- Email: [email protected]
About the physical relevance of similarity solutions of the boundary-layer flow equations describing mixed convection flow along a vertical plate
Fluid Dynamics Research 32 (2003) 1 – 13
About the physical relevance of similarity solutions of the boundary-layer ow equations describing mixed convection ow along a vertical plate Herbert Steinr'uck∗ Institute of Fluid Dynamics and Heat Transfer, Vienna University of Technology, Resselgasse 3, 1040 Vienna, Austria Received 9 September 2002; accepted 22 October 2002 Communicated by M. Oberlack
Abstract Similarity solutions of the boundary-layer equations describing mixed convection ow along a vertical plate exist if the di/erence between the temperature of the plate and the temperature of the ambient uid is inverse proportional to the distance from the leading edge of the plate. A new solution branch of similarity solutions will be presented and it will be discussed which similarity solution is of physical relevance by a numerical solution of the boundary layer equations. Employing a spatial stability analysis similarity solutions are identi1ed which can be attained as asymptotic limit for large distances from the leading edge of the plate. c 2002 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved. PACS: 47.15.Cb; 47.27.Te; 44.25.+f Keywords: Laminar boundary layer; Mixed convection; Similarity solutions
1. Introduction In this paper the mixed convection boundary-layer ow along a vertical plate aligned parallel to the oncoming uniform free stream will be considered. The boundary layer equations modi1ed to take buoyancy e/ects into account can be found in Gebhart et al. (1988) or Bejan (1995). Dimensional analysis shows that similarity solutions are only possible if the temperature di/erence between the surface of the plate and the temperature of the ambient uid is inverse proportional to the ∗
Corresponding author. Tel.: +43-1-58801-32231; fax: +43-1-58801-32299. E-mail address: [email protected] (H. Steinr'uck). URL: http://www. uid.tuwien.ac.at
c 2002 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. 0169-5983/03/$30.00 All rights reserved. doi:10.1016/S0169-5983(02)00138-7
2
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
distance from the leading edge of the plate (Merkin and Pop, 2002). Since such a plate temperature distribution requires an in1nite heat transfer to the uid this similarity solutions can only serve as asymptotic approximations for large distances from the leading edge. A remarkable feature of the similarity solutions reported by Merkin and Pop (2002) is that the temperature pro1les have regions where the temperature is below the ambient temperature although the plate temperature is above the ambient temperature. Such solutions seem to contradict the second law of thermodynamics. Therefore we will discuss whether these similarity solutions will be attained as limiting solutions or not. In Section 2 we introduce the governing equations following Merkin and Pop (2002), but use a slightly di/erent scaling. In Section 3 the similarity solutions are reviewed. Merkin and Pop (2002) report about two solution branches. The upper branch terminates in a singularity at = 0, where is the mixed convection parameter de1ned in Eq. (7). Due to the di/erent scaling used in the present paper the upper solution branch can be continued across = 0. In Section 4 we consider non-self-similar solutions. A plate temperature distribution which behaves asymptotically inverse proportional to the distance from the leading edge will be given. Numerical integration of the boundary layer equation shows that the boundary layer ow either terminates in a Goldstein type singularity or converges to the upper similarity solution or diverges. In Section 5 an eigenvalue analysis clari1es which similarity solutions can be attained as limiting solutions for large distances from the leading edge. Finally the question of uniqueness of the solution of the steady state mixed convection boundary-layer equations is addressed in Section 6. 2. Governing equations The modi1ed boundary-layer equations describing incompressible mixed convection ow along a vertical plate are @uH @vH + = 0; (1) @xH @yH 2 @uH @uH H TH − TH ∞ ) + H @ uH ; uH + vH = gH ( (2) @xH @yH @yH 2 @TH @2 uH @TH + vH = aH uH ; (3) @xH @yH @yH 2 where x, H yH are the Cartesian coordinates parallel and perpendicular to the plate, respectively. The origin of the coordinate system is at the leading edge of the plate. The velocity components parallel and normal to the plate are denoted by uH and v. H The temperature of the uid is denoted by TH . The velocity and temperature of the ambient uid are uH ∞ and TH ∞ , respectively. The properties of the uid are described by the H the kinematic viscosity, aH the thermal di/usivity and H the isobaric expansion coeIcient. The gravity acceleration is denoted by g. H The boundary conditions at the plate are u( H x; H 0) = v( H x; H 0) = 0; TH (x; 0) = TH w (x) H + TH ∞ ; (4) where TH w is the di/erence between the plate temperature and the temperature of the ambient uid. The matching conditions to the outer ow are: (5) TH → TH ∞ ; uH → uH ∞ as yH → ∞:
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
3
We make use of the stream function H in the usual way (uH = @ H =@y; H vH = −@ H =@x) H and introduce the dimensionless variables by √ #(x; ) uH2 H = HuH ∞ xf(x; ; KTH L = ∞ (6) H ); TH − TH ∞ = KTH L x gH H LH with x = x= H LH and = yH uH ∞ = Hx, H where LH is an arbitrary vertical length scale. We de1ne the mixed convection parameter by H xH Grx TH w (x) = lim (7) = lim H H x→∞ Rex2 xH→∞ KT L L and the dimensionless plate temperature distribution dw by H x TH w (x) : dw (x) = H KT L
(8)
Thus by de1nition we have dw (∞) = 1. The dimensionless equations and boundary conditions are 1 (9) f + ff + # = x(f fx − fx f ); 2 1 1 # + f# + #f = x(f #x − fx # ); (10) 2 Pr f(x; 0) = f (x; 0) = 0; f → 1;
#→0
#(x; 0) = dw (x);
(11)
as → ∞:
(12)
The dimensionless numbers Rex = uH ∞ x= H , H Grx = gH H TH w (x) H xH3 = H2 and Pr = = H aH are the local Grashof, the local Reynolds and the Prandtl number, respectively. Derivatives with respect to are denoted with a prime. 3. Similarity solutions In case of dw (x) = 1 the dimensionless equations (9)–(12) permit similarity solutions. Thus (9)–(12) reduce to the ordinary di/erential equations: 1 1 1 # + f# + #f = 0; (13) f + ff + # = 0; 2 Pr 2 f(0) = f (0) = 0;
#(0) = ;
f → 1;
#→0
as → ∞:
(14)
3.1. Forced convection = 0 Note that the present scaling is not appropriate for forced convection boundary ow (=0). In that case the present form of the similarity solution yields # = 0, f = fB , where fB is the solution of the Blasius equation. However, the temperature distribution in the forced convection case is proportional to @#=@. For an analysis of that particular case see Merkin and Pop (2002).
4
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
Fig. 1. Similarity solutions for Pr = 0:2: wall friction coeIcient f (0) and dimensionless temperature gradient # (0) at the plate as a function of the mixed convection parameter .
Fig. 2. Similarity solutions for Pr = 2:0: wall friction coeIcient f (0) and dimensionless temperature gradient # (0) at the plate as a function of the mixed convection parameter .
3.2. Lower and upper solution branches Using a solver for boundary value problems of ordinary di/erential equations three solution branches of the similarity equations are found in general. We call them upper, lower and reverse ow branch. Starting from the forced convection case and varying the lower solution branch is obtained. Depending on the Prandtl number Pr qualitative di/erent behavior is observed. For Pr ¡ Prc = 0:7605 the lower solution branch exists for all positive values of . For negative values of this branch exists only for ¿ c (Pr). For a de1nition of the critical Prandtl number Prc see Merkin and Pop (2002). At = c the lower and upper branch are connected. The upper branch exists for all values of with ¿ c (see Fig. 1). The advantage of the present scaling is that it permits the upper branch to be continued across = 0. In Merkin and Pop (2002) the upper branch terminates in a singularity at = 0. As pointed out by Merkin and Pop (2002) the similarity solutions of the lower solution branch have regions where the temperature di/erence to the ambient uid is of opposite sign than the di/erence between the temperature of the plate and the temperature of the ambient uid (Figs. 4–7). The question if and how this similarity solution will be attained will be discussed in Section 4. The upper solution branch is characterized by a velocity overshoot near the plate. This is induced by a layer of uid which is hotter than the ambient uid. This hot layer is present even if the plate temperature is below or equal to the temperature of the ambient uid.
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
5
Fig. 3. Similarity solutions for Prandtl numbers Pr close to the critical Prandtl number Prc =0:7605: Pr =0:6: dashed-dotted line Prc solid line Pr = 0:9 dashed line: wall friction coeIcient f (0) and temperature gradient # (0) at the plate as a function of the mixed convection parameter . The reverse ow branches are almost indistinguishable.
(a)
(b)
Fig. 4. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = −0:05 and the Prandtl number Pr = 0:2. Upper solution branch: thick line, lower branch dashed line, reverse ow branch: dash dotted line.
( ) (a)
( ) (b)
Fig. 5. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 0 and the Prandtl number Pr = 0:2. Upper solution branch: thick line, lower branch dashed line.
For Pr ¿ Prc the upper and lower solution branch exist for ¡ c (Pr). For ¿ c (Pr) no similarity solutions have been found (see Fig. 2). In Figs. 8–11 velocity pro1les for Pr = 2 are shown. The velocity pro1les corresponding to the solutions on the upper branch have a reverse ow region. Near the plate there is a wall jet followed by a broad reverse ow region. The temperature pro1les according to both solutions have regions where the temperature di/erence between the uid and the ambient uid has opposite sign than the di/erence between the plate temperature and the ambient uid. For Prandtl numbers close to the critical Prandtl number Prc = 0:7605 the positive turning point splits up into three turning points (Pr = 0:9) (Fig. 3). Thus in a very narrow range for four
6
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
( ) (a)
( ) (b)
Fig. 6. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 0:05 and the Prandtl number Pr = 0:2. Upper solution branch: thick line, lower branch dashed line.
(a)
(b)
Fig. 7. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 1 and the Prandtl number Pr = 0:2. Upper solution branch: thick line, lower branch dashed line.
(a)
( ) (b)
Fig. 8. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 0:1, Pr = 2. Upper solution branch: thick line, lower branch: dashed line, lower branch: dash-dotted line.
solutions are possible. According to the discussion in the following section none of them is believed to be of physical relevance. 3.3. Reverse 4ow branch Moreover, for all Prandtl numbers investigated here, a third solution branch with a reverse ow region close to the plate has been found for ¡ 0. Approaching → 0− this solution branch terminates in a singularity. The reverse ow region becomes in1nite (Figs. 4, 11, 10).
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
( ) (a)
7
(b)
Fig. 9. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 0, Pr = 2. Upper solution branch: thick line, lower branch: dashed line, lower branch: dash-dotted line.
(a)
(b)
Fig. 10. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = −0:05, Pr = 2. Upper solution branch: thick line, lower branch: dashed line, reverse ow branch: dash-dotted line.
( ) (a)
( ) (b)
Fig. 11. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = −1, Pr = 2. Upper solution branch: thick line, lower branch: dashed line, reverse ow branch: dash-dotted line.
4. Non-similarity solutions We have shown that up to four similarity solutions can exist. Thus we have to ask, which of them are of physical relevance. Since the associated plate temperature is singular at the leading edge, a similarity solution can only serve as the asymptotic limit for large distances from the leading edge. Due to the second law of thermodynamics or equivalently the maximum principle for the energy equation (see: Gilbarg and Trudinger, 1983) a similarity solution with a change of sign of the
8
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
(a)
(b)
Fig. 12. Non-similarity solutions for Pr = 0:2: f (0) as function of x. (a) = −0:05, dw; max = 5:25 solid line, dw; max = 5:18 dashed line. (b) = 0:05, dw (x) = 1 − (1 − x)2 for x ¡ 1, dw (x) = 1 for x ¿ 1: solid line, dw = 1: dashed lines. The values corresponding to the similarity solutions are indicated by the dotted lines.
temperature di/erence # is only possible if worked out in the appendix. To have a nonsingular plate temperature −2dw; max x; dw (x) = (2dw; max + 2)(x − 1) + 1; 1;
the plate temperature TH w changes sign. The argument is distribution we choose x ¡ 0:5; 0:5 ¡ x ¡ 1;
(15)
x ¿ 1:
Note that the corresponding plate temperature distribution TH w has near the leading the constant value H 2dw; max KTH L for 0 ¡ xH ¡ L=2. 4.1. Pr ¡ Prc Starting at x=0 with the Blasius solution as initial condition we solve the boundary-layer equations numerically by a marching method. In Fig. 12a the solution is shown for = −0:05, Pr = 0:2 and dw; max = 5:18 and 5.24, respectively. After a short increase of f (0) due to the acceleration of the ow induced by initial heating of the plate f (x; 0) decays relatively fast to the value corresponding to the lower solution branch. However, the lower similarity solution turns out to be spatially unstable. In case of dw; max = 5:18 f (x; 0) decreases and the solution 1nally terminates in a Goldstein like singularity. Taking dw; max = 5:24 f (x; 0) increases and 1nally tends to the upper similarity solution. However, it takes a very long distance for the transition from the lower to the upper solution. If the plate is not too long (x ¡ 1020 ) the lower self-similar solution will be observed near the trailing edge of the plate. It has to be noted that great care had been taken to 1nd values of dw; max such that the solution comes close to the lower similarity solution. Taking dw; max too small the solution terminates very soon in a singularity. Taking dw; max too large the solution converges directly to the upper solution. In Fig. 12b solutions for = 0:05, Pr = 0:2 are shown. Here we have assumed no pre-cooling of the plate: dw (x) = 1 − (1 − x)2 for x ¡ 1 and dw (x) = 1 for x ¿ 1. The solution tends directly to the upper similarity solution. The second solution in Fig. 12b (dashed line) corresponds to dw (x) = 1. The computations are started at the leading edge with the lower branch solution. First the similarity solution is reproduced, but due to computational errors the solution deviates from the lower solution and tends to the upper similarity solution. It is a non-similarity
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
(a)
9
(b)
Fig. 13. Non-similarity solutions for Pr =2: f (0) as function of x. (a) =−0:05, dw; max =2:3355 solid line, dw; max =2:336 dashed line. (b) = 0:05, dw; max = 5:25, solid line, dw; max = 5:18 dashed lines. The values corresponding to the similarity solutions are indicated by the dotted lines.
solution connecting two similarity solutions. Therefore, the lower similarity solution is spatially unstable. Similar connecting solutions have been found in mixed convection ow over a horizontal plate (Steinr'uck, 1995). 4.2. Pr ¿ Prc In Fig. 13 solutions for Pr = 2 are shown. The cases = −0:05 (Fig. 13a) and = 0:05 (Fig. 13b) look qualitatively similar. The values of dw; max are chosen such that after an initial cooling or heating section, respectively, the lower similarity solution is almost attained. For x ¿ 1010 the solution again deviates from the lower similarity solution and either terminates in a Goldstein-type singularity or diverges. In case of divergence the asymptotic behavior is given by f(x; ) ∼ cx (x );
# ∼ cx4 !(x );
(16)
where = (Pr) and ; ! are the eigenvalues end eigenfunctions of the non-linear eigenvalue problem 1 2 + − 2 + ! = 0; (17) + 2 1 1 ! + + ! + (1 − 4 ) ! = 0; (18) Pr 2 (0) =
(0) =
(∞) = !(0) = !(∞) = 0;
(0) = 1
(19)
and c is an appropriate constant. This eigenvalue problem is discussed in Merkin and Pop (2002) and in Merkin (1985), respectively. Since the exponent (4 ) of decay of the temperature pro1le is determined by an eigenvalue problem, ( ; !) are a similarity solution of the second kind (Barenblatt, 1996) of the boundary-layer equation. They describe a plume near the wall induced by natural convection only. In Fig. 14 the exponent is shown as a function of the Prandtl number. In case of Pr ¿ Prc we have (Pr) ¿ 0 and thus the natural convection solution (16) decays slower than the mixed convection similarity solution. Therefore, the natural convection solution determines the asymptotic behavior for x → ∞. If Pr ¡ Prc we have ¡ 0 and thus the mixed convection solution decays slower than the natural convection solution and thus determines the
10
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
Fig. 14. Exponent of limiting solution: Pr ¡ Prc : = 0, mixed convection similarity solution, Pr ¿ Prc : = (Pr): natural convection similarity solution.
Fig. 15. Asymptotic behavior of f (x; 0) for x → ∞, Pr = 3:0: comparison with natural convection wall plume f (x; 0) ∼ x3 , (3:0) = 0:020366.
asymptotic behavior. In Fig. 14 the thick line indicates the exponent of decay of the asymptotic approximation of the solution of the mixed convection boundary-layer equation for x → ∞. The asymptotic behavior of a numerical solution of the mixed convection boundary-layer equation for = 0:05, Pr = 3: and dw = max(1; x) represented by f (x; 0) is shown in Fig. 15. It is compared with the asymptotic growth rate x3 . 5. Spatial stability To analyze whether a similarity solution f0 ; #0 is spatially stable, we consider small perturbations of the similarity equations. Thus we linearize the boundary-layer equations at the similarity solution and analyze the evolution of a small perturbation with respect to x. f(x; ) ∼ f0 () + KF(x; );
#(x; ) ∼ #0 () + KD():
(20)
Separation of variables KF(x; ) = x$ F();
KD(x; ) = x$ D();
yields the eigenvalue problem: 1 F + (f0 F + f0 F) = $(f0 F − f0 F); 2
(21)
(22)
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
11
Fig. 16. Eigenvalues of the linearized problem.
1 1 D + (f0 D + #0 F) + #0 F + f0 D = $(f0 D − #0 F); Pr 2
(23)
F(0) = F (0) = D(0) = 0;
(24)
F → 0;
D→0
as → ∞:
If a positive eigenvalue exists a perturbation will be ampli1ed downstream, thus the similarity solution is spatially unstable. A necessary condition for spatial stability is that all eigenvalues of (23) and (24) are negative. (Note that this condition is not suIcient in general). Eigenvalues of (23) and (24) for Pr = 0:2 and 2 are shown in Fig. 16. Consider Pr = 0:2 1rst. Along the lower solution branch a positive eigenvalue exists. It becomes zero at the turning point. Now following the upper branch this eigenvalue becomes negative. This is in agreement with the results from previous section (Fig. 12) where the solution branches o/ from the lower similarity solution and tends to the upper one. For Pr=2 the situation is more complicate. Starting from the Blasius solution a positive eigenvalue is found. Approaching the turning point this eigenvalue seems to increase beyond any bound. A second branch of eigenvalues has been found which are negative along the lower branch and become zero at the turning point. These eigenvalues are positive along the upper branch. In that case no spatially stable similarity solution exists which is agreement with the results of Section 4. 6. Uniqueness of solutions Since a positive eigenvalue associated to the Blasius solution ( = 0, lower branch) exists, one is tempted to conclude that the initial value problem has a non-unique solution with the asymptotic expansion f(x; ) = f0 () + cx$ F(n);
#(x; ) = #0 () + cx$ D(n);
for x 1;
(25)
where (F; D) and $ ¿ 0 are the eigenfunction and eigenvalue of the eigenvalue problem (23) and (24) with F (0) = 1, respectively, and c is an arbitrary constant. The corresponding temperature in the uid would be locally near the leading edge of the plate of the form TH − T∞ ∼ KTH L x$−1 D():
(26)
12
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
Such a temperature distribution is non singular only if $ ¿ 1. Numerical results for Pr = 0:2 and 2 show that this is not the case. Moreover = 0 corresponds to the forced convection ow. Thus multiple solutions would contradict the uniqueness of the Blasius solution shown by Libby and Fox (1964). Note that the situation encountered here is di/erent from the case of mixed convection over a cooled horizontal plate, where multiple solutions seem to be indeed physical meaningful, cf. Steinr'uck (2001). 7. Conclusions New similarity solution of mixed convection ow along a vertical plate have been found. A discussion of the physical relevance of the similarity solutions shows that only the wall jet like similarity solutions corresponding to the upper solution branch serve as asymptotic limit for large distances from the leading edge. The similarity solutions corresponding to the lower branch are not spatially stable. They can be attained for very peculiar wall temperature distributions as intermediate solutions. The solution of the steady state boundary layer equations with a nonsingular wall temperature distribution is unique. In general the asymptotic behavior for x → ∞ is given by the upper similarity solution if Pr ¡ Prc or by a natural-convection wall plume described in Merkin (1985) if Pr ¿ Prc . It is worth to note that the limiting behavior of the solution of the mixed convection boundary-layer equations for x → ∞ for Pr ¡ Prc is described by a similarity solution of the 1rst kind while for Pr ¡ Prc it is described by a similarity solution of the second kind (see: Barenblatt, 1996). Due to the slow spatial convergence to the upper similarity solution it has to be clari1ed under which conditions the similarity solutions discussed in this paper really describe a laminar mixed convection ow along a vertical 1nite plate. Appendix A. A note on the maximum principle Theorem. If a solution (ub ; vb ; Tb ) of the boundary-layer equations, which approximates a solution (un ; vn ; Tn ) of the Navier–Stokes and energy equation√behaves asymptotically as x → ∞ like a similarity solution u ∼ f0 (), v ∼ (f0 () − f0 ())=2 x, T ∼ #0 ()=x where #0 has a change of sign, then the di8erence Tw between the plate temperature and the ambient temperature must have a change of sign either. √ H ReL . Here we make the coordinate yH perpendicular to the plate dimensionless with L= We assume that #0 (0) ¿ 0 and that #0 has a negative minimum #∗ ¡ 0 at =∗ . Since Tb behaves asymptotically like #0 =x for x → ∞ to every & ¿ 0 there exists an X with Tb (x; y) − 1 #0 () ¡ &=x for any x ¿ X: (A.1) x Using that Tb approximates a solution of the Navier–Stokes equation we can conclude that |Tb (x; y) − Tn (x; y)| ¡ &=X
(A.2)
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
13
for any x ¿ X and Rex suIciently large. Taking & ¡ − #∗ =8 we estimate √ √ 1 1 Tn (X; X ∗ ) ¡ |Tb (X; X ∗ ) − #∗ | + |Tb − Tn | + #∗ X X 2& 3 ∗ 1 ¡ (A.3) + #∗ ¡ # : X X 4X We chose a square with its center at the leading edge, side length 2X and two of its sides parallel to the plate. The dimensionless temperature minimum on the boundary of the square is then larger than (1=2X )#∗ + 2&=X ¿ (3=4X )#∗ . √ Thus we have shown that inside the square, namely at (X; ∗ X ) there is a temperature below the minimum value of the temperature at the boundary of the square. The maximum principle Gilbarg and Trudinger (1983) for the energy equation states that the temperature values interior of a closed control surface must be between the minimum and maximum temperature on the boundary of the control surface. We chose as the control surface the boundary of the square and both surfaces of the plate interior of the square. Then by the maximum principle it follows that there must be a location on the surface of the plate where the temperature is below that of the ambient uid. References Barenblatt, G.I., 1996. Scaling, Self-similarity and Intermediate Asymptotics. Cambridge University Press, Cambridge. Bejan, A., 1995. Convection Heat Transfer. Wiley, New York. Gebhart, B., Jaluria, Y., Mahajan, R.L., Sammkia, B., 1988. Bouyancy-induced Flows and Transport. Hemisphere Publishing Corporation, New York. Gilbarg, D., Trudinger, N., 1983. Elliptic Partial Di/erential Equations of Second Order. Springer, Berlin. Libby, P., Fox, H., 1964. Some perturbation solution in laminar boundary-layer theory, part I: The momentum equation. J. Fluid Mech. 17, 433–449. Merkin, J.H., 1985. A note on the similarity solutions for free convection on a vertical plate. J. Eng. Math. 19, 189–201. Merkin, J.H., Pop, I., 2002. Mixed convection along a vertical surface: similarity solutions for uniform ow. Fluid Dyn. Res. 30, 233–250. Steinr'uck, H., 1995. Mixed convection over a horizontal plate: self-similar and connecting boundary-layer ows. Fluid Dyn. Res. 15, 113–127. Steinr'uck, H., 2001. A review of the mixed convection boundary-layer ow over a horizontal cooled plate. GAMM Mitteilungen 24, 127–158.
About the physical relevance of similarity solutions of the boundary-layer ow equations describing mixed convection ow along a vertical plate Herbert Steinr'uck∗ Institute of Fluid Dynamics and Heat Transfer, Vienna University of Technology, Resselgasse 3, 1040 Vienna, Austria Received 9 September 2002; accepted 22 October 2002 Communicated by M. Oberlack
Abstract Similarity solutions of the boundary-layer equations describing mixed convection ow along a vertical plate exist if the di/erence between the temperature of the plate and the temperature of the ambient uid is inverse proportional to the distance from the leading edge of the plate. A new solution branch of similarity solutions will be presented and it will be discussed which similarity solution is of physical relevance by a numerical solution of the boundary layer equations. Employing a spatial stability analysis similarity solutions are identi1ed which can be attained as asymptotic limit for large distances from the leading edge of the plate. c 2002 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved. PACS: 47.15.Cb; 47.27.Te; 44.25.+f Keywords: Laminar boundary layer; Mixed convection; Similarity solutions
1. Introduction In this paper the mixed convection boundary-layer ow along a vertical plate aligned parallel to the oncoming uniform free stream will be considered. The boundary layer equations modi1ed to take buoyancy e/ects into account can be found in Gebhart et al. (1988) or Bejan (1995). Dimensional analysis shows that similarity solutions are only possible if the temperature di/erence between the surface of the plate and the temperature of the ambient uid is inverse proportional to the ∗
Corresponding author. Tel.: +43-1-58801-32231; fax: +43-1-58801-32299. E-mail address: [email protected] (H. Steinr'uck). URL: http://www. uid.tuwien.ac.at
c 2002 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. 0169-5983/03/$30.00 All rights reserved. doi:10.1016/S0169-5983(02)00138-7
2
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
distance from the leading edge of the plate (Merkin and Pop, 2002). Since such a plate temperature distribution requires an in1nite heat transfer to the uid this similarity solutions can only serve as asymptotic approximations for large distances from the leading edge. A remarkable feature of the similarity solutions reported by Merkin and Pop (2002) is that the temperature pro1les have regions where the temperature is below the ambient temperature although the plate temperature is above the ambient temperature. Such solutions seem to contradict the second law of thermodynamics. Therefore we will discuss whether these similarity solutions will be attained as limiting solutions or not. In Section 2 we introduce the governing equations following Merkin and Pop (2002), but use a slightly di/erent scaling. In Section 3 the similarity solutions are reviewed. Merkin and Pop (2002) report about two solution branches. The upper branch terminates in a singularity at = 0, where is the mixed convection parameter de1ned in Eq. (7). Due to the di/erent scaling used in the present paper the upper solution branch can be continued across = 0. In Section 4 we consider non-self-similar solutions. A plate temperature distribution which behaves asymptotically inverse proportional to the distance from the leading edge will be given. Numerical integration of the boundary layer equation shows that the boundary layer ow either terminates in a Goldstein type singularity or converges to the upper similarity solution or diverges. In Section 5 an eigenvalue analysis clari1es which similarity solutions can be attained as limiting solutions for large distances from the leading edge. Finally the question of uniqueness of the solution of the steady state mixed convection boundary-layer equations is addressed in Section 6. 2. Governing equations The modi1ed boundary-layer equations describing incompressible mixed convection ow along a vertical plate are @uH @vH + = 0; (1) @xH @yH 2 @uH @uH H TH − TH ∞ ) + H @ uH ; uH + vH = gH ( (2) @xH @yH @yH 2 @TH @2 uH @TH + vH = aH uH ; (3) @xH @yH @yH 2 where x, H yH are the Cartesian coordinates parallel and perpendicular to the plate, respectively. The origin of the coordinate system is at the leading edge of the plate. The velocity components parallel and normal to the plate are denoted by uH and v. H The temperature of the uid is denoted by TH . The velocity and temperature of the ambient uid are uH ∞ and TH ∞ , respectively. The properties of the uid are described by the H the kinematic viscosity, aH the thermal di/usivity and H the isobaric expansion coeIcient. The gravity acceleration is denoted by g. H The boundary conditions at the plate are u( H x; H 0) = v( H x; H 0) = 0; TH (x; 0) = TH w (x) H + TH ∞ ; (4) where TH w is the di/erence between the plate temperature and the temperature of the ambient uid. The matching conditions to the outer ow are: (5) TH → TH ∞ ; uH → uH ∞ as yH → ∞:
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
3
We make use of the stream function H in the usual way (uH = @ H =@y; H vH = −@ H =@x) H and introduce the dimensionless variables by √ #(x; ) uH2 H = HuH ∞ xf(x; ; KTH L = ∞ (6) H ); TH − TH ∞ = KTH L x gH H LH with x = x= H LH and = yH uH ∞ = Hx, H where LH is an arbitrary vertical length scale. We de1ne the mixed convection parameter by H xH Grx TH w (x) = lim (7) = lim H H x→∞ Rex2 xH→∞ KT L L and the dimensionless plate temperature distribution dw by H x TH w (x) : dw (x) = H KT L
(8)
Thus by de1nition we have dw (∞) = 1. The dimensionless equations and boundary conditions are 1 (9) f + ff + # = x(f fx − fx f ); 2 1 1 # + f# + #f = x(f #x − fx # ); (10) 2 Pr f(x; 0) = f (x; 0) = 0; f → 1;
#→0
#(x; 0) = dw (x);
(11)
as → ∞:
(12)
The dimensionless numbers Rex = uH ∞ x= H , H Grx = gH H TH w (x) H xH3 = H2 and Pr = = H aH are the local Grashof, the local Reynolds and the Prandtl number, respectively. Derivatives with respect to are denoted with a prime. 3. Similarity solutions In case of dw (x) = 1 the dimensionless equations (9)–(12) permit similarity solutions. Thus (9)–(12) reduce to the ordinary di/erential equations: 1 1 1 # + f# + #f = 0; (13) f + ff + # = 0; 2 Pr 2 f(0) = f (0) = 0;
#(0) = ;
f → 1;
#→0
as → ∞:
(14)
3.1. Forced convection = 0 Note that the present scaling is not appropriate for forced convection boundary ow (=0). In that case the present form of the similarity solution yields # = 0, f = fB , where fB is the solution of the Blasius equation. However, the temperature distribution in the forced convection case is proportional to @#=@. For an analysis of that particular case see Merkin and Pop (2002).
4
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
Fig. 1. Similarity solutions for Pr = 0:2: wall friction coeIcient f (0) and dimensionless temperature gradient # (0) at the plate as a function of the mixed convection parameter .
Fig. 2. Similarity solutions for Pr = 2:0: wall friction coeIcient f (0) and dimensionless temperature gradient # (0) at the plate as a function of the mixed convection parameter .
3.2. Lower and upper solution branches Using a solver for boundary value problems of ordinary di/erential equations three solution branches of the similarity equations are found in general. We call them upper, lower and reverse ow branch. Starting from the forced convection case and varying the lower solution branch is obtained. Depending on the Prandtl number Pr qualitative di/erent behavior is observed. For Pr ¡ Prc = 0:7605 the lower solution branch exists for all positive values of . For negative values of this branch exists only for ¿ c (Pr). For a de1nition of the critical Prandtl number Prc see Merkin and Pop (2002). At = c the lower and upper branch are connected. The upper branch exists for all values of with ¿ c (see Fig. 1). The advantage of the present scaling is that it permits the upper branch to be continued across = 0. In Merkin and Pop (2002) the upper branch terminates in a singularity at = 0. As pointed out by Merkin and Pop (2002) the similarity solutions of the lower solution branch have regions where the temperature di/erence to the ambient uid is of opposite sign than the di/erence between the temperature of the plate and the temperature of the ambient uid (Figs. 4–7). The question if and how this similarity solution will be attained will be discussed in Section 4. The upper solution branch is characterized by a velocity overshoot near the plate. This is induced by a layer of uid which is hotter than the ambient uid. This hot layer is present even if the plate temperature is below or equal to the temperature of the ambient uid.
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
5
Fig. 3. Similarity solutions for Prandtl numbers Pr close to the critical Prandtl number Prc =0:7605: Pr =0:6: dashed-dotted line Prc solid line Pr = 0:9 dashed line: wall friction coeIcient f (0) and temperature gradient # (0) at the plate as a function of the mixed convection parameter . The reverse ow branches are almost indistinguishable.
(a)
(b)
Fig. 4. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = −0:05 and the Prandtl number Pr = 0:2. Upper solution branch: thick line, lower branch dashed line, reverse ow branch: dash dotted line.
( ) (a)
( ) (b)
Fig. 5. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 0 and the Prandtl number Pr = 0:2. Upper solution branch: thick line, lower branch dashed line.
For Pr ¿ Prc the upper and lower solution branch exist for ¡ c (Pr). For ¿ c (Pr) no similarity solutions have been found (see Fig. 2). In Figs. 8–11 velocity pro1les for Pr = 2 are shown. The velocity pro1les corresponding to the solutions on the upper branch have a reverse ow region. Near the plate there is a wall jet followed by a broad reverse ow region. The temperature pro1les according to both solutions have regions where the temperature di/erence between the uid and the ambient uid has opposite sign than the di/erence between the plate temperature and the ambient uid. For Prandtl numbers close to the critical Prandtl number Prc = 0:7605 the positive turning point splits up into three turning points (Pr = 0:9) (Fig. 3). Thus in a very narrow range for four
6
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
( ) (a)
( ) (b)
Fig. 6. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 0:05 and the Prandtl number Pr = 0:2. Upper solution branch: thick line, lower branch dashed line.
(a)
(b)
Fig. 7. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 1 and the Prandtl number Pr = 0:2. Upper solution branch: thick line, lower branch dashed line.
(a)
( ) (b)
Fig. 8. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 0:1, Pr = 2. Upper solution branch: thick line, lower branch: dashed line, lower branch: dash-dotted line.
solutions are possible. According to the discussion in the following section none of them is believed to be of physical relevance. 3.3. Reverse 4ow branch Moreover, for all Prandtl numbers investigated here, a third solution branch with a reverse ow region close to the plate has been found for ¡ 0. Approaching → 0− this solution branch terminates in a singularity. The reverse ow region becomes in1nite (Figs. 4, 11, 10).
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
( ) (a)
7
(b)
Fig. 9. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = 0, Pr = 2. Upper solution branch: thick line, lower branch: dashed line, lower branch: dash-dotted line.
(a)
(b)
Fig. 10. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = −0:05, Pr = 2. Upper solution branch: thick line, lower branch: dashed line, reverse ow branch: dash-dotted line.
( ) (a)
( ) (b)
Fig. 11. Velocity pro1les (a) and temperature pro1les (b) of the similarity solution for = −1, Pr = 2. Upper solution branch: thick line, lower branch: dashed line, reverse ow branch: dash-dotted line.
4. Non-similarity solutions We have shown that up to four similarity solutions can exist. Thus we have to ask, which of them are of physical relevance. Since the associated plate temperature is singular at the leading edge, a similarity solution can only serve as the asymptotic limit for large distances from the leading edge. Due to the second law of thermodynamics or equivalently the maximum principle for the energy equation (see: Gilbarg and Trudinger, 1983) a similarity solution with a change of sign of the
8
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
(a)
(b)
Fig. 12. Non-similarity solutions for Pr = 0:2: f (0) as function of x. (a) = −0:05, dw; max = 5:25 solid line, dw; max = 5:18 dashed line. (b) = 0:05, dw (x) = 1 − (1 − x)2 for x ¡ 1, dw (x) = 1 for x ¿ 1: solid line, dw = 1: dashed lines. The values corresponding to the similarity solutions are indicated by the dotted lines.
temperature di/erence # is only possible if worked out in the appendix. To have a nonsingular plate temperature −2dw; max x; dw (x) = (2dw; max + 2)(x − 1) + 1; 1;
the plate temperature TH w changes sign. The argument is distribution we choose x ¡ 0:5; 0:5 ¡ x ¡ 1;
(15)
x ¿ 1:
Note that the corresponding plate temperature distribution TH w has near the leading the constant value H 2dw; max KTH L for 0 ¡ xH ¡ L=2. 4.1. Pr ¡ Prc Starting at x=0 with the Blasius solution as initial condition we solve the boundary-layer equations numerically by a marching method. In Fig. 12a the solution is shown for = −0:05, Pr = 0:2 and dw; max = 5:18 and 5.24, respectively. After a short increase of f (0) due to the acceleration of the ow induced by initial heating of the plate f (x; 0) decays relatively fast to the value corresponding to the lower solution branch. However, the lower similarity solution turns out to be spatially unstable. In case of dw; max = 5:18 f (x; 0) decreases and the solution 1nally terminates in a Goldstein like singularity. Taking dw; max = 5:24 f (x; 0) increases and 1nally tends to the upper similarity solution. However, it takes a very long distance for the transition from the lower to the upper solution. If the plate is not too long (x ¡ 1020 ) the lower self-similar solution will be observed near the trailing edge of the plate. It has to be noted that great care had been taken to 1nd values of dw; max such that the solution comes close to the lower similarity solution. Taking dw; max too small the solution terminates very soon in a singularity. Taking dw; max too large the solution converges directly to the upper solution. In Fig. 12b solutions for = 0:05, Pr = 0:2 are shown. Here we have assumed no pre-cooling of the plate: dw (x) = 1 − (1 − x)2 for x ¡ 1 and dw (x) = 1 for x ¿ 1. The solution tends directly to the upper similarity solution. The second solution in Fig. 12b (dashed line) corresponds to dw (x) = 1. The computations are started at the leading edge with the lower branch solution. First the similarity solution is reproduced, but due to computational errors the solution deviates from the lower solution and tends to the upper similarity solution. It is a non-similarity
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
(a)
9
(b)
Fig. 13. Non-similarity solutions for Pr =2: f (0) as function of x. (a) =−0:05, dw; max =2:3355 solid line, dw; max =2:336 dashed line. (b) = 0:05, dw; max = 5:25, solid line, dw; max = 5:18 dashed lines. The values corresponding to the similarity solutions are indicated by the dotted lines.
solution connecting two similarity solutions. Therefore, the lower similarity solution is spatially unstable. Similar connecting solutions have been found in mixed convection ow over a horizontal plate (Steinr'uck, 1995). 4.2. Pr ¿ Prc In Fig. 13 solutions for Pr = 2 are shown. The cases = −0:05 (Fig. 13a) and = 0:05 (Fig. 13b) look qualitatively similar. The values of dw; max are chosen such that after an initial cooling or heating section, respectively, the lower similarity solution is almost attained. For x ¿ 1010 the solution again deviates from the lower similarity solution and either terminates in a Goldstein-type singularity or diverges. In case of divergence the asymptotic behavior is given by f(x; ) ∼ cx (x );
# ∼ cx4 !(x );
(16)
where = (Pr) and ; ! are the eigenvalues end eigenfunctions of the non-linear eigenvalue problem 1 2 + − 2 + ! = 0; (17) + 2 1 1 ! + + ! + (1 − 4 ) ! = 0; (18) Pr 2 (0) =
(0) =
(∞) = !(0) = !(∞) = 0;
(0) = 1
(19)
and c is an appropriate constant. This eigenvalue problem is discussed in Merkin and Pop (2002) and in Merkin (1985), respectively. Since the exponent (4 ) of decay of the temperature pro1le is determined by an eigenvalue problem, ( ; !) are a similarity solution of the second kind (Barenblatt, 1996) of the boundary-layer equation. They describe a plume near the wall induced by natural convection only. In Fig. 14 the exponent is shown as a function of the Prandtl number. In case of Pr ¿ Prc we have (Pr) ¿ 0 and thus the natural convection solution (16) decays slower than the mixed convection similarity solution. Therefore, the natural convection solution determines the asymptotic behavior for x → ∞. If Pr ¡ Prc we have ¡ 0 and thus the mixed convection solution decays slower than the natural convection solution and thus determines the
10
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
Fig. 14. Exponent of limiting solution: Pr ¡ Prc : = 0, mixed convection similarity solution, Pr ¿ Prc : = (Pr): natural convection similarity solution.
Fig. 15. Asymptotic behavior of f (x; 0) for x → ∞, Pr = 3:0: comparison with natural convection wall plume f (x; 0) ∼ x3 , (3:0) = 0:020366.
asymptotic behavior. In Fig. 14 the thick line indicates the exponent of decay of the asymptotic approximation of the solution of the mixed convection boundary-layer equation for x → ∞. The asymptotic behavior of a numerical solution of the mixed convection boundary-layer equation for = 0:05, Pr = 3: and dw = max(1; x) represented by f (x; 0) is shown in Fig. 15. It is compared with the asymptotic growth rate x3 . 5. Spatial stability To analyze whether a similarity solution f0 ; #0 is spatially stable, we consider small perturbations of the similarity equations. Thus we linearize the boundary-layer equations at the similarity solution and analyze the evolution of a small perturbation with respect to x. f(x; ) ∼ f0 () + KF(x; );
#(x; ) ∼ #0 () + KD():
(20)
Separation of variables KF(x; ) = x$ F();
KD(x; ) = x$ D();
yields the eigenvalue problem: 1 F + (f0 F + f0 F) = $(f0 F − f0 F); 2
(21)
(22)
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
11
Fig. 16. Eigenvalues of the linearized problem.
1 1 D + (f0 D + #0 F) + #0 F + f0 D = $(f0 D − #0 F); Pr 2
(23)
F(0) = F (0) = D(0) = 0;
(24)
F → 0;
D→0
as → ∞:
If a positive eigenvalue exists a perturbation will be ampli1ed downstream, thus the similarity solution is spatially unstable. A necessary condition for spatial stability is that all eigenvalues of (23) and (24) are negative. (Note that this condition is not suIcient in general). Eigenvalues of (23) and (24) for Pr = 0:2 and 2 are shown in Fig. 16. Consider Pr = 0:2 1rst. Along the lower solution branch a positive eigenvalue exists. It becomes zero at the turning point. Now following the upper branch this eigenvalue becomes negative. This is in agreement with the results from previous section (Fig. 12) where the solution branches o/ from the lower similarity solution and tends to the upper one. For Pr=2 the situation is more complicate. Starting from the Blasius solution a positive eigenvalue is found. Approaching the turning point this eigenvalue seems to increase beyond any bound. A second branch of eigenvalues has been found which are negative along the lower branch and become zero at the turning point. These eigenvalues are positive along the upper branch. In that case no spatially stable similarity solution exists which is agreement with the results of Section 4. 6. Uniqueness of solutions Since a positive eigenvalue associated to the Blasius solution ( = 0, lower branch) exists, one is tempted to conclude that the initial value problem has a non-unique solution with the asymptotic expansion f(x; ) = f0 () + cx$ F(n);
#(x; ) = #0 () + cx$ D(n);
for x 1;
(25)
where (F; D) and $ ¿ 0 are the eigenfunction and eigenvalue of the eigenvalue problem (23) and (24) with F (0) = 1, respectively, and c is an arbitrary constant. The corresponding temperature in the uid would be locally near the leading edge of the plate of the form TH − T∞ ∼ KTH L x$−1 D():
(26)
12
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
Such a temperature distribution is non singular only if $ ¿ 1. Numerical results for Pr = 0:2 and 2 show that this is not the case. Moreover = 0 corresponds to the forced convection ow. Thus multiple solutions would contradict the uniqueness of the Blasius solution shown by Libby and Fox (1964). Note that the situation encountered here is di/erent from the case of mixed convection over a cooled horizontal plate, where multiple solutions seem to be indeed physical meaningful, cf. Steinr'uck (2001). 7. Conclusions New similarity solution of mixed convection ow along a vertical plate have been found. A discussion of the physical relevance of the similarity solutions shows that only the wall jet like similarity solutions corresponding to the upper solution branch serve as asymptotic limit for large distances from the leading edge. The similarity solutions corresponding to the lower branch are not spatially stable. They can be attained for very peculiar wall temperature distributions as intermediate solutions. The solution of the steady state boundary layer equations with a nonsingular wall temperature distribution is unique. In general the asymptotic behavior for x → ∞ is given by the upper similarity solution if Pr ¡ Prc or by a natural-convection wall plume described in Merkin (1985) if Pr ¿ Prc . It is worth to note that the limiting behavior of the solution of the mixed convection boundary-layer equations for x → ∞ for Pr ¡ Prc is described by a similarity solution of the 1rst kind while for Pr ¡ Prc it is described by a similarity solution of the second kind (see: Barenblatt, 1996). Due to the slow spatial convergence to the upper similarity solution it has to be clari1ed under which conditions the similarity solutions discussed in this paper really describe a laminar mixed convection ow along a vertical 1nite plate. Appendix A. A note on the maximum principle Theorem. If a solution (ub ; vb ; Tb ) of the boundary-layer equations, which approximates a solution (un ; vn ; Tn ) of the Navier–Stokes and energy equation√behaves asymptotically as x → ∞ like a similarity solution u ∼ f0 (), v ∼ (f0 () − f0 ())=2 x, T ∼ #0 ()=x where #0 has a change of sign, then the di8erence Tw between the plate temperature and the ambient temperature must have a change of sign either. √ H ReL . Here we make the coordinate yH perpendicular to the plate dimensionless with L= We assume that #0 (0) ¿ 0 and that #0 has a negative minimum #∗ ¡ 0 at =∗ . Since Tb behaves asymptotically like #0 =x for x → ∞ to every & ¿ 0 there exists an X with Tb (x; y) − 1 #0 () ¡ &=x for any x ¿ X: (A.1) x Using that Tb approximates a solution of the Navier–Stokes equation we can conclude that |Tb (x; y) − Tn (x; y)| ¡ &=X
(A.2)
H. Steinr+uck / Fluid Dynamics Research 32 (2003) 1 – 13
13
for any x ¿ X and Rex suIciently large. Taking & ¡ − #∗ =8 we estimate √ √ 1 1 Tn (X; X ∗ ) ¡ |Tb (X; X ∗ ) − #∗ | + |Tb − Tn | + #∗ X X 2& 3 ∗ 1 ¡ (A.3) + #∗ ¡ # : X X 4X We chose a square with its center at the leading edge, side length 2X and two of its sides parallel to the plate. The dimensionless temperature minimum on the boundary of the square is then larger than (1=2X )#∗ + 2&=X ¿ (3=4X )#∗ . √ Thus we have shown that inside the square, namely at (X; ∗ X ) there is a temperature below the minimum value of the temperature at the boundary of the square. The maximum principle Gilbarg and Trudinger (1983) for the energy equation states that the temperature values interior of a closed control surface must be between the minimum and maximum temperature on the boundary of the control surface. We chose as the control surface the boundary of the square and both surfaces of the plate interior of the square. Then by the maximum principle it follows that there must be a location on the surface of the plate where the temperature is below that of the ambient uid. References Barenblatt, G.I., 1996. Scaling, Self-similarity and Intermediate Asymptotics. Cambridge University Press, Cambridge. Bejan, A., 1995. Convection Heat Transfer. Wiley, New York. Gebhart, B., Jaluria, Y., Mahajan, R.L., Sammkia, B., 1988. Bouyancy-induced Flows and Transport. Hemisphere Publishing Corporation, New York. Gilbarg, D., Trudinger, N., 1983. Elliptic Partial Di/erential Equations of Second Order. Springer, Berlin. Libby, P., Fox, H., 1964. Some perturbation solution in laminar boundary-layer theory, part I: The momentum equation. J. Fluid Mech. 17, 433–449. Merkin, J.H., 1985. A note on the similarity solutions for free convection on a vertical plate. J. Eng. Math. 19, 189–201. Merkin, J.H., Pop, I., 2002. Mixed convection along a vertical surface: similarity solutions for uniform ow. Fluid Dyn. Res. 30, 233–250. Steinr'uck, H., 1995. Mixed convection over a horizontal plate: self-similar and connecting boundary-layer ows. Fluid Dyn. Res. 15, 113–127. Steinr'uck, H., 2001. A review of the mixed convection boundary-layer ow over a horizontal cooled plate. GAMM Mitteilungen 24, 127–158.