Boussinesq approaching

The equations of thermal convection ( Boussinesq equations, the Boussinesq approximation ) in the Boussinesq - Oberbek approximation are the most popular model for describing convection in liquids and gases.

The model includes the Navier - Stokes equation , the heat equation and the incompressibility equation . The main idea of the approximation consists in taking into account the dependence of density on temperature . Namely, in the system of convection equations, this dependence is taken into account only with mass forces :

$\rho _{0}\left({\frac {\partial {\vec {v}}}{\partial t}}+({\vec {v}}\cdot \nabla ){\vec {v}}\right)=-\nabla p+\eta \Delta {\vec {v}}+\rho (T){\vec {g}},$ ${\ displaystyle \ rho _ {0} \ left ({\ frac {\ partial {\ vec {v}}} {\ partial t}} + ({\ vec {v}} \ cdot \ nabla) {\ vec { v}} \ right) = - \ nabla p + \ eta \ Delta {\ vec {v}} + \ rho (T) {\ vec {g}},}$ ${\ displaystyle \ rho _ {0} \ left ({\ frac {\ partial {\ vec {v}}} {\ partial t}} + ({\ vec {v}} \ cdot \ nabla) {\ vec { v}} \ right) = - \ nabla p + \ eta \ Delta {\ vec {v}} + \ rho (T) {\ vec {g}},}$

${\frac {\partial T}{\partial t}}+{\vec {v}}\cdot \nabla T=\chi \Delta T,$ ${\ displaystyle {\ frac {\ partial T} {\ partial t}} + {\ vec {v}} \ cdot \ nabla T = \ chi \ Delta T,}$ ${\ frac {\ partial T} {\ partial t}} + {\ vec v} \ cdot \ nabla T = \ chi \ Delta T,$

$\operatorname {div} {\vec {v}}=0,$ ${\ displaystyle \ operatorname {div} {\ vec {v}} = 0,}$ $\ operatorname {div} {\ vec v} = 0,$

Where ${\vec {v}}$ ${\ displaystyle {\ vec {v}}}$ ${\ vec {v}}$ - flow rate, $T$ ${\ displaystyle T}$ $T$ - absolute temperature $p$ ${\ displaystyle p}$ $p$ - pressure $\eta$ ${\ displaystyle \ eta}$ $\ eta$ - dynamic viscosity $\chi$ ${\ displaystyle \ chi}$ $\ chi$ - coefficient of thermal diffusivity , ${\vec {g}}$ ${\ displaystyle {\ vec {g}}}$ ${\ vec g}$ - acceleration of gravity .

Often, a linear approximation is applied to the dependence of density on temperature:

$\rho (T)=\rho _{0}(1-\beta \theta )$ ${\ displaystyle \ rho (T) = \ rho _ {0} (1- \ beta \ theta)}$ ${\ displaystyle \ rho (T) = \ rho _ {0} (1- \ beta \ theta)}$ ,

Where $\beta$ ${\ displaystyle \ beta}$ $\ beta$ - coefficient of volume expansion , $\theta =T-T_{0}$ ${\ displaystyle \ theta = T-T_ {0}}$ ${\ displaystyle \ theta = T-T_ {0}}$ - temperature deviation from the equilibrium state, $\rho _{0}$ ${\ displaystyle \ rho _ {0}}$ $\ rho _ {0}$ - fluid density at some equilibrium temperature $T_{0}$ ${\ displaystyle T_ {0}}$ $T_ {0}$ . Insofar as $\beta$ ${\ displaystyle \ beta}$ $\ beta$ and the temperature deviation is usually relatively small, the linear approximation has acceptable accuracy in most of the problems under study.

Substitution of the linear dependence of the density and renormalization of pressure make it possible to exclude the term $\rho _{0}{\vec {g}}$ ${\ displaystyle \ rho _ {0} {\ vec {g}}}$ ${\ displaystyle \ rho _ {0} {\ vec {g}}}$ . Finally, the problem of convection of an incompressible fluid in the Boussinesq approximation takes the following form:

${\frac {\partial {\vec {v}}}{\partial t}}+({\vec {v}}\cdot \nabla ){\vec {v}}=-{\frac {1}{\rho _{0}}}\nabla p+\nu \Delta {\vec {v}}-\beta \theta {\vec {g}},$ ${\ displaystyle {\ frac {\ partial {\ vec {v}}} {\ partial t}} + ({\ vec {v}} \ cdot \ nabla) {\ vec {v}} = - {\ frac { 1} {\ rho _ {0}}} \ nabla p + \ nu \ Delta {\ vec {v}} - \ beta \ theta {\ vec {g}},}$ ${\ displaystyle {\ frac {\ partial {\ vec {v}}} {\ partial t}} + ({\ vec {v}} \ cdot \ nabla) {\ vec {v}} = - {\ frac { 1} {\ rho _ {0}}} \ nabla p + \ nu \ Delta {\ vec {v}} - \ beta \ theta {\ vec {g}},}$

${\frac {\partial \theta }{\partial t}}+{\vec {v}}\cdot \nabla \theta =\chi \Delta \theta ,$ ${\ displaystyle {\ frac {\ partial \ theta} {\ partial t}} + {\ vec {v}} \ cdot \ nabla \ theta = \ chi \ Delta \ theta,}$ ${\ displaystyle {\ frac {\ partial \ theta} {\ partial t}} + {\ vec {v}} \ cdot \ nabla \ theta = \ chi \ Delta \ theta,}$

$\operatorname {div} {\vec {v}}=0,$ ${\ displaystyle \ operatorname {div} {\ vec {v}} = 0,}$ $\ operatorname {div} {\ vec v} = 0,$

here $\nu$ ${\ displaystyle \ nu}$ $\ nu$ - kinematic viscosity .

The above convection problem in various settings has been repeatedly studied. The most widely known is the Rayleigh - Benard problem of convection in a flat liquid layer. Under certain conditions, an exact solution to the problem is possible, for example, for laminar convection in a vertical layer when heated from the side (sometimes found under the name "Gershuni problem").

Literature

Ostroumov G.A. Free thermal convection in the conditions of an internal problem. Moscow - Leningrad. Gostekhizdat. - 1952.
Landau L.D., Lifshits E.M. Course in Theoretical Physics. T. 6. Hydrodynamics. — M.: Science. — 1988. — 736 pp. — § 56
Gershuni G.Z., Zhukhovitsky E.M. Stability of convective flows. - M.: Nauka. - 1989.
Gershuni G.Z., Zhukhovitsky E.M. Convective stability of an incompressible fluid.
Kriegel AM On the applicability of the approximation of free convection to atmospheric turbulence // Bulletin of the Leningrad State. University. — Ser. 7. — 1991. — Issue 2 (14). — S.107-110.

Boussinesq approaching

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Literature

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