Investigation Pulsation Motion of the Liquid in the Flat Channels

Transcription

1 International Journal of Science and Qualitative Analsis 8; 4(3: ttp:// doi:.648/j.ijsqa.843. ISSN: (Print; ISSN: (Online Investigation Pulsation Motion of te Liquid in te Flat Cannels Fakriddin Abdikarimov, Fotima Abdikarimova, Temur Kasanov, Madamin Abdirimov, Azamat Rajabov Department of Applied Matematics, Psics-Matematics Facult, Urgenc State Universit, Urgenc Cit, Uzbekistan address: To cite tis article: Fakriddin Abdikarimov, Fotima Abdikarimova, Temur Kasanov, Madamin Abdirimov, Azamat Rajabov. Investigation Pulsation Motion of te Liquid in te Flat Cannels. International Journal of Science and Qualitative Analsis. Vol. 4, No. 3, 8, pp doi:.648/j.ijsqa.843. Received: April 5, 8; Accepted: Ma 9, 8; Publised: June, 8 Abstract: Pulsation motion of Newton liquid in flat cannels as begun to be investigated as a result of te development of te liquid mecanics. Te function of te pressure gradient due to time is given on matematic modeling of pulsation motion of liquid and as a result, te rest of dimension mill be defined in relating to time. Wile investigating te process of Navier- Stokes, it is made lazier incessant equation according to te usage of teir replacement of te tasks. For tis, te terms, wic are given in te sstems of te equations are transformed into new dimensions and te value of te terms, wic are contrasted wit eac oter is defined in te sstem of equation. Te terms, wic ave acquired boundless small quantit is omitted witout counting. Tis article discusses te pulsating fluid flows in te flat tube. Here it is determined te dependenc of te speed of te pulsating movement of fluid on te vibration parameter. Kewords: Newton Liquids, Continuit, Vibration Parameter, Pulsing Movements. Introduction Te problems associated wit te movement of blood in large arteries ave long attracted te attention of researcers. However, onl in te last decades te teor in tis area as been able to acieve significant progress, since it became possible to take into account te complex mecanical properties of te walls in te formulation of te problem, and wen solving it, use modern means of matematics and computers. Te crudest approac to te teoretical description of te pulsating blood flow in te arteries relies on te assumption tat te blood is an incompressible viscous liquid wit a constant propert and moves laminar in an infinitel long clindrical tube of circular cross-section wit rigid walls under te action of a pressure gradient armoniousl varing wit time. Te propagation of pulse pressure waves along elastic vessels is analzed. Of great interest is te stud of te pulsating flow of viscous fluid, in particular blood, in pipes of viscoelastic material. Pulsating fluid flows in pipes wit various mecanical properties of te wall (rigid, elastic, elasticall permeable are considered. Some drodnamic regularities in te pulsating flow (te nature of te pressure cange, te fluid flow are given. Te task in te sstem of te difficult equation is simplified. Based on tese, we look at pulsation motion of Newton fluid in te flat pipe, wic as enoug lengt. Figure. Te motion of Newton liquid in a flat pipe. Te plates of te flat pipe are located in distance, and te arrow is lined from te middle part and te arrow is crossed wit it in a perpendicular. Te flow of te liquid is considered as smmetrical to te arrow and as a

2 7 Fakriddin Abdikarimov et al.: Investigation Pulsation Motion of te Liquid in te Flat Cannels result, te flow is focused directl towards and arrows, but te flow towards te arrow is not taken into consideration.. Metod Now te equation of motion of te liquid towards te investigated flow is expressed a Descartes coordination sstem b te equations of Navier-Stokes and te incessant equation of te liquid. Te dimensions are replaced as following: t = Tt, x = Lx, = µ LU u = U u = V V = = ε U L, υ υ, p = p As a result, te following equation sstem, wic is expressed b new equation sstem is formed: u u u u u α + ε u + υ = + ε + t x x x υ υ υ υ υ ε α ε u + υ ε ε t + x = + + x u υ + = x Here time, and orizontal and vertical coordination, and vertical and cross velocities, ω pressure, α = R te appropriate number of te ν V frequenc, given b Renold ε = = << small L U quantities. Subsequentl, we don t need to take into consideration te sign in te form ( as it is undertaked onl re curring dimensions. 3. Result Taking into account te smmetr of te flow, at te section and considering tat, it is placed frontier terms for orizontal and vertical velocit. Te following approac will be reliable because of frontier terms in pipe center. = да, υ =, u = u = да, =, υ = ( ( (3 B dropping te terms of small quantities, used in te equation, mentioned above, it will be discovered te following simple equation. u u ρ = + µ t x Here, te gradient of te pressure armonic scales of time. x (4 sould consist of = A i t + realae ω = A + Acosωt x In tat case, we tr to find te solution to te equation like u = u + uei ω t and as a result te above mention equation will be tis form. u = A µ u iω u = A ν µ Here, te solution to te first equation will be related to Puazel sum and te second one, being in a trigonometric form, will be defined as follows: 4. Discussion А u = ( µ ( i A c α u = ( ( µ iα c( iα Using te first equalit of te sstem (7, average velocit will be considered on te surface of te section. 3 3 A A A µ 3 µ 3 µ Q = ud = ( d = ( = U (5 (6 (7 (8 3 A Q 3µ = = = A (9 3µ And total saluting is mode b counting te quantities of and

3 International Journal of Science and Qualitative Analsis 8; 4(3: c( i iωt А (( α A i t u u ue ( ( ω = + e А µ iα c( iα = + ( c( iα real( u 3A i t ( e U = ω A i ( real ( iα c( α Te final solution is made b dividing real part of te equalit ( c( iα real u A ( U A i ( 3 = ( real ( (cos ωt + i sin ωt iα c( α ( real( u 3 A M N = ( sin ωt + cos ωt (3 U A E E ere coefficients suc as,,,,,,,, are mode as te following. α k = A = ckcosk C = ckcosk B = sksink D = sksink α k = M = C( C A + D( D B N = C( D B DC ( A E = ( C + D α (4 (5 Figure. Distribution of velocit beond te cross-section of te cannel (. Figure 3. Distribution of velocit beond te cross-section of te cannel ( 4.

4 7 Fakriddin Abdikarimov et al.: Investigation Pulsation Motion of te Liquid in te Flat Cannels Figure 4. Distribution of velocit beond te cross-section of te cannel ( 5. Digital accounting as been carried out wit te elp of te determined formula and te law of distribution as been identified beond te surface of te cannel of te section of te velocit on te different quantit of time. Figure 5. Distribution of velocit beond te cross-section of te cannel (all states. 5. Conclusion Being obvious from te grap, wen =5, te distribution of te cross-section of velocit will be removed from parabolic distribution, and goes to te M-distribution. Largescale fluid movement is in te = σ section of te cross-section witout te center of te cannel. Tis will depend on te parameter of lengt and on te parameter of vibration. Wen <<, will be σ, te motion of maximum velocit will be observed at te center and in cases >, will be σ, ig quantities of velocit are undergone around cannel sides. References [] Ambartsuman S. A., Movsisan L. A. To te problem of pulse wave propagation. Mecanics of polmers pp [] Valtneris A. D. Influence of te viscosit of blood on te velocit of propagation of a pulse wave. Academ of Sciences of Latvia pp [3] Landau L. D., Lifsitz E. M. Teoretical psics. T. VI. Hdromecanics. Moscow. Nauka, 986. p. 736.

5 International Journal of Science and Qualitative Analsis 8; 4(3: [4] Leonov A. I. On te slow flow of a viscous fluid in a pipe wit simple walls. RAS. OTN and mecanics and engineering P [5] Loitsansk L. G. Mecanics of fluid and gas. - Moscow: Nauka, p. [6] Medvedev P. V. On metods for determining te propagation velocit of a pulse wave, Tr. Leningrad. Institute is perfect. Doctors. Issue S [7] Navruzov K. N. Periodic flows in a pipe wit slowl canging walls// Investigations on te mecanics of liquids. - Taskent, 985. pp. -. [8] Navruzov K. N. Pulsating flow of elastic fluid in a circular clindrical tube. Uzbekistan, Problems of Mecanics,, 5. [9] Navruzov K. N, Kakberdiev Z. B. Dnamics of non- Newtonian fluids, Taskent, Fan ",, 46 p. [] Navruzov K. N, Yakubov B. S. Flows of a viscous incompressible fluid in a flat tube wit periodicall varing walls. "Problems of mecanics", 7 3. [] [Eleven]. Targ S. M. Te main problems of te teor of laminar flows. - Moscow: Gostekizdat, p. [] Fazullaev D. F., Navruzov K. N. Hdrodnamics of pulsating flows, - Taskent, "Fan", 986, 9 p. [3] Fazullaev D. F., Navruzov K. N, Radzabova R. Ya. Pulsating and peristaltic flow of fluid in pipes: Abstract. All- Union. Conf. on mecanics of continuous media. - Т, 979. P. 3. [4] Atabek H. B. End effects. In: Pulsatile blood flow (ed. E. O. Attinger. New York. Mc. Graw-Hill, 964. [5] Scomb T. W. Flow in a cannel wit pulsating walls. Fluid Mecanics, 978, vol. 88 Part, pp [6] Abdikarimov F. B. Periodic flows of liquid in a flat tube wit a ig frequenc of oscillation. Problems of Mecanics, Taskent, 3-4, 3, pp [7] Navruzov K. N., Abdikarimov F. B.// Hdrodnamics of pulsating blood flows.// Lambert Academic Publising, ISBN: , Heinric-Böcking-st. 6-8, 66, Saarbrucken, German, 4. [8] E. P. Valueva, M. S. Purdin. Te pulsating laminar flow in a rectangular cannel. Termopsics and Aeromecanics November 5, Volume, Issue 6, pp