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 1

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. 2

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. 3

Equations for convection v i,t + v j v i,j = − 1 ρp ,i + αgk i T + ν ∆ v i (1) v i,i = 0 . (2) ρc p ( T t + v i T ,i ) = − q i,i (3) 4

Heat flux equation? Christov 2009 τ ( q i,t + v j q i,j − q j v i,j ) = − q i − κT ,i , (4) Fox 1969? See Straughan and Franchi 1984 � ∂q i ∂q i − 1 2 v i,j q j + 1 � τ ∂t + v j 2 v j,i q j = − q i − κT ,i ∂x j 5

boundary conditions v i = 0 , z = 0 , d, T = T L , z = 0 , T = T U , z = d, T L > T U , both constants. 6

basic (conduction) solution ¯ ¯ v i = 0 , T = − βz + T L , q i = (0 , 0 , κβ ) ¯ β is the temperature gradient, β = T L − T U . d 7

To analyse instability introduce perturbations ( u i , θ, π, q i ) v i = ¯ v i + u i , T = ¯ T + θ , p = ¯ p + π , q i = ¯ q i + q i . 8

non-dimensional perturbation equations are Cattaneo-Christov model u i,t + u j u i,j = − π ,i + Rk i θ + ∆ u i , u i,i = 0 , (5) Pr ( θ t + u i θ ,i ) = Rw − q i,i , 2 CPr ( q i,t + u j q i,j − q j u i,j ) = − q i + 2 CRu i,z − θ ,i . 9

linearized perturbation equations are Cattaneo-Fox model u i,t = − π ,i + Rθk i + ∆ u i , u i,i = 0 , (6) Prθ ,t = Rw − q i,i , 2 CPrq i,t = CR ( u i,z − w ,i ) − q i − θ ,i . 10

Comparison, Cattaneo-Christov model 2 CPrq i,t = − q i + 2 CRu i,z − θ ,i Cattaneo-Fox model 2 CPrq i,t = CR ( u i,z − w ,i ) − q i − θ ,i 11

Linearized instability analysis Put u i ( x , t ) = e σt u i ( x ) , θ ( x , t ) = e σt θ ( x ) , q i ( x , t ) = e σt q i ( x ) , π ( x , t ) e σt π ( x ) . 12

Cattaneo-Christov σu i = − π ,i + Rk i θ + ∆ u i , u i,i = 0 , (7) σPrθ = Rw − q i,i , 2 σCPrq i = − q i + 2 CRu i,z − θ ,i . 13

Eliminate π and put Q = q i,i σ ∆ w = R ∆ ∗ θ + ∆ 2 w (8) σPrθ = Rw − Q 2 σCPrQ = − Q − ∆ θ , ∆ ∗ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 14

solve (8) by a D 2 Chebyshev tau method Boundary conditions w = w z = θ = 0 , z = 0 , 1 . (9) 15

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 ) , 16

( D 2 − a 2 ) W − χ = 0 , ( D 2 − a 2 ) χ − Ra 2 Θ = σχ, (10) ( D 2 − a 2 )Θ + Q = − 2 σCPrQ, Q − RW = − σPr Θ . 17

Expand W, χ, Θ and Q in terms of Chebyshev polynomials, e.g. W ( z ) = � N n =0 w n T n ( z ), N odd. Due to the fact that T n ( ± 1) = ( ± 1) n , n ( ± 1) = ( ± 1) n − 1 n 2 , T ′ the boundary conditions (9) become, w 0 + w 2 + w 4 + . . . + w N − 1 = 0 , (11) w 1 + w 3 + . . . + w N = 0 18

with a similar representation for θ n . Also, w 1 + 3 2 w 3 + 5 2 w 5 + . . . + N 2 w N = 0 , (12) 4 w 2 + 4 2 w 4 + . . . + ( N − 1) 2 w N − 1 = 0 .

A x = σB x , where the ( N + 1) × ( N + 1) matrices A and B are given by 19

4 D 2 − a 2 I   − I 0 0 BC 1 0 . . . 0 0 . . . 0 0 . . . 0     BC 2 0 . . . 0 0 . . . 0 0 . . . 0          4 D 2 − a 2 I  − Ra 2 I 0 0     BC 3 0 . . . 0 0 . . . 0 0 . . . 0       A = BC 4 0 . . . 0 0 . . . 0 0 . . . 0         4 D 2 − a 2 I 0 0 I       0 . . . 0 0 . . . 0 BC 5 0 . . . 0     0 . . . 0 0 . . . 0 BC 6 0 . . . 0         − RI 0 0 I 20

  0 0 0 0 0 . . . 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       B = 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0         0 0 0 − 2 CPrI      0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0      0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0         0 0 − PrI 0 21

x = ( w 0 , . . . , w N , χ 0 , . . . , χ N , Θ 0 , . . . , Θ N , Q 0 , . . . , Q N ) 22

C a Ra σ 1 2 . 1 × 10 − 2 3 . 12 1707 . 765 0 2 . 2 × 10 − 2 3 . 12 1707 . 765 0 2 . 2 × 10 − 2 4 . 87 1728 . 151 − 2 . 275 2 . 3 × 10 − 2 3 . 12 1707 . 765 0 2 . 3 × 10 − 2 4 . 87 1647 . 279 +2 . 371 2 . 4 × 10 − 2 3 . 12 1707 . 765 0 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 23

From the table it is evident that for values of C below a transition value C T , with C T ∈ [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 C T there is a bifurcation and the dominant eigenvalue changes. Convection is then by oscillatory convection, σ 1 � = 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. 24

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 of the critical Rayleigh number which means thermal convection occurs more easily. Thus, the properly invariant heat flux law of Christov leads to an important effect in the field of thermal convection. 25

Cattaneo-Fox 26

C a Ra σ 1 0 3.12 1707.765 0 10 − 4 3.12 1711.180 0 10 − 3 3.09 1742.393 0 10 − 2 2.89 2113.893 0 1 . 2 × 10 − 2 2.84 2113.969 0 1 . 4 × 10 − 2 2.80 2321.775 0 1 . 4 × 10 − 2 5.01 2696.505 -3.703971 1 . 5 × 10 − 2 2.77 2378.814 0 1 . 5 × 10 − 2 5.00 2497.573 -3.873272 1 . 550214 × 10 − 2 2.760 2408.291 0 1 . 550214 × 10 − 2 4.994 2408.291 3.932125 1 . 6 × 10 − 2 2.75 2438.093 0 1 . 6 × 10 − 2 4.99 2325.822 -3.981396 1 . 8 × 10 − 2 2.70 2563.800 0 1 . 8 × 10 − 2 4.97 2044.356 4.086621 2 . 0 × 10 − 2 2.65 2699.881 0 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 = σ r + iσ 1 and all values in the table are when σ r = 0. Oscillatory convection corresponds to σ 1 � = 0. The Prandtl number is fixed with value Pr = 6. 27

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 = C T = 1 . 550214 × 10 − 2 , we witness a striking transition. For C > C T , 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. 28