Ön gerilmeli elastik tüp içerisinde pulsatif akımın incelenmesi
An investigation of pulsatile flow in a prestressed thin elastic tube
- Tez No: 68899
- Danışmanlar: PROF. DR. HİLMİ DEMİRAY
- Tez Türü: Yüksek Lisans
- Konular: Mühendislik Bilimleri, Engineering Sciences
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Mekanik Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 102
Özet
ÖZET Bu çalışmada, özellikle biyomühendislikte uygulama alanı bulan, içi sıkışmaz bir viskoz akışkan ile dolu ön gerilmeli elastik bir tüp içerisinde harmonik dalga yayılımı problemi incelenmiştir. Birinci bölümde kısaca konunun tarihsel gelişiminden söz edilmiş, bu konuda yapılmış deneysel ve teorik çalışmalar anlatılmıştır. Damar malzemesinin sıkışmaz, elastik ve izotrop olduğu ve bir ön statik şekil değiştirmeye maruz kaldığı kabul edilmiş ve bununla ilgili temel denklemler Bölüm 2' de verilmiştir. Pulsatif kan akımı sırasında, bu statik ön şekil değiştirmeler üzerine küçük dinamik yer değiştirmelerin süperpoze edildiği hareket denklemleri Bölüm 3' de verilmiştir. Bu matematiksel türetimin uygulaması olarak Bölüm 4' de ön gerilmeye maruz ve içerisinde viskoz akışkan bulunan ince tüplerde pulsatif akım problemi incelenmiştir, önce Bölüm 4.1' de sabit yarıçaplı silindirik bir tüp içerisinde harmonik dalga yayılımı problemine yer verilmiştir. Biyolojik uygulamalarda, damar yarıçapının dalga boyuna göre çok küçük olduğu gözönünde bulundurularak dispersiyon bağıntısı mümkün olan yerlerde analitik diğer hallerde de nümerik olarak incelenmiş ve sonuçlar bir kısım grafikler üzerinde gösterilmiştir. Daha sonra Bölüm 4.2' de eksen boyunca yarıçapı yavaş değişen tüplerde pulsatif akım problemi incelenmiş ve uzun dalga boyu için alan denklemlerine bir pertürbasyon serisi çözümü sunulmuştur. Sayısal incelemeden sonra, sonuç ve öneriler kısmında, elde edilen sonuçlar maddeler halinde sıralanmış, bu konuda yapılan diğer çalışmalar ile karşılaştınlmıştır. VI
Özet (Çeviri)
SUMMARY Propagation of harmonic waves in initially stressed (or unstressed) cylindrical elastic (or viscoelastic) tubes filled with a viscous (or inviscid) fluid is a problem of interest since the time of Thomas YOUNG [1] who first studied the pulse speed in human arteries. Treating the artery as an elastic thin tube and the blood as an incompressible inviscid fluid, Moens - Korteweg (1809) studied the wave propogation in such a medium and obtained the wave mode, known in the current literature as Moens-Korteweg wave speed. Witzig is the first one who took the viscosity into account but ignored the effects of Poisson 's ratio and viscoelasticity. In 1954, Morgan and Kiely studied the same problem by assuming that the artery as a linear elastic tube and the blood as a Navier-Stokes fluid and obtained the dispersion relation. In 1966, Anliker and Maxwell studied the non-symmetrical wave motion by treating the artery as a thin walled elastic tube and the blood as a viscous fluid and obtained the cut-off frequency. In these works either the effects of initial stress are neglected or taken into account in ad-hoc manner. Physiological studies indicate that for a healty human being the systolic pressure is about 120 mm Hg and the diastolic pressure is around 80 mm Hg. Furthermore, the arteries are subjected to an axial stretch ratio, which is about 1.5. Thus, large blood vessels are subjected to a static mean pressure which is about 100mm Hg and the axial stretch ratio. In the course of blood flow a pressure increment ± 20mm Hg is added by the left ventricle on this large initial static deformation. The initial stress of arterial wall material had been taken into account first by Atabek and Lew, in 1966. Since, in those years, nonlinear constitutive relations for arteries were not known in functional form, these initial stresses were not incorporated into the initial static and into the incremental dynamical deformations. As a result of this, they treated the coefficients of incremental stress- strain relations as some constants, although they depend on the initial deformations. The effect of initial deformation was properly taken into consideration by Rachev in 1980, but he treated the artery as a purely cylindirical thin membrane. In essence, the arterial geometry is not cylindirical, it is rather a conical shell. Having observed some of the drawbacks of the previous works on this subject, in the present work, we have presented a theoretical analysis of wave propagation in a prestressed thin elastic conical tube filled with a viscous fluid and studied some special cases. vuBasic Equations: LEquation of Membranes Let us consider a thin conical tube subjected to axially symmetric external forces exerted by the flow of a viscous fluid inside of the tube. Denoting the membrane forces along the meridional curve by T^ and the circumferential curve by T2 (see Figures 1 and 2), the equation of motion of the tube in cylindirical polar coordinates may be given by a_ dz Tlr'+^ H 0 dz (ro + U) AT + A(r + u)p =ph rj 1 - - ) a (1) d dz T ;(ro + u) + Al (ro + U)1 ^o o \ - ^ I a V dz (2) where p is the mass density, h is the thickness of the tube after finite static deformation, r0 is the radius of the tube after finite static deformation, u and w are the radial and axial displacements superimposed on initial static deformation, P and P are the external forces exerted by the fluid, a and a are the components of acceleration vector in the cylindirical polar coordinates, and they are given by a = r a2 O U at 8ü _ d\ t- + a - r + 2 v 2 z dz z dz at r + v a2 2 d u z a, 2 dz a = 1 - - - z v dz a2w.,, 5w^ a2w (. 5wV a2w a2w + 21-- + i az azat az az2 a*2 (3) Other quantities are defined by : A = v = 1 + f au r + -Iİ/2 az 1- aw az \~J aw (4) V1UHere prime denotes the differentiation with respect to axial coordinate z. The external forces applied by the fluid are given by AP = dv P - 2p, - L + p. v ör ' (dv dv } r + z dz dx t du r + - 0 dz + AY (5) AP = -Plr'3 0 dz. (dv dv\ T + Z dz dr dv f + 2n dz i du r + -. ° dz + AY where P is the fluid pressure, p. is the viscosity, v and v are the velocity components of the fluid body. The membrane forces T and T may be given, constitutionally as, T = H gZ 1 x dx 2 1 T = H SL' 2 X dX 2 2 (6) where S is the strain energy density function X and A, are the stretch ratios along the meridional and the circumferential curves. 2.Equation of Fluids Treating the blood as an incompressible Newtonian fluid, the governing equations in the cylindrical coordinates may be given as follows: f dv dv dv ^ ~ + V - - + v r 3P Ut* T dr z dz) d t 'd v dv d v ^ - f + v - - + v - - \d\ r dx z dz + V- dzv \ dv v d2v r +- *---+ + - r Kdx2 r dx r2 dx*2j ap - + u. dz v (7) ' d 2v i d v 52v z_, j_ z_, z_ v5r2 i dx 5t*2y (8) and the ^compressibility condition IXÖV V öv L + _L + L = O ÖT T d Z (9) where p is the mass density and P is the pressure function of the fluid. The boundary conditions to be satisfied by the field quantities are stated as du - r + v at r=rn+u d Z = V 1- d w d z) \~x d w 17 - V (10) Equations of Incremental Motions Assuming that the incremental deformation superimposed on the initial static deformation is small, from (1)- (10), the incremental equations and the associated boundary conditions are given as follows d_ dz.-, ^.. m°~ 3, ÖU ^0.,“0., 2, OW IR smi+TR cos $ - + T sın 6 u - T R sın 4> cos d z w COS *o r ^0 0 a*2 (11) _a_ ÖZ u 0 2 2 5w E R cos a V r r ar dz1 (15) av z at av 0 + v z + V ar av r ar z +v° av z az z + V av1 az y ap az ^a2v z ar2 + 1 dv. r ar a v z + z az^ (16) and the boundary conditions XIdu dw - V r=R”+ U 5vu r_ dr r=R“ = V r=R”+ U z_ 5r (18) r=R" Propagation of harmonic waves As one of the applications of the present theoretical analysis, in this part we shall study the propagation of harmonic waves in a prestressed fluid filled cylindrical tube filled with a viscous fluid. For this particular case, the tube will be assumed to be cylindrical. Seeking a harmonic wave type of solution to the field equations we obtain the following dispersion relation
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