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1 53.546: 59.6 DOI:.459/mmph : ; ; - ; ;. - - [ 3] [3 6] ρu P = λ l d P l d λ ρ u [7 8]. - - : [9 ]. -.. «..»

2 - -.. l. z ( ur uϕ uz ) uz u r = u ϕ =. [4] u z u z ν u z + u P z = r < r < < t T t z r r r ρ z u z u = z = P = P = () z ϕ ρ r ρ ϕ u z P ρ ν = µ ρ µ. () uz r t P r ϕ. - P. z P P( t) u( r t) = uz ( r t) = z l () - u ν u P( t) = r + < r < < t T () t r r r ρ l P( t). ρ ν P( t). () u t= = ψ ( r) (3) r = u r= =. (4) r. (3) (4). Q( t). () π rudr = Q( t). (5) 6 Ser. Mathematics. Mechanics. Physics 7 vol. 9 no. pp. 5

3 ... «..» π ψ ( r) dr = Q(). r t T u( r t ) () (3) (5). { } () (5) - [3 6]. (5) - () (5) - -. () (5). (5) t u dq π r dr =. t dt u t () ν u P( t) dq π r r + dr = r r r ρ l dt. (4) u ( ) ν r P t dq r= + =. r ρ l π dt r u r= () u dq r= = P( t). (6) r πν dt ρlν u( r t ) () (3) (4) (6). () (4) (6) -. {( ri t ) : ri i r t t i... n... m} ω = = = = = { r t T} r = n r t = T m t. () ω [7] ui ui ν ui+ ui ui u i = ri + / ri / + P t ri r r r ρl ui u( ri t ) P = P( t ) ri ± / = ri ± r / i = 3... n = 3... m. (3) : ui = ψ i i = n. (4) (6) -. : r/ r/ ν r/ u u u u = + P t r ρl n n n / Q Q un un n / ν rn / u u r r = P t π t r ρl 7

4 ψ = ψ ( r ) Q = Q( t ). i i : t aiui ciui + bi ui+ = ( ui + P ) i = 3... n = 3... m (7) ρl η ui = ψ i i n θu η n θun η u u = (8) = + (9) = + () ν tr ν tr ν tr a ; b ; c a b ; ; i / i+ / / i = i = i = i + i + θ = ri r ri r 5r/ r + ν tr/ 5r/ ru + 5 P r/ r t / ρl η = ; 5r/ r + ν tr/ ( ) π n / n n / = 5( rn / ) r + ν trn / 5 r Q Q / + 5( r ) ru 5 P r r t / ρl ; θ ν tr n / = 5( rn / ) r + ν trn / (7) () = m u( x t) - i.. u + i = n = m. ( ) [8]. (7) () - = m i αi+ + βi+ u = u i + i = n n... u i i i = 3... n :. θβn + η n = θα n α = θ bi β = η αi+ = ai βi + ui + t P / ρl βi+ = i =... n. ci aiαi ci aiαi. -. : u r= = g( t) u( r t) () (4). - (5) Q( t) () (4) (6). - r = t =. = 6 ; 3 µ = ; = / 3 ; ψ ( r) = / ; 5 P( t) = ; g( t ) = / ; l = 5. ; u t u u( r t ) /. ( ) u r t. 8 Ser. Mathematics. Mechanics. Physics 7 vol. 9 no. pp. 5

5 .. t = 6 t = 9 t = 5 t = 8 r t u u t u u t u u t u u µ d du P r r u ust = < r < r dr dr l du r= dr π rudr = Q Q P Q P u( r) = + ( r ). () π 8µl Q P () ρ = / 3 ; 3 µ = ; Q = /c; P l = / ; = 6. ; u - /c; u st () /c. Q P «..»

6 ... - /....: /......: / : /....: /....: /.. : Neto C. Boundary slip in Newtonian liquids: a review of experimental studies / C. Neto D. Evans E. Bonaccurso // eports on Progress in Physics. 5. Vol. 68 no.. P Lauga E. Microfluidics: the no-slip boundary condition in Handbook of Experimental Fluid Dynamics / E. Lauga M.P. Brenner H.A. Stone. New York: Springer 6. P /... : ao I.J. The effect of the slip boundary condition on the flow of fluids in a channel / ao I.J. K.. aagopal // Acta Mechanica Vol. 35. P / // (8) Volker J. Slip with friction and penetration with resistance boundary conditions for the Navier Stokes equation numerical tests and aspects of the implementation / J. Volker // J. Computational and Applied Mechanics.. Vol. 47 Issue. P Cannon J.. The solution of heat equation subect to the specification of energy / J.. Cannon // Quart. Appl. Math Vol.. P /.. // /.. // /....: / /......: Series Mathematics. Mechanics. Physics 7 vol. 9 no. pp. 5 DOI:.459/mmph7 NUMEICAL METHOD FO SOLVING A NONLOCAL POBLEM ON PIPELINE TANSPOTATION OF VISCOUS LIQUID K.M. Gamzaev Azerbaian State Oil and Industry University Baku Azerbaian xan.h@rambler.ru The paper deals with a process of unsteady axisymmetric flow of incompressible viscous liquid in the cylindrical pipeline described by the nonlinear system of Navier-Stokes differential equations. The set of equations is transformed into one linear parabolic equation with an initial and natural boundary Ser. Mathematics. Mechanics. Physics 7 vol. 9 no. pp. 5

7 ... «..» condition on the pipeline axis. We face a problem for determining velocity distribution in a cross-section of the pipeline based on the desired law of time variation of the pressure drop along the pipeline. As in case of liquid flow in pipes it s practically impossible to define interaction models of fluid with a solid boundary the boundary condition on the pipe wall is considered as unknown. For the problem accuracy an additional condition in the form of integral flow characteristic is specified. In other words the law of time variation of volumetric flow rate in the pipeline is specified. This problem is related to nonlocal problems with an integral condition for partial differential equations. The specified integral condition is differentiated in time and the obtained ratio with the help of the differential equation is transformed into a local boundary condition. As a result the set task is altered to a direct problem with local conditions. The finite difference method is applied for numerical solution of the boundary value problem. For this purpose we create a discrete analog of the problem in the form of an implicit difference scheme using the integral method. A computational algorithm of solving the obtained difference equation system is suggested. Numerical experiments for test problems have been conducted to check the efficiency of practical application of the suggested computational algorithm. The computational algorithm has also been tried on the data of steady flow of viscous incompressible liquid in the pipeline. Keywords: pipeline transportation; viscous liquid; axisymmetric flow; non-local problem with integral condition; finite difference method. eferences. Lur'e M.V. Matematicheskoe modelirovanie protsessov truboprovodnogo transporta nefti nefteproduktov i gaza (Mathematical modeling of processes of oil oil products and gas pipeline transportation). Moscow Neft i gaz Publ p. (in uss.).. Leonov E.G. Isaev V.I. Gidroaeromekhanika v burenii (Aerohydromechanics in drilling). Moscow Nedra Publ p. (in uss.). 3. Basniev K.S. Dmitriev N.M. ozenberg G.D. Neftegazovaia gidromekhanika (Oil and gas fluid mechanics). Moscow Izhevsk Institut kompiuternykh issledovanii Publ p. (in uss.). 4. Loitsianskii L.G. Mekhanika zhidkosti i gaza (Mechanics of liquid and gas). Moscow Nauka Publ p. (in uss.). 5. abinovich E.Z. Gidravlika (Hydraulics). Moscow Nedra Publ p. (in uss.). 6. Shlikhting G. Teoriia pogranichnogo sloia (Boundary layer theory). Moscow Nauka Publ p. (in uss.). 7. Neto C. Evans D. Bonaccurso E. Boundary slip in Newtonian liquids: a review of experimental studies. eports on Progress in Physics 5 Vol. 68 no. pp DOI:.88/ /68//5 8. Lauga E. Brenner M.P. Stone H.A. Microfluidics: The No-Slip Boundary Condition. New York Springer 6 pp DOI:.7/ _9 9. Yankov V.I. Pererabotka voloknoobrazuyushchikh polimerov. Osnovy reologii polimerov i techenie polimerov v kanalakh (Processing of fiber-forming polymers. Basics of polymer rheology and flow of polymers in channels). Moscow Izhevsk: NITs egulyarnaya i khaoticheskaya dinamika Institut komp'yuternykh issledovaniy Publ p. (in uss.).. ao I.J. aagopal K.. The effect of the slip boundary condition on the flow of fluids in a channel. Acta Mechanica 999 Vol. 35 pp DOI:.7/BF Borzenko E.I. Diakova O.A. Shrager G.. Studying the slip phenomenon for a viscous fluid flow in a curved channel. Vestn. Tomsk. Gos. Univ. Mat. Mekh. 4 no. (8) pp (in uss.).. Volker J. Slip with friction and penetration with resistance boundary conditions for the Navier Stokes equation numerical test s and aspects of the implementation. J. Computational and Applied Mechanics Vol. 47 Issue pp DOI:.6/s377-47() Cannon J.. The solution of heat equation subect to the specification of energy. Quart. Appl. Math. 963 Vol. no. pp DOI:.9/qam/ Ionkin N.I. Differentsial'nye uravneniya 977 Vol. 3 no. pp (in uss.). 5. Samarskiy A.A. Differentsial'nye uravneniya 98 Vol. 6 no. pp (in uss.). 6. Nakhusheva V.A. Differentsial'nye uravneniya matematicheskikh modeley nelokal'nykh protsessov (Differential equations of mathematical models of nonlocal processes). Moscow Nauka Publ p. (in uss.).

8 7. Samarskiy A.A. Teoriya raznostnykh skhem (Theory of difference schemes). Moscow Nauka Publ p. (in uss.). 8. Samarskiy A.A. Vabishchevich P.N. Chislennye metody resheniya obratnykh zadach matematicheskoy fiziki (Numerical methods of solving inverse problems of mathematical physics). Moscow LKI Publ p. (in uss.). eceived July 8 6 Ser. Mathematics. Mechanics. Physics 7 vol. 9 no. pp. 5