On the stability of a forced-free boundary layer flow with viscous heating

Fluid Dynamics Research 31 (2002) 65 – 78

On the stability of a forced-free boundary layer "ow with viscous heating Eunice W. Mureithi ∗ , David P. Mason School of Computational and Applied Mathematics, University of Witwatersrand, Private Bag 3, WITS 2050, Johannesburg, South Africa Received 26 September 2001; received in revised form 10 May 2002; accepted 11 May 2002 Communicated by J.M. Hyun

Abstract The inviscid instability of an accelerating forced-free convection boundary layer with viscous dissipation is investigated. The boundary layer equations for the "ow with free-stream velocity U∞ xn admit self-similar solutions for n = 1 only. The “overshooting” of the free-stream value, characteristic of accelerating buoyant boundary layers, is found to increase with increasing viscous heating. The scaled temperature pro7le is also found to exceed its plate value close to the plate where the viscous dissipation e9ects are strongest. For Eckert number Ec = 0 only one single inviscid unstable mode exists. For the "ow with viscous dissipation, three unstable modes have been identi7ed. The secondary modes may be associated with the combined e9ect of thermal buoyancy and viscous heating. There is evidence of these modes crossing at relatively higher wavenumbers. The disturbance growth rate is found to increase with increasing buoyancy and viscous heating. c 2002 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.
Keywords: Boundary layer; Viscous dissipation; Buoyancy; Super-velocities; In"ectional points; Eigenvalues; Eigenfunctions

1. Introduction Studies on thermal boundary layers have been on the increase due to their wide range of applications in engineering. The e9ects of thermal buoyancy on the stability of boundary layers include the early work of Gage and Reid (1968), who investigated the e9ects of thermal strati7cation in plane Poiseuille "ow, Gage (1971) who looked at strati7ed boundary layer "ows, and Wu and Cheng (1976) who used a quasi-parallel approach to look at the stability of a thermally strati7ed Blasius boundary layer to longitudinal vortices. Later work includes that of Hall and Morris ∗

Corresponding author. Tel.: 27-011-717-6127; fax: 27-011-403-9317. E-mail address: [email protected] (E.W. Mureithi).

c 2002 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. 0169-5983/02/$22.00  All rights reserved. PII: S 0 1 6 9 - 5 9 8 3 ( 0 2 ) 0 0 0 8 8 - 6

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(1993) who considered the linear stability of a heated boundary layer to disturbances in the form of longitudinal vortices and Hall (1993), who extended the work of Hall and Morris (1993) to the non-linear regime. Recent work by Mureithi et al. (1997) has investigated the e9ects of thermal buoyancy on the stability of upper branch modes of the curve of neutral stability using the method of matched asymptotic expansions. Their results indicate that at the large thermal strati7cation limit the 7ve-zoned disturbance structure of Smith and Bodonyi (1982) becomes modi7ed to a two-layered structure consisting of a boundary layer and a lower passive viscous wall layer. They showed further that the basic streamwise velocity has a characteristic of overshooting its free-stream value, attaining a well de7ned maximum value in the interior of the boundary layer. Little has been reported on the e9ect of viscous dissipation on the stability of a boundary layer. This may be because in most practical "ows the e9ect is negligible. There are some "ows, however, such as sonic "ows with small temperature di9erences between the plate and the free-stream and high-speed "ows of viscous "uids, where the e9ect of viscous dissipation may not be negligible and needs to be included in the energy balance equation. The early studies of the e9ect of viscous dissipation on a boundary layer "ow include the work of Gebhart (1962) who looked at free convection "ow over vertical surfaces and Gebhart and Mollendorf (1969) who obtained similarity type solutions for the same "ow with exponential wall temperature variation. A study of the e9ects of viscous dissipation on the stability of plane Poiseuille "ow where the lower and the upper plates are isothermally heated and cooled, respectively, was conducted by Cheng and Wu (1976). Their results indicate that viscous dissipation is destabilizing, the e9ect increasing with increase of Prandtl number. Recently, Barletta (1998) has looked at mixed convection in a fully developed channel "ow, where it is shown that for the case of asymmetric heating, viscous dissipation enhances the e9ect of "ow reversal in the case of downward "ow with the reverse occurring for the upward "ow. Here we wish to investigate the e9ect of viscous dissipation on a forced-free convection boundary layer over a horizontal "at plate. In Section 2 we formulate the problem. In Section 3 the steady boundary layer equations are presented and their similarity form is derived. In Section 4 the linearized disturbance equations for a locally parallel "ow are obtained and a parallel-inviscid stability analysis is performed. In Section 5 the numerical solution is obtained and the results are discussed. Finally, in Section 6 we draw some conclusions.

2. Formulation of problem Consider the two-dimensional "ow of a viscous, incompressible "uid over a heated semi-in7nite "at plate. The horizontal plate coincides with the x∗ -axis and the y∗ -axis is normal to the plate. The dimensional continuity, Navier–Stokes and energy balance equations governing the "ow, under the Boussinesq approximation, with dimensional quantities denoted by an asterisk, are @u∗ @v∗ + = 0; @x∗ @y∗
∗  ∗ ∗ @u @p∗ ∗ @u ∗ @u = − 0 + u + v + ∇∗2 u∗ ; @t ∗ @x∗ @y∗ @x∗

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67




 ∗ ∗ @v∗ @p∗ ∗ @v ∗ @v ˆ ∗ − T0 )) + ∇∗2 v∗ ; = − 0 + u + v − g0 (1 − (T @t ∗ @x∗ @y∗ @y∗
∗  ∗ ∗ @T ∗ @T ∗ @T = ∇∗2 T ∗ + ∗ ; +u +v 0 c p @t ∗ @x∗ @y∗ where g is the acceleration due to gravity, ∇∗2 is the two-dimensional Laplacian and ∗ is the dissipation function, expressed as
∗
∗ 2
∗ 2  @u @v∗ 2 @u @v ∗ + +2 +2 ;  = @y∗ @x∗ @x∗ @y∗ 0 is the density of the "uid at some properly chosen constant reference temperature T0 ;  is the coeKcient of shear viscosity, ˆ is the coeKcient of volume expansion,  is the coeKcient of thermal conductivity and cp is the coeKcient of speci7c heat of the "uid at constant pressure. Let U∞ denote the scaling for the free-stream speed, L a typical length scale measured from the plate’s leading edge and T∞ the free-stream temperature. The corresponding velocity, pressure and temperature 7elds of the "ow are taken to be (u∗ ; v∗ ) = U∞ (u; v);

2 p ∗ = 0 U∞ p;

T ∗ = T∞ + (T0 − T∞ )T

and the time, t ∗ , is non-dimensionalized using L=U∞ . The non-dimensional form of the governing equations of motion of the "uid become @v @u + = 0; @x @y @u @u @p 1 2 @u +u +v =− + ∇ u; @t @x @y @x Re Lg @v @v @v @p ˆ 0 − T∞ )) + GT + 1 ∇2 v; − 2 (1 + (T +u +v =− @t @x @y @y U∞ Re @T @T 1 @T Ec +u +v = ∇2 T + : @t @x @y Pr Re Re

(2.1)

The parameters of the "ow are the Reynolds number, Re, buoyancy parameter, Gr, the Prandtl number, Pr and the Eckert number, Ec, de7ned as Re =

U∞ L ; 

Gr =

ˆ 0 − T∞ ) L3 g(T ; 2

Ec =

2 U∞ ; cp (T0 − T∞ )

Pr =

cp ; 

where  = =0 . The Grashof number, Gr, is a measure of the e9ects of free convection or buoyancy in a given "ow. The Eckert number is the viscous dissipation parameter. We also de7ne G=

Gr Re2

which is a buoyancy related parameter.

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E.W. Mureithi, D.P. Mason / Fluid Dynamics Research 31 (2002) 65 – 78

3. The steady boundary layer equations In the limit of large Reynolds number, a velocity boundary layer forms. We will assume that the Prandtl number is of order unity so that the thermal boundary layer will approximately be of the same thickness as the velocity boundary layer. We have the scalings y = Re−1=2 Y;

v = Re−1=2 v; M

T = TM ;

u = u;

˜ p = P(y) − 12 Ue (x)2 + p(x; M Y );

where gL d P˜ ˆ 0 − T∞ )] = − 2 [1 + (T dy U∞ is the pressure gradient term which balances the gravitational forces, Ue is the free-stream speed scaled by U∞ and p(x; M Y ) is the buoyancy induced pressure. Substituting the foregoing expansions into the governing equations and taking the formal limit Re → ∞, we obtain the buoyant boundary layer equations @u @vM + = 0; @x @Y u

@u dUe @pM @u @2 u ; + vM = Ue − + @x @Y dx @x @Y 2

@pM = G Re−1=2 TM ; @Y @TM 1 @2 TM @TM + vM = u + Ec @x @Y Pr @Y 2




@u @Y

2

;

(3.2)

which are solved subject to the boundary conditions u = vM = TM − TM p (x) = 0 u → Ue (x);

TM → 0

at Y = 0;

(3.3)

as Y → ∞;

(3.4)

where TM p (x) is the non-dimensional prescribed temperature distribution on the plate. The boundary layer stream-wise velocity is accelerated by the e9ect of the favourable pressure gradient @p=@x M ¡ 0, which is negative. The e9ect is further magni7ed by the presence of the buoyancy induced pressure gradient @p=@Y M . If G Re1=2 , then the buoyancy induced pressure gradient is negligible. Consider now similarity solutions with Ue (x) = xn and TM p (x) = x(5n−1)=2 . The boundary layer equations (3.2) and the boundary conditions (3.3) and (3.4) admit a family of self-similar solutions of the form u = xn f ( );

TM = x(5n−1)=2 h( );

pM = x2n q( );

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69

where = Yx(n−1)=2 is the similarity variable. The resulting similarity form of the boundary layer equations is f = n(f )2 − 12 (n + 1)ff + n(2q − 1) + 12 (n − 1) q ; h =

Pr [(5n − 1)hf − (n + 1)h f] − x(n−1)=2 Pr Ec(f )2 ; 2

q = G0 h;

(3.5)

where f ; h, and q are, respectively, the scaled functions for the stream-wise velocity, temperature and the pressure distribution in the boundary layer. The boundary conditions to be satis7ed are f(0) = f (0) = 0;

h(0) = 1;

f (∞) = 1;

h(∞) = 0;

q(∞) = 0:

(3.6)

G Re−1=2 .

The primes denote di9erentiation with respect to and G0 = If G0 = O(1), then the asymptotically small buoyancy induced pressure gradient becomes of order one. When G0 = 0, the above equations have self-similar solutions for all values of n. A similarity solution which accommodates both the buoyancy and the viscous heating e9ects is possible only if n = 1. For these Falkner–Skan type "ows, the case when n = 1 physically represents stagnation point "ow. The solution of the self-similar boundary layer equations (3.5) together with their boundary conditions (3.6) is discussed in Section 5. 4. The stability equations The stability analysis is carried out by assuming that the "ow is locally parallel and the disturbance amplitude is small. The perturbed "ow can be expressed as a superposition of the basic "ow and of a "uctuating quantity. Under the parallel "ow approximation, vM = 0 we have q(x; y; t) = q(y) M + Q(x; y; t); where qM stands for the basic "ow terms and Q denotes the perturbations. The linearized disturbance equations become @U @V + = 0; @x @y du @P 1 2 @U @U +V =− + ∇ U; +u @t @x dy @x Re @P 1 2 @V @V +u =− + G& + ∇ V; @t @x @y Re d TM 1 @& Ec du @& +u +V = ∇2 & + 2 @t @x dy Pr Re Re dy




@V @U + @x @y

 ;

(4.7)

which are the equations governing the small amplitude perturbations, with U; V; P and & being the stream-wise velocity, normal velocity, pressure and the temperature perturbations, respectively.

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E.W. Mureithi, D.P. Mason / Fluid Dynamics Research 31 (2002) 65 – 78

Under the parallel "ow approximation, we have the boundary layer scalings (X; Y; () = Re1=2 (x; y; t) and for the thermally coupled temperature and momentum 7elds the buoyancy parameter expands as G = Re1=2 G0 . We seek traveling wave type solutions where the perturbations take the form (U; V; P; &)(X; Y; () = (U0 ; V0 ; P0 ; &0 )(Y )exp[i)X − i!(]; where ) and ! are the wave number and the wave frequency, respectively. For temporal stability analysis, ) is assumed to be real-valued while ! is complex-valued. We say that the waves are temporally ampli7ed if !i ¿ 0, where !i is the imaginary part of !. Otherwise the waves are said to be damped. Substituting the above expansions into the system of equations (4.7) yields an eigensystem of the form 1 (u − c)(D2 − )2 )V0 − u V0 = (D2 − )2 )2 V0 + i)G0 &0 ; (4.8) i)Re1=2 1  1 2u Ec (D2 − )2 )&0 + 2 1=2 (D2 + )2 )V0 ; (u − c)&0 + TM V0 = 1=2 i) i)Re Pr ) Re (4.9) where d : dY This is a sixth order Orr-Sommerfeld type equation with c = !=) as the complex wave-speed. The boundary conditions to be satis7ed are D=

V0 = V0 = &0 = 0

on Y = 0

and the requirement that the disturbances decay exponentially as Y → ∞. We now carry out a parallel-inviscid stability analysis where, in the formal limit Re → ∞, the inverse powers of the Reynolds number are neglected. Hence, the sixth order eigensystem (4.8) and (4.9) reduces to the Taylor–Goldstein equation,    d 2 V0 u G0 TM 2 V0 = 0; − ) + − dY 2 u − c (u − c)2 which is solved subject to the boundary conditions V0 = 0

at Y = 0; ∞:

The basic "ow terms are determined by the self-similar boundary layer equations (3.5). At this limit, the viscous dissipation term does not explicitly appear in the disturbance equations. The Taylor– Goldstein equation is transformed to similarity form by rede7ning c = xn cˆ and ) = x(n−1)=2 ). ˆ This yields 
G0 g  d 2 V0 f 2 −  V0 = 0 − )ˆ +  (4.10) d 2 (f − c) ˆ (f − c) ˆ2 where

is the similarity variable. This equation is solved subject to the boundary conditions V0 = 0

at

= 0; ∞:

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71

The eigenvalues and eigenfunctions are determined by numerical integration of the Taylor–Goldstein equation as discussed in the next section. 5. Numerical methods and discussion of results The boundary layer equations considered here admit similarity solutions only for n = 1. The similar equations (3.5) together with the boundary conditions (3.6) were solved using a fourth order Runge–Kutta scheme with an Newton iteration. 1.4 G0=5 1.2 G0=3 1

G0=1

G0=0

f

′

0.8

0.6

0.4

0.2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3.5

4

4.5

5

scaled normal distance, η 1.2

1

h

0.8

0.6

0.4

G0=0

G0=5

0.2

0 0

0.5

1

1.5

2

2.5

3

scaled normal distance, η

Fig. 1. Plots of similarity solution f ( ) and h( ) for Ec = 1 and for G0 = 0; 1; 3; 5.

72

E.W. Mureithi, D.P. Mason / Fluid Dynamics Research 31 (2002) 65 – 78 1.4 Ec=1 1.2 Ec=0 1

f

′

0.8

0.6

0.4

0.2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

2.5

3

3.5

4

4.5

5

scaled normal distance, η

1.2

1

h

0.8

0.6 Ec=1 0.4 Ec=0 0.2

0

0

0.5

1

1.5

2

scaled normal distance, η

Fig. 2. Plots of similarity solution f ( ) and h( ) for G0 = 5 and Ec = 0; 0:5; 1.

The results presented are for gas "ow (Pr = 0:7). Fig. 1 shows the e9ect of increasing the buoyancy related parameter G0 . As G0 increases there is increase in the “super-velocities”, that is, the velocities that exceed the free-stream value of unity, in the boundary layer. The addition of the viscous dissipation term is seen to enhance the velocity overshoot as shown in Fig. 2. The stream-wise velocity pro7les have in"ection points, and hence the basic "ow is susceptible to inviscid type of instabilities. We note that for Ec = 1 and G0 ¿ 5 there is a small region adjacent to the plate where the scaled temperature pro7le exceeds its plate temperature. This is because the viscous

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73

1.4

disturbance growth rate, α c

i

1.2

1

Ec=0 0.8 Ec=1

Ec=0.5

0.6

0.4

0.2

0 0

0.5

1

1.5

2

wavenumber, α

2.5

3

3.5

4

3.5

4

1.2

1.15

Ec=0 Ec=0.5

1.1

Ec=1

wavespeed, cr

1.05

1

0.95

0.9

0.85

0.8

0.75 0

0.5

1

1.5

2

wavenumber, α

2.5

3

Fig. 3. Plots of disturbance growth rate and wave speed versus the wave number for G0 = 5 and for Ec = 0, 0.5, 1.

dissipation e9ects are strongest nearer to the plate. The results presented are for order one values of Eckert and Prandtl numbers as it was diKcult to obtain numerical solutions for higher values of these parameters. The Taylor–Goldstein equation (4.10), together with the appropriate boundary conditions, constitutes an eigenvalue problem for complex cˆ as a function of the real-valued ). ˆ We can write an eigenrelation of the form cˆ = c(); ˆ G0 ; Pr; Ec), where cˆ = cˆr + icˆi . We iterate on cˆr and cˆi until

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E.W. Mureithi, D.P. Mason / Fluid Dynamics Research 31 (2002) 65 – 78 1.4

disturbance growth rate, αci

1.2

G0=5

1

G0=4 0.8 G =3 0

0.6 G0=2 0.4

G0=1

0.2

0

0

0.5

1

1.5

1

1.5

2

2.5

3

3.5

4

2

2.5

3

3.5

4

wavenumber, α

1.2

1.1

1

wavespeed, cr

G =5 0

0.9 G =4 0 G =3 0

0.8

G =2 0

G0=1

0.7

0.6

0.5

0

0.5

wavenumber, α

Fig. 4. Plots of disturbance growth rate and wave speed versus the wave number for Ec = 1 and for G0 = 1, 2, 3, 4, 5.

V0 = Vr + iVi = 0 on Y = 0. The stability equation (4.10) was solved by 7rst discretizing it using a second order centered di9erence scheme. Figs. 3 and 4 show that increasing Ec and G0 results in an increase in the disturbance growth rate and wave speed. Fig. 5 shows that as )ˆ increases the eigenfunction becomes concentrated closer to the wall about the location where the stream-wise velocity attains its maximum value. For the "ow with viscous dissipation, we have identi7ed three unstable modes. Fig. 6 shows the eigenvalues for

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75

1 α=4

0.8

Real part of eigenfunction, V

r

α=0.5 0.6 α=1 α=2

0.4

α=3 0.2

0

-0.2

-0.4

0

1

2

3

4

5

6

7

8

9

10

5

6

7

8

9

10

scaled normal distance, η

0.2

Imaginary part of eigenfunction, V

i

0

-0.2 α=0.5 α=1

-0.4 α =2

-0.6

α =3

-0.8

α =4

-1 -1.2 -1.4

0

1

2

3

4

scaled normal distance, η

Fig. 5. Plots of the real and the imaginary parts of the eigenfuctions for Ec = 1; G0 = 5 and ) = 0:5; 1; 2; 3; 4.

the three modes for the case when Ec = 1. For the case Ec = 0 only a single inviscid unstable mode exists. We may therefore conclude that the additional secondary modes arise due to the combined e9ect of thermal buoyancy and viscous dissipation. From Fig. 6(b), we see that modes cross at higher wave number. Fig. 7 shows the three associated eigenfunctions for G0 = 5; ) = 2 and Ec = 1. The mode crossing at higher wave numbers may explain why the eigenfunction shown in Fig. 5 for ) = 3 and 4 appear as though they are second modes.

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E.W. Mureithi, D.P. Mason / Fluid Dynamics Research 31 (2002) 65 – 78 1.4

1st

disturbance growth rate, α ci

1.2

1

2nd

0.8

3rd

0.6

0.4

0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

3

3.5

4

wavenumber, α 1.25

1.2

wavespeed, c

r

1.15

1.1

3rd 2nd

1.05

1

1st 0.95

0.9

0.85

0

0.5

1

1.5

2

2.5

wavenumber, α

Fig. 6. Plots of disturbance growth rate, )ci and wave speed, cr versus ) for the 7rst three modes for G0 = 5; Ec = 1 and ) = 2.

E.W. Mureithi, D.P. Mason / Fluid Dynamics Research 31 (2002) 65 – 78

77

0.6

solidus

first mode second mode third mode

..........

Real part of eigenfunction, V

r

0.4

0.2

0

-0.2

-0.4

-0.6

0

1

2

3

4

5

6

scaled normal distance, η

7

8

9

10

9

10

0.4

solidus

first mode second mode third mode

..........

Imaginary part of eigenfunction, V

i

0.2

0

-0.2

-0.4

-0.6

-0.8

0

1

2

3

4

5

6

scaled normal distance, η

7

8

Fig. 7. Plots of real and imaginary parts of the eigenfunctions for the 7rst three modes for G0 = 5; Ec = 1 and ) = 2.

6. Conclusions We have investigated the inviscid instability of a forced-free convection boundary layer with viscous dissipation. The buoyant boundary layer equations with viscous dissipation admit similarity type solutions for n=1 only. The basic stream-wise velocity has “super-velocities” in the body of the buoyant boundary layer which increase with increasing Ec. The rapid rise in temperature adjacent to the plate shows that viscous dissipation e9ects are greatest there. Our results indicate that viscous dissipation increases buoyancy forces.

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E.W. Mureithi, D.P. Mason / Fluid Dynamics Research 31 (2002) 65 – 78

A parallel-inviscid analysis yields a stability equation of the Taylor–Goldstein type. At this inviscid limit the viscous dissipation term drops out from the stability equation and hence its e9ect is felt only through the basic "ow. For "ow with viscous dissipation we have identi7ed three unstable modes which are found to cross at higher wave number. The disturbance growth rate increases with increasing Ec for a 7xed valued of G0 . Viscous dissipation is destabilizing. The spatial localization of the eigenfunctions about the position where the basic stream-wise velocity has its maximum value is in agreement with the results of Mureithi et al. (1997). This may suggest the existence of an inner viscous layer, which could be analyzed using a weakly nonlinear theory as outlined in the work of Denier and Mureithi (1996). Acknowledgements The research work of E.W.M. was supported by a Postdoctoral Research Fellowship, awarded by the University of Witwatersrand. D.P.M. would like to thank the National Research Foundation, Pretoria, South Africa for 7nancial support. References Barletta, A., 1998. Laminar mixed convection with viscous dissipation in a vertical channel. Int. J. Heat Mass Transfer 41, 3501–3513. Cheng, K.C., Wu, R., 1976. Viscous dissipation e9ects on convective instability and heat transfer in plane Poiseuille "ow heated from below. Appl. Sci. Res. 32, 327–346. Denier, J.P., Mureithi, E.W., 1996. Weakly nonlinear wave motions in a thermally strati7ed boundary layer. J. Fluid Mech. 315, 293–316. Gage, K.S., 1971. The e9ect of stable thermal strati7cation on the stability of viscous parallel "ows. J. Fluid Mech. 47, 1–20. Gage, K.S., Reid, W.H., 1968. The stability of thermally strati7ed plane Poiseuille "ow. J. Fluid Mech. 33, 21–32. Gebhart, B., 1962. The e9ect of viscous dissipation in natural convection. J. Fluid Mech. 14, 225–232. Gebhart, B., Mollendorf, J., 1969. Viscous dissipation in external natural convection "ows. J. Fluid Mech. 38, 97–107. Hall, P., 1993. Streamwise vortices in heated boundary layers. J. Fluid Mech. 252, 301–324. Hall, P., Morris, H., 1993. On the instability of boundary layers on heated "at plates. J. Fluid Mech. 245, 367–400. Mureithi, E.W., Denier, J.P., Stott, J.A.K., 1997. The e9ect of buoyancy on upper branch Tollmien–Schlichting waves. IMA J. Appl. Math. 58, 19–50. Smith, F.T., Bodonyi, R.J., 1982. Nonlinear critical layers and their development in streaming-"ow stability. J. Fluid Mech. 118, 165–185. Wu, R.S., Cheng, K.C., 1976. Thermal instability of Blasius "ow along horizontal plates. Int. J. Heat Mass Transfer 105, 907–913.