Convection with the Cattaneo heat flux law Brian Straughan - - PowerPoint PPT Presentation

Convection with the Cattaneo heat flux law Brian Straughan Department of Mathematics Durham University DH1 3LE U.K. email: brian.straughan@durham.ac.uk

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Thermal convection Cattaneo, C.: Sulla conduzione del calore. Atti Sem. Mat. Fis.

• Univ. Modena 3, 83-101 (1948).

Straughan, B., Franchi, F.: B´ enard convection and the Cattaneo law of heat conduction. Proc. Roy. Soc. Edinburgh A 96, 175– 178 (1984).

• C. Christov. On frame indifferent formulation of the Maxwell -

Cattaneo model of finite - speed heat conduction. Mech. Res. Communications 36 (2009) 481–486.

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Applications Vadasz, J.J., Govender, S. and Vadasz, P. Heat transfer en- hancement in nano-fluids suspensions: possible mechanisms and

• explanations. Int. J. Heat Mass Transfer 48 (2005), 2673–2683.

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Equations for convection vi,t + vjvi,j = −1 ρp,i + αgkiT + ν∆vi (1) vi,i = 0. (2) ρcp(Tt + viT,i) = −qi,i (3)

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Heat flux equation? Christov 2009 τ(qi,t + vjqi,j − qjvi,j) = −qi − κT,i , (4) Fox 1969? See Straughan and Franchi 1984 τ

∂qi

∂t + vj ∂qi ∂xj − 1 2vi,jqj + 1 2vj,iqj

• = −qi − κT,i

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boundary conditions vi = 0, z = 0, d, T = TL, z = 0, T = TU, z = d, TL > TU, both constants.

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basic (conduction) solution ¯ vi = 0, ¯ T = −βz + TL, ¯ qi = (0, 0, κβ) β is the temperature gradient, β = TL − TU d .

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To analyse instability introduce perturbations (ui, θ, π, qi) vi = ¯ vi + ui, T = ¯ T + θ, p = ¯ p + π, qi = ¯ qi + qi.

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non-dimensional perturbation equations are Cattaneo-Christov model ui,t + ujui,j = −π,i + Rkiθ + ∆ui, ui,i = 0, Pr(θt + uiθ,i) = Rw − qi,i, 2CPr(qi,t + ujqi,j − qjui,j) = −qi + 2CRui,z − θ,i . (5)

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linearized perturbation equations are Cattaneo-Fox model ui,t = −π,i + Rθki + ∆ui , ui,i = 0, Prθ,t = Rw − qi,i , 2CPrqi,t = CR(ui,z − w,i) − qi − θ,i . (6)

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Comparison, Cattaneo-Christov model 2CPrqi,t = −qi + 2CRui,z − θ,i Cattaneo-Fox model 2CPrqi,t = CR(ui,z − w,i) − qi − θ,i

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Linearized instability analysis Put ui(x, t) = eσtui(x), θ(x, t) = eσtθ(x), qi(x, t) = eσtqi(x), π(x, t)eσtπ(x).

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Cattaneo-Christov σui = −π,i + Rkiθ + ∆ui, ui,i = 0, σPrθ = Rw − qi,i, 2σCPrqi = −qi + 2CRui,z − θ,i . (7)

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Eliminate π and put Q = qi,i σ∆w = R∆∗θ + ∆2w σPrθ = Rw − Q 2σCPrQ = −Q − ∆θ , (8) ∆∗ = ∂2/∂x2 + ∂2/∂y2

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solve (8) by a D2 Chebyshev tau method Boundary conditions w = wz = θ = 0, z = 0, 1. (9)

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Introduce χ = ∆w. Write w, χ, θ and Q in the form w = W(z)f(x, y), χ = χ(z)f(x, y), θ = Θ(z)f(x, y), Q = Q(z)f(x, y),

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(D2 − a2)W − χ = 0, (D2 − a2)χ − Ra2Θ = σχ, (D2 − a2)Θ + Q = −2σCPrQ, Q − RW = −σPrΘ. (10)

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Expand W, χ, Θ and Q in terms of Chebyshev polynomials, e.g. W(z) = N

n=0 wnTn(z), N odd.

Due to the fact that Tn(±1) = (±1)n, T ′

n(±1) = (±1)n−1n2,

the boundary conditions (9) become, w0 + w2 + w4 + . . . + wN−1 = 0, w1 + w3 + . . . + wN = 0 (11)

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with a similar representation for θn. Also, w1 + 32w3 + 52w5 + . . . + N2wN = 0, 4w2 + 42w4 + . . . + (N − 1)2wN−1 = 0. (12)

Ax = σBx, where the (N + 1) × (N + 1) matrices A and B are given by

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A =

                         

4D2 − a2I −I BC1 0 . . . 0 0 . . . 0 0 . . . 0 BC2 0 . . . 0 0 . . . 0 0 . . . 0 4D2 − a2I −Ra2I BC3 0 . . . 0 0 . . . 0 0 . . . 0 BC4 0 . . . 0 0 . . . 0 0 . . . 0 4D2 − a2I I 0 . . . 0 0 . . . 0 BC5 0 . . . 0 0 . . . 0 0 . . . 0 BC6 0 . . . 0 −RI I

                         

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B =

                         

0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 I 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 −2CPrI 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 −PrI

                         

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x = (w0, . . . , wN, χ0, . . . , χN, Θ0, . . . , ΘN, Q0, . . . , QN)

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C a Ra σ1 2.1 × 10−2 3.12 1707.765 2.2 × 10−2 3.12 1707.765 2.2 × 10−2 4.87 1728.151 −2.275 2.3 × 10−2 3.12 1707.765 2.3 × 10−2 4.87 1647.279 +2.371 2.4 × 10−2 3.12 1707.765 2.4 × 10−2 4.87 1573.613 +2.444 3.0 × 10−2 4.85 1240.442 −2.617 3.2 × 10−2 4.84 1158.610 −2.624 3.4 × 10−2 4.84 1086.887 −2.627 4.0 × 10−2 4.83 916.600 −2.586 10−1 4.80 356.918 +1.949

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From the table it is evident that for values of C below a transition value CT, with CT ∈ [2.2 × 10−2, 2.3 × 10−2], stationary convec- tion is the mechanism by which thermal convection starts. The wavenumber a = 3.12 in this regime. Once C increases beyond CT there is a bifurcation and the dominant eigenvalue changes. Convection is then by oscillatory convection, σ1 = 0, with a dif- ferent, and larger wavenumber. This means that the convection cells become narrower. As C increases further the convection cells continue to become narrower and the Rayleigh number de- creases.

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To sum up, we have found that the Christov model coupled with the Cattaneo one leads to a very interesting effect in thermal convection. For very small Cattaneo number convection is by stationary convection only and the convection cells have a fixed aspect ratio. As C increases a threshold is reached and convec- tion then switches to oscillatory convection (Hopf bifurcation) with narrower cells. Further increase in the Cattaneo number leads to further narrowing of the convection cells and lowering

• f the critical Rayleigh number which means thermal convection
• ccurs more easily.

Thus, the properly invariant heat flux law

• f Christov leads to an important effect in the field of thermal

convection.

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Cattaneo-Fox

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C a Ra σ1 3.12 1707.765 10−4 3.12 1711.180 10−3 3.09 1742.393 10−2 2.89 2113.893 1.2 × 10−2 2.84 2113.969 1.4 × 10−2 2.80 2321.775 1.4 × 10−2 5.01 2696.505

• 3.703971

1.5 × 10−2 2.77 2378.814 1.5 × 10−2 5.00 2497.573

• 3.873272

1.550214 × 10−2 2.760 2408.291 1.550214 × 10−2 4.994 2408.291 3.932125 1.6 × 10−2 2.75 2438.093 1.6 × 10−2 4.99 2325.822

• 3.981396

1.8 × 10−2 2.70 2563.800 1.8 × 10−2 4.97 2044.356 4.086621 2.0 × 10−2 2.65 2699.881 2.0 × 10−2 4.96 1823.474 4.119397 1.0 × 10−1 4.87 342.568

• 2.479659

The numerical values given in the table represent instability val- ues at the threshold where the conduction solution becomes (lin- early) unstable. The eigenvalue σ = σr + iσ1 and all values in the table are when σr = 0. Oscillatory convection corresponds to σ1 = 0. The Prandtl number is fixed with value Pr = 6.

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We see that the stationary convection behaviour witnessed asymp- totically by Straughan & Franchi for small C, presists for two fixed surfaces. As C increases (C small) the critical Rayleigh number Ra likewise increases. (The critical wavenumbers are calculated to two decimal places apart from at the Cattaneo number transition). However, at C = CT = 1.550214 × 10−2, we witness a striking transition. For C > CT, a Hopf bifurcation oc- curs and convection switches from stationary convection to one where oscillatory convection is dominant. The critical Rayleigh number then begins to rapidly decrease. Also, the wave num- ber increases and this means the transition is accompanied by a switch from a larger to a narrower convection cell. Mathemat- ically, the transition is manifest by the lowest critical Rayleigh number value switching from one eigenvalue σ(1) to another σ(2). The table shows clearly how the second eigenvalue begins to dominate as C moves through the transition values.

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