Abstract.: The steady mixed convection boundary layer flow over a vertical surface with prescribed heat flux is revisited in this Note. The subset of solutions which can be obtained with the aid of the Reynolds analogy is discussed in a close relationship with the dual solutions reported by Merkin and Mahmood [1] for impermeable, and more recently by Ishak et al. [2], for permeable surface

Magyari, E.

Pop, I.

RERO DOC Digital Library

Z. angew. Math. Phys. 60 (2009) 324–3330044-2275/09/020324-10DOI 10.1007/s00033-008-7131-1c© 2008 Birkha¨user Verlag, BaselZeitschrift fu¨r angewandteMathematik und Physik ZAMPThe Reynolds analogy for the mixed convectionover a vertical surface with prescribed heat fluxE. Magyari∗ and I. PopAbstract. The steady mixed convection boundary layer flow over a vertical surface with pre-scribed heat flux is revisited in this Note. The subset of solutions which can be obtained with theaid of the Reynolds analogy is discussed in a close relationship with the dual solutions reported byMerkin and Mahmood [1] for impermeable, and more recently by Ishak et al. [2], for permeablesurfaces.Mathematics Subject Classification (2000). 76R05, /6R10.Keywords. Boundary layer, mixed convection, Reynolds analogy, dual solutions,suction/injection.1. IntroductionThe steady mixed convection boundary layer flow of an incompressible fluid overan impermeable or permeable vertical surface with prescribed temperature, or withprescribed heat flux is governed in the Boussinesq approximation by the continuity,momentum and energy equations, [1], [2],ux + vy = 0 (1)uux + vuy = υuyy + UUx + gβ(T − T∞) (2)uTx + vTy = αTyy (3)where the subscripts denote partial derivatives with respect to the coordinates xand y, and U(x) = U0·(x/L)m is the free stream velocity. The boundary conditionsadopted are [1], [2],u∣∣y=0= 0, v∣∣y=0= v0 ·( xL)m−12, u∣∣y→∞ = U(x)−k dTdy∣∣∣∣y=0= q0 ·( xL) 5m−32, T∣∣y→∞ = T∞.(4)∗Corresponding author.Vol. 60 (2009) The Reynolds analogy for the mixed convection 325With the aid of the stream function ψ (u = ψy, v = −ψx) and the similaritytransformations,ψ =√υLU0( xL)m+12f(η), η =√U0υL( xL)m−12y (5)T = T∞ +∆T( xL)2m−1θ(η). (6)Eqs. (1)–(4) reduce to the system of ordinary differential equationsf ′′′ +m+ 12ff ′′ +m(1− f ′2) + λθ = 0 (7)θ′′Pr+m+ 12fθ′ + (1− 2m)f ′θ = 0 (8)subject to the boundary conditionsf(0) = f0, f ′(0) = 0, f ′(∞) = 1 (9)θ′(0) = −1, θ(∞) = 0. (10)In the above equations L, ∆T = (q0/k)(υL/U0)1/2 and U0 specify the length,temperature and velocity scales, respectively, the primes denote differentiationswith respect to the similarity variable η, and λ = Gr/Re2 is the mixed convectionparameter with Gr = gβ∆TL3/υ2, and Re = U0L/υ the Grashof and Reynoldsnumbers, respectively. The other notations, as well as the choice of the coordinatesystem coincide with those of [1] and [2].In References [1] and [2], the structure of solution space of the boundary valueproblem (7)–(10) has been investigated for impermeable (f0 = 0) and both im-permeable and permeable (f0 6= 0) surfaces, respectively. The main issue of thepresent Note is to discuss the subset of solutions of this boundary value problemwhich can be obtained with the aid of the Reynolds analogy.2. The Reynolds analogyThe Reynolds analogy of the momentum and energy balance equations is basedon the simple observation that for a uniform free stream (m = 0, i.e., Ux = 0)and in the absence of buoyancy forces, Eqs. (2) and (3) possess for Pr = 1 (i.e.,for υ = α) the same form for u and T , i.e. Eqs. (2) and (3) reduce in fact to asingle equation only. Consequently, for certain velocity and temperature boundaryconditions, there always exists a temperature solution which is a linear functionof streamwise velocity u.This idea can also be extended for the case of the full Eqs. (2) and (3), with thedistinction that the coefficients of the linear function mentioned should firstly betaken as functions of the streamwise coordinate x. Indeed, writing T = A(x)u +B(x), the asymptotic conditions u∣∣y→∞ = U(x) and T∣∣y→∞ = T∞ immediately326 E. Magyari and I. Pop ZAMPimplyT = T∞ +A(x)(u− U). (11)Substituting Eq. (11) in Eqs. (2) and (3) (with α = υ), one easily sees that theresulting equations become identical when A = const. ≡ A0, and (Ux−gβA0)(U−u) = 0. Having in mind that U = U0 · (x/L)m, the latter condition requires m = 1and implies further A0 = U0/(gβL). Accordingly, Eq. (11) becomesT = T∞ +U20gβL(uU0− xL)= T∞ +∆Tλ(uU0− xL)(12)and, as a consequence of this relationship, Eqs. (2) and (3) reduce to the sameformuux + vuy = υuyy +U0Lu. (13)Furthermore, Eqs. (5) and (6) give for m = 1u = U0xLf ′(η), v = −√U0υLf(η), T = T∞ +∆TxLθ(η) (14)and thus equations (12) and (13) becomeT = T∞ +∆TxLf ′(η)− 1λ, (15)andf ′′′ + ff ′′ + f ′(1− f ′) = 0 (16)respectively. Comparing the third Eq. (14) to Eq. (15), one obtains that (withinthe Reynolds analogy) the similar temperature and velocity fields θ and f ′ corre-sponding to m = 1 and Pr = 1 are related to each other by the simple relationshipθ(η) =f ′(η)− 1λ(Reynolds analogy). (17)3. Solution and discussionAs shown above, for the subset of solutions {f ′, θ} = {f ′, (f ′ − 1)/λ} given bythe Reynolds analogy only the single Eq. (16) has to be solved, instead of thesystem of coupled Eqs. (7) and (8). The corresponding solutions describe themixed convection stagnation point flow (m = 1 ) of a fluid with Prandtl numberPr = 1. The goal of the present Section is the investigation of the correspondingboundary value problem (16), (9) in detail.We first notice that Eqs. (17) and (10) yield for the dimensionless wall temper-ature θ(0) and skin friction f ′′(0) the simple relationshipsθ(0) = − 1λand f ′′(0) = −λ. (18)Vol. 60 (2009) The Reynolds analogy for the mixed convection 327Figure 1. Plot of f ′(∞) ' f ′(10) as a function of λ = −f ′′(0) for f0 = 0. The two intersectionpoints of the curve f ′(10) with the horizontal line 1 (marked by dots) correspond to theλ-values λ1 = −0.7314105 and λ2 = +0.3026979, respectively. The associated similar velocityprofiles f ′(η) are plotted in Figure 2.Therefore, the task is to find the values of the mixed convection parameter λ, forwhich the solution of the initial value problemf ′′′ + ff ′′ + f ′(1− f) = 0,f(0) = f0, f ′(0) = 0, f ′′(0) = −λ(19)satisfies (for a specified value of f0) the asymptotic condition f ′(−∞) = 1.Being faced here with an initial value problem with an additional condition, thesolution is easily found. The procedure is illustrated in Figure 1 where f ′(∞) ∼=f ′(10) has been plotted as a function of λ for f0 = 0 (impermeable surface), andthis curve has been intersected with the straight line f ′(10) = 1 according tothe asymptotic condition. Two intersection points of the curve f ′(10) with thehorizontal line 1 were found, namely at λ1 = −0.7314105 and λ2 = +0.3026979,respectively. The procedure has then to be repeated subsequently for f ′(12),f ′(15), etc. until the values of λ1 and λ2 do not change within the desired precision.These solutions, corresponding to (f0, λ) = (0, λ1) and (f0, λ) = (0, λ2) are plottedin Figure 2. The results of similar calculations performed for f0 = −0.1 (injection)and f0 = +0.1 (suction) are collected (together with the case f0 = 0) in Table 1.We see that the left and right intersection points of the curve f ′(∞) with thestraight line f ′(∞) = 1, having for a given f0 the respective abscisses λLeft(f0) < 0and λRight(f0) > 0, specify two solution branches of the boundary value problem(16), (9). In the asymptotic limit f0 → ∞ of large suction velocities, the de-328 E. Magyari and I. Pop ZAMPλ f ′′(0) θ(0) Branchf0 Left/Right Left/Right Left/Right f′′(0); θ(0)−0.1−0.664197 +0.664197 +1.505577 U ; L+0.2504476 −0.2504476 −3.9928512 L; L0−0.7314105 +0.7314105 +1.3672212 U ; L+0.3026979 −0.3026979 −3.3036238 L; L+0.1−0.8006085 +0.8006085 +1.2490499 U ; L+0.3619513 −0.3619513 −2.7628026 L; LTable 1. The left and right values of λ, f ′′(0) and θ(0) corresponding to three specified valuesof the suction/injection parameter f0. The symbols U and L refer to the upper and lowerbranch of solutions according to Figures 1 and 3 of Reference [2], respectively.Figure 2. Plot of similar velocity profiles f ′(η) corresponding to the skin frictionsf ′′(0) = −λ1 = 0.7314105 and f ′′(0) = −λ2 = −0.3026979, respectively, associated with thesame value f0 = 0 of the suction/injection parameter.Vol. 60 (2009) The Reynolds analogy for the mixed convection 329Figure 3. Plot of f ′(∞) ' f ′(13) as a function of λ for f0 = −0.65. The six points of the curvef ′(13) with the horizontal line 1 (marked by dots) correspond to the λ-values λ1 = −0.3369764,λ2 = −0.1056770, λ3 = −0.0393022, λ4 = +0.0098145, λ5 = +0.0370746 and λ6 = +0.0676765respectively. The associated similar velocity profiles f ′(η) are plotted in Figure 4.pendence of the corresponding skin friction coefficients f ′′(0) = −λ on f0 can beestimated as follows:In the case of the left branch we first substitute in Eqs. (16) and (9)η =ξf0, f(η) = f0 +F (ξ)f0(Left branch). (20)This substitution transforms Eqs. (16) and (9) in...F +(1 +Ff20)F¨ +F˙ (1− F˙ )f20= 0,F (0) = 0, F˙ (0) = 0, F˙ (∞) = 1(21)where the dot denotes differentiation with respect to the scaled variable ξ.The transformed Eqs. (21) suggest the power series expansion of the formF (ξ) = F0(ξ) +F1(ξ)f20+ . . . (22)In this way, we obtain to the zeroth and the first order approximations in 1/f20the linear boundary value problems...F 0 + F¨0 = 0,F0(0) = 0, F˙0(0) = 0, F˙0(∞) = 1(23)330 E. Magyari and I. Pop ZAMPand ...F 0 + F¨1 = −F0F¨0 − F˙0(1− F˙0),F0(0) = 0, F˙0(0) = 0, F˙0(∞) = 0.(24)As expected, the zeroth order approximation (23) yields the asymptotic suctionprofileF˙0 = 1− e−ξ, F0 = ξ − 1 + e−ξ. (25)For the solution of the first order correction (24) we obtain by elementary calcu-lationsF1 = 2−(2 + 2ξ +12ξ2)e−ξ (26)such thatF (ξ) = ξ − 1 + e−ξ + 1f20[2−(2 + 2ξ +12ξ2)e−ξ]. (27)Accordingly, the skin friction coefficient f ′′(0) = f0F¨ (0) of the left solution branchresults asf ′′(0) = f0 +1f0+ . . . (Left branch, f0 →∞). (28)In the case of the right branch, it is convenient to substitute in Eqs. (16) and (9)η =ξf0, f(η) = f0φ(ξ) (Right branch). (29)This substitution transforms Eqs. (16) and (9) in...φ + φφ¨− φ˙2 + 1f20φ˙ = 0,φ(0) = 1, φ˙(0) = 0, φ˙(∞) = 1f20.(30)The transformed Eqs. (30) suggest again a power series expansion of the formφ(ξ) = φ0(ξ) +φ1(ξ)f20+ . . . (31)However, in contrast to the left branch, in the leading order approximation theproblem (30) is still non-linear...φ0 + φ0φ¨0 − φ˙20 = 0,φ0(0) = 1, φ˙0(0) = 0, φ˙0(∞) = 0.(f0 →∞) (32)This problem could be solved only numerically, yielding for the scaled skin frictioncoefficient the value φ¨0(0) = −0.06. Consequently, we arrive at the result that, tothe leading order, the skin friction coefficient f ′′(0) = f30 φ¨(0) of the right solutionbranch scales for large suction velocities asf ′′(0) = −0.04f30 + . . . (Right branch, f0 →∞). (33)Vol. 60 (2009) The Reynolds analogy for the mixed convection 331The higher order approximations can also be determined only numerically in thiscase. It is worth mentioning here that for the left branch the leading order ap-proximation f ′′(0) ∼= f0 obtained from Eq. (28), gives a quite accurate resultalready for f0 = 10, namely f ′′(0) = 10, compared to the “exact” numerical resultf ′′(0) ∼= 10.0976. For the right branch by contrast, the leading order approxima-tion (33) gives an accurate result only around f0 = 104, namely f ′′(0) = −4 · 1010,compared to the “exact” numerical result f ′′(0) = −4.0779 · 1010.The left and right branches discussed above can be continued also for negativevalues of f0, corresponding to a lateral injection of the fluid. A qualitative analy-sis of the boundary value problem (16), (9) shows that the skin friction coefficientf ′′(0) goes to zero smoothly for both solution branches as f0 → ∞. However,the decrease of f ′′(0) is very fast, so that it can be followed by numerical calcu-lations to limited values of f0 < 0 only. Namely, one obtains f ′′Left(0) = 0.001636and f ′′Right(0) = −0.000024 already at f0 < −1.8. At f0 < −1.9, the left valuereduces to f ′′Left(0) = 0.000076 and the right one is not viewable any more. Thisbehaviour, however, is not the only peculiarity of the injection case. In con-trast to the increasing suction, with increasing lateral injection, also the shapeof the curve f ′(∞) shown in Figure 1 changes dramatically, becoming more andmore complex. A cascade of bifurcations, leading to further solution branchesof the boundary value problem (16), (9) with both positive and negative valuesof the skin friction coefficient f ′′(0) occur. This feature is illustrated in Figure3 where f ′(∞) ∼= f ′(13) has been plotted as function of λ for f0 = −0.65. Inthis case six intersection points of the curve f ′(13) with the horizontal line 1were found, namely at λ1 = −0.3369764, λ2 = −0.1056770, λ3 = −0.0393022,λ4 = +0.0098145, λ5 = +0.0370746 and λ6 = +0.0676765, respectively. Thesesolutions, corresponding to (f0, λ) = (−0.65, λn), n = 1, 2, . . . , 6 are plotted inFigure 4.A remarkable property of the subset of solutions resulting from the Reynoldsanalogy is that, according to Eqs. (18), the product of the dimensionless walltemperature θ(0) and the skin friction f ′′(0) always equals unity,θ(0)f ′′(0) = 1. (34)As a consequence, the representative points of the solutions in the characteris-tic parameter plane {θ(0), f ′′(0)} of the present prescribed-flux boundary valueproblem are all located on the branches of an equilateral hyperbola. This fea-ture is illustrated in Figure 5, where the points {θ(0), f ′′(0)} corresponding to the6 + 6 = 12 solutions collected in Table 1 and Figure 3 are shown.4. Summary and conclusionsThe subset of solutions of the title problem which can be obtained with the aidof the Reynolds analogy for the stagnation flow (m = 1) of a fluid with Pr = 1,has been investigated in a close relationship with the solutions reported by Merkin332 E. Magyari and I. Pop ZAMPFigure 4. Plot of similar velocity profiles f ′(η) corresponding to the skin frictions f ′′(0) = −λn,n = 1, 2, . . . , 6, associated with the six λ-values specified in Figure 3 and the same valuef0 = −0.65 of the injection parameter.Figure 5. The representative points corresponding to the 6 + 6 = 12 solutions collected in Table1 and Figure 3 are all located on the branches of an equilateral hyperbola θ(0)f ′′(0) = 1.Vol. 60 (2009) The Reynolds analogy for the mixed convection 333and Mahmood [1] and more recently by Ishak et al. [2]. Our main result showsthat the representative points of this subset of solutions in the parameter plane{θ(0), f ′′(0)} are always located on the branches of an equilateral hyperbola. TheReynolds analogy-approach applies both to the assisting and opposing mixed flowregimes. Among the solutions gained in this way, flow profiles with and withoutbackflow regions were found. The mathematical advantage of the Reynolds analogyconsists of the fact that it decouples the mixed convection momentum and energybalance equations by a linear relationship between the temperature field T andthe streamwise velocity component u of the flow.AcknowledgementWe are indebted to a reviewer for valuable suggestions concerning the asymptoticlimit of massive suction.References[1] J. H. Merkin and T. Mahmood, Mixed convection boundary layer similarity solutions: pre-scribed wall heat flux, J. Appl. Math. Phys (ZAMP) 40 (1989), 51–68.[2] A. Ishak, J. H. Merkin, R. Nazar and I. Pop, Mixed convection boundary layer flow over apermeable vertical surface with prescribed wall heat flux, J. Appl. Math. Phys (ZAMP) 59(2008), 100–123.E. MagyariInstitut fu¨r HochbautechnikETH Zu¨richWolfgang-Pauli Str.1CH-8093 Zu¨richSwitzerlande-mails: magyari@hbt.arch.ethz.ch and magyari@bluewin.chI. PopFaculty of MathematicsUniversity of ClujR-3400 Cluj, CP 253Romania(Received: November 1, 2007; revised: February 29, 2008)Published Online First: October 10, 2008To access this journal online:www.birkhauser.ch/zamp

The Reynolds analogy for the mixed convection over a vertical surface with prescribed heat flux

http://doc.rero.ch/record/315351/files/33_2008_Article_7131.pdf

The Reynolds analogy for the mixed convection over a vertical surface with prescribed heat flux

Abstract

Similar works

Full text

Available Versions

RERO DOC Digital Library