, - [1]. - [2]., - - 2:10:20,. -,. -,. -,, -,. -, 1/3,,, [5] -. -

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1 DOI: /mmph ,..,., , -. : ; ; - ;.. -., - -,. - [1]. - []., -- :10:0,. -. -,. -, -. [3 5], - :0:40,. -,, -,. -,.., 1/3,,,...,, -.,, [5] «..» 017, 9,,

2 ,,,. - -., :.,. y T h g d z -d l ) ). 1. (). () ε.,, 1.. -, (. 1, ). -,, d:l:h = :0:40; d, l, h,. -,., [6] = 0 (1 εt), ε. : 1) ). -. -,, - 3. [6] : 1 + ( ) = p + ( 1 kt ) k ( T ) k T rot + RaT, (1) t Pr Pr T + ( ) T = T, div = 0. () t, p, T,,, -. - :,, 3 : Ra = gβθ d 0χ, Pr = 0 χ, k = ε, 0 - ;,,, d, g Bulleti of the South Ural State Uiversity Ser. Mathematics. Mechaics. Physics, 017, vol. 9, o., pp

3 ..,... «..» 017, 9,, : L = l d, H = h d.,,, -, : d, d / 0, /d, 0 /d, (ρ - ). Pr = 6,. (1) (),, = y, y =. - =,y,t cos π z.. ( ) ( ) : T = y + T, ( ) ( ) 1-1 cos ( ) T = θ,y,t + aπ π z. : = ( ) T z a T T, a = λ m /λ f D, λ m, λ f 0,, D -, T 0, -. z,, : ϕ 8 ψ ϕ ψ ϕ 16a ϕ ϕ π = + 1+ ky k1+ θ ϕ + + t Pr 4 y y y 16a θ θ π ψ θ ψ θ π ψ + k 1+ ϕ y y 4 y y ψ θ ψ θ 16a ϕθ k 1 ϕ θ ϕ a θ 3 y y y y ϕ ψ θ ψ θ ψ θ ( + a) θ Ra 16, (3) + a y y y y y π ( ) a 4 + a θ 8a ψ θ ψ θ + a ψ Pr1+ 1 = π π a t y y π ψ ψ ϕ = + ; y ( ) θ θ ( ) a 4 + a a + a θ π, (4) y t = 0: ψ = ϕ = θ = 0; (5) y = 0, H: θ = 0, ψ = ψ = 0 ; = 0, L: ψ = ψ = 0 ; (6) y : θ = 0 ; θ : = 0. y.., [4] ( 1,1),. k : 4 π L H m m 1 Ram = 1+ k L H L H 49

4 1 π + 4a a 1 m ( + ) + 4 a a L H m, y. - k = 0 (a = 0) (a = ) a = 0,,. - Ra 0, (3) (4). (3) (6).. [7, 8] 8 16a π ϕt = ψϕ ψϕ + 1+ ky k1+ θ ϕ + ϕ yy ϕ + Pr y y 4 16a π π + k1+ ( θ + θ yy ) ϕ + ψθ + ψθ ψ 4 y y 4 y 16a aϕθ + ψθ + ψθ k1+ ϕ θ ϕ θ + 3 y y y y ( + a) ϕ + ψ Ra θ + ψ yyθ yy + ψ θ θ + 16a y y y π, (7) ( ) a 4 + a 8a + a Pr1+ θ 1 t = + a + ψ θ ψ θ ψ + π π y y π ( ) ( ) ( ) a 4 + a a + a + 1+ θ + θ θ yy, (8) π i in, y = 0, H: θ i,1 = 0, θ i, N = 0, ψ 1 = ψ = 0 j N j, = 0, L: ψ1, = ψ 1, = 0 ϕ ϕ 1, j = ψ, j i,1 = ψ i, y i, N i, N 1 y h, ϕ = ψ N1, j N1 1, j h, ϕ = ψ θ 1, j = 0, θ 1, = 0 N j ; h ; (9) h ; (10) θ 1, j = θ, j, θn1, j = θn1 1, j. ϕ = ψ + ψ ; i, j, N1, N yy y. ϕ [8]. + 1 fi, j fi, j fi+ 1, j fi 1, j fi+ 1, j fi, j fi, j fi 1, j f t =, f =, f =, f =. τ h h h τ, h, h y, y., - [8].,,.,.,.,... -, Bulleti of the South Ural State Uiversity Ser. Mathematics. Mechaics. Physics, 017, vol. 9, o., pp

5 .., FORTRAN-90. (7), (8) (9), (10) ε = 0 (- )...., ) - Ra = 1,6; ) ) Ra =,7 Ra 0.8, (., ). -, -, (. 3,, [4])., -,,, ε = 0 Ra = 1,9,. (,1) Ra 1,0 (. 1, ) (1,1),.. (1,1). -, - Ra 1,5 (. 3, )., -, [4],.. «-»,.,,. -,, -, (., ;, )..,, -,. - -., -.,,. 5.. ε = 0,0 1/, 0.,,. «..» 017, 9,,

6 Ra 1,. - (. 4, ).,, - (. 4, ).., ε,,.. 3., [4]: ) ) Ra = 0,9 1,4; ) Ra = 1,5. 4. ;) ) Ra = 1, Ra = 1,4;) ) Ra = 1,8, Ra = 1,6 -,.,. (. 4, ). - -., (. 3, 4, )., (. 4, ; 4, ) Bulleti of the South Ural State Uiversity Ser. Mathematics. Mechaics. Physics, 017, vol. 9, o., pp

7 ..,..., -., - -.,, - -,,. 1.,.. /..,.. // ,.. /..,..,.. // :..., ,.. - /..,.. //., Babushki, I.A. Eperimetal ad theoretical ivestigatio of trasitioal covective flows i Hele Shaw cell / I.A. Babushki, V.A. Demi, D.V. Aferov // Proceedig of Iteratioal Coferece. «Advaced Problems i Thermal Covectio». Perm, Russia, 004. P /..,..,.. // ,.. /..,..,....:, ,.. /....:, ,.. /... : -. -, Bulleti of the South Ural State Uiversity Series Mathematics. Mechaics. Physics 017, vol. 9, o., pp DOI: /mmph17006 THE EFFECT OF TEMPERATURE DEPENDENCE OF THE VISCOSITY ON STATIONARY CONVECTIVE FLOWS IN HELE SHAW CELL V.A. Demi, M.I. Petukhov Perm State Natioal Research Uiversity, Perm, Russia Federatio demi@psu.ru The results of direct umerical simulatio of statioary covective flows i a vertical Hele Shaw cell uder the uiform heatig from below are preseted i this paper. The calculatios have bee fulfilled for realistic values of the heat-trasfer coefficiet o vertical wide boudaries ad model thermal coditios o arrow vertical walls. The approimatio of plae trajectories has bee applied to calculate the flows i the Hele Shaw cell. The liear stability aalysis is eecuted for the situatio whe the viscosity depeds o the temperature. A aalytical formula for critical values of Rayleigh umber has bee deduced which determie the threshold of covectio i depedece o parameters of the problem. It has bee show that the umerical simulatio imitatig the full-scale eperimet gives adequate descriptio of the trasitio from oe-vorte statioary flow to the two-vorte steady regime whe the depedece of viscosity o the temperature is take ito accout i mathematical model. The equatios system of thermal covectio i Boussiesq approimatio was solved by the method of fiite. «..» 017, 9,,

8 differeces at the PGU-Tesla supercomputer of the Research Academic Ceter Parallel ad Distributed Calculatios i Perm State Uiversity. The fields of stream fuctio i vertical sectio have bee calculated which cofirm the effect of the vortices ceters displacemet to the bottom of the cavity. Keywords: Hele Shaw cell; heatig from below; statioary regimes; o-homogeeity of viscosity. Refereces 1. Puti G.F., Tkacheva E.A. Izvestia RAN, Mekhaika Zhidkosti i Gaza, 1979, o. 1, pp (i Russ.).. Lyubimov D.V., Puti G.F., Cheratyskiy V.I. Gidrodiamika, 1977, Issue 10, pp (i Russ.). 3. Babushki I.A., Demi V.A. Izvestia RAN, Mekhaika Zhidkosti i Gaza, 006, o. 3, pp (i Russ.). 4. Babushki I.A., Demi V.A., Aferov D.V. Eperimetal ad theoretical ivestigatio of trasitioal covective flows i Hele Shaw cell. Proceedig of Iteratioal Coferece Advaced Problems i Thermal Covectio, Perm, Russia, 004, pp Babushki I.A., Demi V.A., Glazki I.V., Platoova A.N., Puti G.F. Izvestia RAN, Mekhaika Zhidkosti i Gaza, 009, o. 5, pp (i Russ.). 6. Gershui G.Z., Zhukhovitskiy E.M., Nepomyashchiy A.A. Ustoychivost' kovektivykh techeiy (The stability of covective flows). Moscow, Nauka Publ., 1989, 30 p. (i Russ.). 7. Samarskiy A.A Teoriya razostykh skhem (Theory of differece schemes). Moscow, Nauka Publ., 1977, 656 p. (i Russ.). 8. Tarui E.L. Vychislitel'yy eksperimet v zadachakh svobodoy kovektsii (Computatioal eperimet i free covectio problems). Irkutsk, izd-vo Irkut. U-ta Publ., 1990, 8 p. (i Russ.). Received February 10, Bulleti of the South Ural State Uiversity Ser. Mathematics. Mechaics. Physics, 017, vol. 9, o., pp