Influence of the dispersion on boundary conditions

for continuous models

E.V.Prozorova

St. Petersburg State University, Russia

This paper gives an account ofљ newљ terms forљ equationsљ of continuous mechanics,љ whichљљ is determinedљљ the influence of anљ angular momentum in elementaryљ volume.љљ In classical caseљ the angularљ momentum variationљ is disregarded. Makingљ this suppositionљ we actually is assumed the symmetric press tensor.љ Mathematically it leads toљ that љљequilibrium conditions are postulatedљ asљљ equilibriumљ conditions of forces. The choice ofљ equilibrium conditions asљљ equilibriumљ conditions of angular momentum leads to new equations. For conditions of equilibrium of forces are received theљ special formulation of continuous mechanics equations with the angular momentum.љљ Observing experimental facts leads to hypothesis that spatial gradients are very importance. The connection between gradients of the physical values and modified equations withљ angular momentumљ and some experience might be followed.љ Original (classical) formulation conservation laws was basedљ of exclusive of elementary volume. Elementary volume appearing open system exchanges with the surroundings by components of the physics values ( mass, liner moment , energy, angular momentum) over all directions. The new Navie-Stokes equations were received from the modified Boltzmann equations. The solution of the Boltzmann equation is invariantљ under macroparametrs of the equilibrium function. This fact and new equationsљ areљ used in our work for solution of the Gilbert problem.

љIn the previous papers we investigatedљ the following problems of noncompressible liquid: infinite and half-infinite plates,љ infinite and half-infinite tubes, the Falkner-Skan problem in usually formulation, infinite thin jet, nonstationary boundary layer for little time. The Blasius problem was solved asљ numerically and with analytical methods.љ Forљ the infinite plate case the motionless thin film was detectedљ and the Prandtl formula was received from theљ suggested equation.љ The order of the new equations (for the density and for the linear moment)љ is more than in classical case. If we deal with continualљљ medium the externalљљљ boundaryљ condition for boundary layerљљ can be determined as the value rotor velocity or as value normal velocity.љ That is for the vertical velocity. For the longitudinal velocity it is need to put friction. For the rarefied gas the boundary conditions would be included the value gas flow besides the classical boundary conditionsљ as order of modifiedљ equation is more.

For the classical caseљ near the surfaceљ in transition regime the Knudsen layer is considered . This layer has the lengthљљ of orderљљљ free path.љљ In point of fact theљ we have macroparametrs.љ It is better to solve the Navie-Stokes equations everywhere besides thin layer near surface andљ to look for solution in thin layer separately.љ In theљ workљ the equation for two-particle function was received near the surface. Inside flow the Chapman-Enskogљ method is given the solution which is asymptotic convergent to the Boltzmann equation if the Knudsen number is converging to the zero. However at any little the Knudsen number near the surface we have regime where the sum does not the solution of equation. We suggest closer definition of the solution by introducing of macroparameters for equilibrium function.љљ Usually the boundary conditions are established at solidљ surface thatљ the Navie-Stokesљ solutionљ coincidesљ with the Boltzmann equation solution at external boundary of the Knudsenљ layer.

The character of the interaction gas and surface moleculesљ determines the boundary layer conditions for distribution function at lower boundary of the Knudsen layer.љ The experiences of the numerical solutions tell us about nearness gasdynamics fields for continuousљ and discrete medium under similar boundary conditions. N-particle distribution function is suggested to use for the relaxation problem by including the surface atoms in ensemble.љ The regime of the stream is established that even for particles without structure it is need to take into account.љљљљљљљљљљ

At distance of some σ ( the interaction radiusљ )љ the molecules of external boundary do not collide with gas molecules but collide with surface molecules. For this case interaction gas and solid molecules is defined by force of gas and solid molecules and may considered bothљ classical and quantum. Inљ concerned thin layer the quantity of molecules is a little. To solute aerodynamics problems for the transition region of the stream is suggested by using the Navie-Stokes solution everywhere besides the layer of order some radius of molecules interaction. The macroparametersљ ofљ theљ Navie-Stokes solution is used for the Chapman-Enckogљљ distributionљљ functionљ at external boundary with iteration after definition them into layer near solid.љ

The Blasius problem was solved asљ numerically and with analytical methods.љ Hereљ this task is reserved for similar positing.љљљ Theљ Prandtlљ problemљљљљ ofљ compressionљ layer by two rough plates was investigated for twoљ dimension.

The results of theљ Blasius problem solution point out on existence unmonotonous profile of longitudinal velocity component and oscillations in district where usually we unstable layer.

In conclusion the specifiedљљ problems areљ mentioned.