially large at infinity. From this we can obtain a complementary function F 2 which has 2β 3 β 5 2 ζ + · · · − B0 ζ − ζ + ··· F 2 ∼ A0 1 + for ζ small (43) 9 90 where A0 =
2 2 3 3 1 3
−2/3 β 9
1.4580 β −2/3
and B0 =
1 2 3 3 2 3
1.3189 and
24 1 β 2/3 F 2 ∼ C0 ζ 4 − ζ log ζ + · · · 0.07704 β 2/3 as ζ → ∞, where C0 = β 3 9 (44) Hence our solution is F2 =
b12 + b2 ζ + a 2 F 2 2β
(45)
for constants a2 and b2 . On applying the boundary conditions in (41) and using (43) we find b2 that 1 + A0 a2 = 0 and −2B0 a2 = b1 giving 2β 2 2 β A0 1/3 3 b1 = = 3(3β) (46)
1 1.1055 β 1/3 B0 3 and then β A0 9(3β)1/3 a2 = − 2 = − 4 2B0
3 23
−0.4191 β 1/3 13
(47)
The constant b2 is not determined at this stage. We can continue to O(X ) here we find
βζ 2 5βζ 5β 5 5 F3 + F3 − F3 = a2 b1 ζ F 2 − F 2 − b1 b2 , F3 (0) = 0, 3 3 3 3 3 (48) F3 (0) = b2
F3 −
where b1 and a2 are given, respectively, in (46) and (47) and noting that F 2 (0) = 0. Equation (48) has the complementary functions ζ and (βζ 5 −60ζ 2 ) and a particular integral of b1 b2 /β for the constant term on the right-hand side of (48). To deal with the terms in F 2 we note that
ζ F2 −
5 16C0 3 4B0 2β A0 2 F2 ∼ ζ + · · · for ζ large and ∼ ζ+ ζ +· · · for ζ small 3 3 3 9 (49)
where C0 is given in (44). This gives rise to a particular integral φ3 which has 4b1 a2 C0 3 ζ + · · · as ζ → ∞ β
(50)
2a2 b1 A0 2 ζ + O(ζ 4 ) for ζ small 9
(51)
φ3 ∼ and has φ3 ∼
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Then F3 = b3 ζ + a3 (βζ 5 − 60ζ 2 ) +
b1 b2 + φ3 β
(52)
b1 A 0 . Satisfying the boundary conditions on ζ = 0 gives b2 = 0 and then a3 = a2270 The outer boundary condition is not satisfied at this stage so Eqs. (35) and (36) with expansion (37) give the solution in an inner region and to complete the solution we require an outer region. To discuss this we use (34) to write the solution in the inner region for ζ large as
β 4b1 a2 C0 3 ψ = y 2 + a2 C0 y 4 + · · · + X 2/3 b1 y + (53) y + · · · ) + O(X log X, X ) 2 β
where the constants in (53) are given above. Expansion (53) suggests that, in the outer region, we put ψ(X, y) = ψ0 (y) + X 2/3 ψ1 (y) + · · ·
(54)
The function ψ0 is not determinable from this procedure only that ψ0 (y) ∼
β 2 y + a2 C0 y 4 + · · · as y → 0, ψ0 (y) → 1 as y → ∞ 2
(55)
At O(X 2/3 ) we have ψ0 ψ1 − ψ0 ψ1 = 0 giving ψ1 =
b1 ψ β 0
(56)
on using expression (53). The reason why we are not able to determine fully the solution in the outer region arises from the fact that this is an asymptotic expansion in the sense described in Stewartson (1955). The leading-order term ψ0 in expansion (54) has to reflect how the solution evolved from the leading edge at x = 0 and is not determined purely by ‘local’ considerations near the singularity at X = 0. This feature is not entirely unexpected as similar behaviour is seen in the Goldstein–Stewartson separation theory Stewartson (1958). From (34), (37) we have τw ∼ −(1 + γ ) +
β A0 2/3 β A0 2/3 X + · · · , θw ∼ −1 + X + ··· B0 B0
(57)
as X → 0 (or as x → xs ). Also from the outer boundary condition in (8), ψ ∼ y − δ(x) as y → ∞ and in Fig. 2c we plot δ against x for the same values of γ . From (54) and (56), δ ∼ δ0 −
b1 2/3 X + · · · as X → 0 β
(58)
where δ0 = lim (y − ψ0 (y)) with δ0 depending on γ . Thus we expect τw , θw and δ to y→∞
approach, respectively, the limits of −(1 + γ ), −1 and δ0 in a singular way as X → 0 (or as x → xs ). The results plotted in Fig. 2a, b show that, at least for the smaller values of γ i.e. for values of γ a little below the critical value γc , the numerical solution can proceed a little way beyond its asymptotic limit, as given by (57) and (58) before breaking down. For larger values of β, i.e. γ further from γc , the numerical solution appears to break down a little before the singularity is reached, as can be seen in the plots for γ = −4.0. The singularity seen in the numerical solution as x → xs is not fully apparent from the plots in Fig. 2, which seem to suggest a weak singularity at xs with the term of O(X ) playing a significant role.
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3/2
Fig. 3 A plot of u w against x for γ = −4.0 obtained from the numerical solution to (24) and (25)
7 Conclusions We have considered the mixed convection boundary-layer flow of a uniform stream over a vertical surface with the surface heated convectively. We assumed a constant surface heat transfer coefficient h s to extend a previous study (Lok et al. 2013) where a variable coefficient was taken so that the system could be reduced to similarity form. Here this reduction is not possible and we started by considering the flow and heat transfer near the leading edge. How the flow then developed from the leading edge depended on the parameter γ , defined in (9), which, through our non-dimensionalisation (5), can be regarded as a mixed convection parameter, the larger the value of γ the more significant is the applied heating. When we examined the nature of the flow at large distances from the leading edge, we saw that there was a critical value γc of γ , γc −1.3541 which had been determined in a previous problem (Klemp and Acrivos 1976; Merkin 1980, 1985). For γ > γc our numerical integrations of the full equations showed that the solution advanced to the appropriate asymptotic solution, as given by (27). This gave a boundary-layer flow the whole length of the surface for all aiding flow, γ > 0, and for opposing flow, γ < 0, provided that γ > γc , see Fig. 1. For opposing flow with γ < γc , our numerical solutions broke down at a finite value x s of x, where xs depends on γ and decreasing as |γ | is increased, and being unable to be continued any further, see Fig. 2. This break down in the numerical solution is similar to that reported in Merkin (1980) and, from (12) and the results given in Merkin (1980), we can estimate that xs ∼ 0.2875 |γ |−2 for |γ | large. This gives xs 0.018 for γ = −4.0, a little greater than the value we found of approximately 0.012. From (57) u w ∼ (β A0 /B0 ) (xs − x)2/3 as x → xs 3/2 and hence a plot of u w against x should show straight line behaviour as x approaches x s . 3/2 In Fig. 3 we plot u w against x for γ = −4.0 and x close to xs . This plot does indicate a straight line form near x = xs as predicted. However, the slope of the curve is somewhat greater than (β A0 /B0 )3/2 2.02 arising in the analysis. Acknowledgments Eqs. (22) and (23).
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We wish to thank a referee for pointing out how to extend our small x analysis to obtain
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