Completely passive natural convection

Transcription

1 Early View publication on wileyonlinelibrary.com (issue an page numbers not yet assigne; citable using Digital Object Ientifier DOI) ZAMM Z. Angew. Math. Mech., 1 6 (2011) / DOI /zamm Completely passive natural convection M. Miklavčič an C. Y. Wang Department of Mathematics, Michigan State University, E. Lansing, MI 48824, USA Receive 2 February 2010, revise 22 October 2010, accepte 10 January 2011 Publishe online 22 February 2011 Key wors Viscous, convection, flow, passive. We show that a unique, nontrivial, natural convection state exists uner the Boussinesq approximation an completely passive bounary conitions. 1 Introuction Natural convection is a basic process which is important in a wie variety of practical applications [1,2]. In essence, a heate flui expans an rises from buoyancy ue to ecrease ensity. Numerous papers have been written on natural or mixe convection in vertical ucts heate on the sie. The effects of ae viscous issipation heat have been consiere [3 8]. The flui motions in all previous works are riven by either an applie pressure graient or by applie heating of the walls. We consier a long uct, with no pressure graient, an with the temperature of the walls the same as the ambient reference temperature. In such a completely passive environment there is no energy input, no temperature ifference, an no motion. But is there a possibility of nontrivial flui motion? The purpose of the present Note is to investigate whether a completely passive vertical uct (no applie pressure graient or applie bounary heat) can support sustaine flui motion. The iea is simple. Flui motion causes issipative heat, heat causes buoyancy, an buoyancy causes flui motion. 2 Parallel plates Fig. 1 shows vertical parallel plates with gap with 2L. In the fully evelope state the momentum an energy equations, uner the well-accepte Boussinesq approximation, are [9] (p. 72 an p. 323) μ 2 w y 2 + ρgβ(t T a)=0, (1) ( ) k 2 T w 2 y 2 + μ y =0. (2) Here μ is the viscosity of the flui, ρ is the ensity, g is the gravitational acceleration, β is the coefficient of thermal expansion, k is the thermal iffusivity, T is the temperature, an T a is the same ambient temperature on the walls. Normalize the lateral coorinate y, which is place at the symmetry axis, by L, the axial velocity w by k/(ρgβl 2 ) an rop the primes. A normalize temperature θ is efine by T T a θ = μk/(ρgβl 2. Then Eqs. (1), (2) give θ + 2 w y 2 =0, 2 θ y 2 + ( w y =0. (3) (4) (5) Corresponing author milan@math.msu.eu cywang@math.msu.eu

2 2 M. Miklavčič an C. Y. Wang: Completely passive natural convection Fig. 1 Flow between parallel plates. Eliminating θ, we obtainthe nonlinearequation 4 w y 4 = ( w. y (6) w an θ are zero on the wall, hence w(1) = 0, Symmetry implies w (0) = 0, y 2 w (1) = 0. y2 3 w (0) = 0. y3 (7) (8) Of course, one solution to Eqs. (6) (8) is the trivial solution w =0, or no motion. We will now show that Eqs. (6) (8) have also a unique nontrivial analytic solution! An analytic solution of (6) must be in the form w(y) = a n y n (9) an a n have to satisfy the recurrence relation (n +4)(n +3)(n +2)(n +1)a n+4 = n (k +1)(n k +1)a k+1 a n k+1 for n 0. (10) Eq. (8) implies a 1 = a 3 =0an hence (10) implies that if n>0 an a n 0then n =4k +2for some k 0. Ifa 2 =0 then (10) implies that all a n =0for n>0 which forces w =0. Hence assume λ =2a 2 0an efine Thus (9) becomes b k =(4k +2)λ k 1 a 4k+2 for k 0. (11) w(y) =a 0 + b k λ k+1 4k +2 y4k+2, (12)

3 ZAMM Z. Angew. Math. Mech. (2011) / 3 n 1 4n(16n 2 1)b n = b k b n 1 k for n 1; b 0 =1. (13) In orer to satisfy (7) we nee to have a 0 = F (λ) = b k λ k+1, an F (λ) =0 (14) 4k +2 λ k (4k +1)b k. (15) Using (13) it is easy to fin a zero of F to be λ = (16) an hence (14) implies w(0) = a 0 = , w (0) = 2a 2 = λ = (17) This establishes existence of the nontrivial solution of Eqs. (6) (8). The total flow rate per with, normalize by k/(ρgβl), is q = 1 1 w(y)y = (18) To prove uniqueness of the analytic solution, we nee to show that F given by Eq. (15) has only one real zero. To o this it is sufficient to show that F (λ) > 0 for all λ. Eq. (13) implies b n > 0 for all n 0, hence we nee to show F (λ) > 0 for λ<0 only. Let h(t) =16t 3/4 F ( t) for t>0. Note that Eq. (15) implies an hence h(t) = 16 ( 1) n nt n 1/4 (4n +1)b n (19) h (t) =t 1/4 ( t) n 1 4n(16n 2 1)b n. (20) Using (13) we obtain h (t) =t 1/4 which can be rewritten as n 1 ( t) n 1 b k b n 1 k (21) Therefore ( h (t) =t 1/4 ( t) n b n. (22) 16t 3/4 F ( t) =h(t) = t ( s 1/4 ( s) n b n s > 0 (23) 0 which proves F (λ) > 0 for λ<0 an hence the uniqueness of the nontrivial solution.

4 4 M. Miklavčič an C. Y. Wang: Completely passive natural convection 3 Circular uct Consier a circular uct, Fig. 2. A similar normalization gives ( 2 r ( W W =. (24) r r r The corresponing bounary conitions are W r (0) = 0, W (1) = 0, ( 2 r r ) W (0) = 0, (25) r r ( 2 r r ) W (1) = 0. (26) r W 1 r Fig. 2 (online colour at: ) Flow in a circular uct. Of course, one solution to Eqs. (24) (26) is the trivial solution W =0, or no motion. We will now show, just like in the previous section, that Eqs. (24) (26) have also a unique nontrivial analytic solution. An analytic solution of (24) has to be in the form W (r) = A n r n (27) an A n have to satisfy the recurrence relation (n +2 (n +4 A n+4 = n (k +1)(n k +1)A k+1 A n k+1 for n 0. (28) (25) implies A 1 = A 3 =0an hence (28) implies that if n>0 an A n 0then n =4k +2for some k 0. IfA 2 =0 then (28) implies that all A n =0for n>0 which forces W =0. Hence assume λ =2A 2 0an efine B k as in (11). We have again W (r) =A 0 + B k λ k+1 4k +2 r4k+2, (29)

5 ZAMM Z. Angew. Math. Mech. (2011) / 5 n 1 32n 2 (2n +1)B n = B k B n 1 k for n 1; B 0 =1. (30) In orer to satisfy (26) we nee to have A 0 = B k λ k+1 4k +2 an G(λ) =0, (31) G(λ) = λ k (2k +1)B k. (32) Using (30) it is easy to fin a zero of G to be λ = (33) which shows existence of the solution for Eqs. (24) (26) with W (0) = A 0 = , W (0) = 2A 2 = λ = (34) The total flow rate per with, normalize by k/(ρgβ),is Q =2π 1 0 w(r)rr= (35) W in Fig. 2 is actually an accurate representation of W. In orer to show uniqueness it is enough to show that G (λ) > 0 for all λ. Leth(λ) =32λG (λ) an note that hence h (λ) = λ n 1 32n 2 (2n +1)B n = 32λG (λ) =h(λ) = λ 0 λ n 1 n 1 ( B k B n 1 k = λ n B n (36) ( s n B n s (37) proving G (λ) > 0 for all λ an hence uniqueness. 4 Results an iscussion Fig. 3 shows the unique velocity profile for completely passive flow between parallel plates an in the circular uct. We mention that in orer to attain such a state, one must start the flow somehow, either with an initial heat input or with a primer pump. Our propose passive pump is relate to the thermo-siphon, heat is utilize for natural convection. The major ifference is that the thermo-siphon requires external heat input while our pump is completely passive. Does the completely passive pump violate the secon law of thermoynamics? The answer is no. The system is not close since flui enters from the bottom an exits at the top with an irreversible thermal expansion. Grante that the theory may have efects, such as entrance effects an non exactness of the Boussinesq approximation etc, our analysis shows such a perpetual passive state exists, thus can be maintaine with little effort even with the efects.

6 6 M. Miklavčič an C. Y. Wang: Completely passive natural convection W r W Parallel Plates w Circular Duct W r References Fig. 3 Comparison of flow in circular uct an parallel plates. [1] S. Ostrach, Laminar Flow with Boy Forces, in: High Spee Aeroynamics an Jet Propulsion, Vol. 4, eite by F. K. Moore (Princeton University Press, Princeton, N.J., 1964). [2] Y. Jaluria, Natural Convection (Pergamon, Oxfor, 1980). [3] B. Gebhart, Effects of viscous issipation in natural convection, J. Flui Mech. 14, (1962). [4] M. S. Rokerya an M. Iqbal, Effects of viscous issipation on combine free an force convection through vertical concentric annuli, Int. J. Heat Mass Transf. 14, (1971). [5] Y. Joshi an B. Gebhart, Effect of pressure stress work an viscous issipation in some natural convection flows, Int. J. Heat Mass Transf. 24, (1981). [6] A. Barletta, Laminar mixe convection with viscous issipation in a vertical channel, Int. J. Heat Mass Transf. 41, (1998). [7] E. Zanchini, Effect of viscous issipation on mixe convection in a vertical channel with bounary conitions of the thir kin, Int. J. Heat Mass Transf. 41, (1998). [8] A. Barletta, Combine force an free convection with viscous issipation in a vertical circular uct, Int. J. Heat Mass Transf. 42, (1999). [9] F. M. White, Viscous Flui Flow, 3r eition (McGraw-Hill, Boston, 2006).