THE HEATED LAMINAR VERTICAL JET IN A LIQUID WITH POWER-LAW TEMPERATURE DEPENDENCE OF DENSITY. V. A. Sharifulin.

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1 THE HEATED LAMINAR VERTICAL JET IN A LIQUID WITH POWER-LAW TEMPERATURE DEPENDENCE OF DENSITY 1. Introduction V. A. Sharifulin Perm State Technical Universit, Perm, Russia sharifulin@perm.ru Water laer near C is a peculiar eample of convectional sstem with non-uniform stratification. Anomalous heat etension is well known to takes place in this area. At Ti = C water densit is maimum, in the interval from melting point T = to T = C, water behavior is anomalous: water densit increases with the increase of temperature; for T > Ti densit decreases with the increase of temperature. In the temperature interval 8 C, temperature dependence of densit can be approimated with high degree of accurac b smmetric parabola with its maimum at the point of inversion. In a wider range of parameters the water behavior is described b power-law temperature dependence of densit as follows: ρ( T) 1 β T T i. The problem of a heated vertical jet of an incompressible liquid in a boundar laer was first investigated analticall and numericall b Brand & Lahe [1]. Analtical solutions were obtained for the certain values of Prandtl number b similarit methods. Mollendorf at al. [] numericall investigated the problem for the case of power-law temperature dependence of densit corresponding to the water behavior for normal pressure and temperature near C. M.A Goldshtik, V. N. Shtern & N. I. Yavorsk [] treated the jet problem for cubic temperature dependence of densit. In their work the authors used standard Bussinesq equations without boundar-laer approimation. The free jet behavior, both numericall and analticall with small parameter epansion method, were investigated. One can see that free jet problem attracts constant interest of investigators. Although from short review above follows necessit of analtical solution account for the case of liquids with power-law temperature dependence of densit. And the present stud serves to improve completeness of the problem in this wa. Analtical solutions for the certain values of Prandtl number and arbitrar values of inversion rate were found b similarit methods same to that emploed b Brand & Lahe [1].. The two-dimensional jet A sstem of rectangular co-ordinates (, ) is chosen such that the -ais coincides with the smmetr ais of the jet. The boundar-laer equations, epressing the conservation laws of mass, momentum, and energ are: v v + =, v v v v v g T + = ν + α θ, (.1) (.)

2 σ v θ + v θ θ. = ν (.) In these equations, v and v are velocit components, and θ is a dimensionless temperature difference related to the local temperature, T(, ) and to the temperature far from the jet, T b θ = T T T (.) Other constants appearing in the equations are: ν the kinematic viscosit; β, the coefficient of volume epansion; g, the acceleration of gravit; and σ, the Prandtl number. Boundar conditions to be satisfied are: v θ at =, v = = = ; (.5) that at =, v = θ = (.6) The continuit equation (.) implies the eistence of a stream function. ψ (, ), Such v = ψ, v = ψ (.7) The partial differential equations are reduced to ordinar equations b means of the following transformation: + η = a, (.8) + (, ) a f ψ = ν η (.9) ( ) (, ) θ = aν gβt τ η (.1) The arbitrar constant, a, is included so as to make η, f ( η ) and τ η dimensionless. It can be chosen so as to match the mathematical solution to a particular phsical case. The governing equations for the problem become [] f + f f ( f ) + τ =, (.11) + + σ τ + f τ =. (.1) + Boundar conditions in terms of f and τ are f = f = τ =, (.1) τ. f = = (.1)

3 The velocit components are related to f b + v = a ν f, (.15) + aν v = ( f ( ) ηf ), (.16) (, ) ( ) θ = aν gβt pη. (.17) Eact solutions for the sstem (.11), (.1), which satisf the boundar conditions (.1) and (.1), have been found for σ σ = + 9. If τ is assumed related to f b equation (.11) becomes = and 5 τ = ( f ) + b ( f ) + ff +, (.18) which has the integral f + b+ ff + b+ ( f ) =, (.19) + + f + b+ ff =. (.) + The same assumption forτ, equation (.18), is now substituted in the energ equation (.1), with the result, b Pr 1 b 5 Pr bf f ( ff ) f f + ff =.(.1) ( ) Inspection of this equation shows that for certain values of b and σ an function which satisfies (.) also satisfies(.1). The appropriate values of b and σ are: case () i b =, σ = ; 5 + case ( ii) b =, σ =. 9 ( + ) In case (i), the coefficient of the first bracketed epression in (.1) vanishes, and the second bracketed epression becomes identical with(.), which is, in this f + ff = + (.) The solution of (.) and the corresponding epression for τ from (.18) are

4 f = tanh η ( + ), (.) ( ) sec h ( ) ( + ) 95 τ = η +. (.) In case (ii), the bracketed epressions in (.1) become identical to the left sides of (.19) and (.). The solution for this case is f 1 = tanh η 6 (.5) τ = sec h η ( ) (.6) From the solutions for f, f and p, one ma compute the velocit components and the temperature from equations (.), (.1) and (.). However, the constant, a, and the location of the origin of have not as et been specified. Both of these quantities are related to the initial conditions of the jet, and can be chosen so as to match the mathematical solution to a particular phsical eperiment. Suppose that at some station, the velocit and the temperature profiles are measured. Then at some, sa =, the mathematical solution must match the measured profiles, at least in an average sense. For instance, if W is the measured volume flow rate, and E is the integral of the measured temperature profile (and is thus a measure of the thermal energ in the jet), one can select a and so as to make the corresponding integrals of the mathematical solution equal to the measured quantities, v, d W, = (.7) T( ) T d = E (.8),. When equations (.15) and (.17) are substituted in (.7) and(.8), the results ma be solved for a and : + W ν a =, (.9) ( + ) E W = τ ( η) dη ( gβν ), (.) 1 1 W E W a = τ η dη ( gβν ) ν. (.1)

5 . The aismmetric jet Analsis of the aismmetric heated jet is completel analogous to the twodimensional problem. The -ais is now the jet ais, and r is the radial co-ordinate. With the other quantities defined as before, the boundar-laer equations are: ( rv ) + ( rv ) =, v v ν v v + vr = r + gβt θ, r θ θ ν θ v + vr = r. σ r Boundar conditions to be satisfied are: v θ at r = ; vr = = = ; v is finite; at r = ; v = θ =. r (.1) (.) (.) The continuit equation is again integrated b means of a stream function, and introduction of a similarit transformation, v 1 ψ 1 ψ =, vr = r r, (.) η = ar, (.5) ( r, ) f ψ = ν η, (.6) (, ) leads to the ordinar differential equations [] θ r = aν gβt τ η, (.7) ( 1 ) ( f ) f ff ff f + + ητ =, η η η η (.8) ητ + σ f τ =. (.9) Boundar conditions on f and τ are: f = f = τ =, (.1) τ is finite, (.11) f τ = lim η η =. (.1) η The velocit components in terms of f are

6 v 1 aν f = (.1) η ν vr = f η f r (.1) Eact solutions for the sstem (.8), (.9), are possible for certain Prandtl numbers. If one assumes f bη =, (.15) b + η then integration of (.9) provides τ : b. τ c b η σ = + (.16) Formulas (.15) and (.16) satisf equation (.8) onl for the following choices of σ, b and c : + case ( i) σ =, b =, c = 18 ; + ( + ) case ( ii) σ =, b =, c = 51( ). From the solutions for f andτ, one ma compute the velocit components and the temperature from equations(.1), (.1) and(.7). The constant a, and the location of the origin of ; can be chosen in an identical wa with that of the two-dimensional jet. Suppose that at some, sa =, the velocit and temperature profiles are measured. Then if W is the measured volume flow rate and E is the integral of the measured temperature distribution, one can select a and so as to make the corresponding integrals of the mathematical solution equal to the measured quantities: π v (, r) rdr = W, (.17) π T( r) T rdr = E. (.18), When equations (.7) and (.1) are substituted into (.17) and (.18), the resulting equations ma be solved for a and, W = πν f ( ), (.19) ( 1) ( )( ) 1 1 E + W π πν f ( ) a= gβ τ η ηdη ν. (.) The value of now locates the origin of the co-ordinate sstem, and the mathematical solutions ma be assumed to represent the actual jet flow for.

7 Acnolegment Author would like to epress the gratitude to V.N. Stern and F. Busse for the discussion and interest to work correspondingl. References 1. Brand R. S., Lahe F. J. J. Fluid Mech., 9, 5 (1967).. Mollendorf R. S., Johnson R. S., Gebhart B., J. Fluid Mech., 11, 69 (1981)..Goldshtik M.A., Shtern V.N., Yavorsk N.I. Viscous Flows with Paradoical Features. - Nauka, Novosibirsk, USSR, 1989, 66 p.