İçi viskoz akışkan ile dolu öngerilmeli viskoelastik tüplerde harmonik dalga yayılımı

Abstract

Bu çalışmada, özellikle biomühendislikte önemli uygulama alam bulan, içi sıkışmaz viskoz bir akışkan ile dolu öngerilmeli viskoelastik bir tüp içerisinde harmonik dalga yayılımı problemi incelenmiştir. Birinci bölümde kısaca konunun tarihsel gelişiminden söz edilmiş, bu konuda yapılmış, deneysel ve teorik çalışmalar anlatılmıştır. Büyük ön şekil değiştir melerde davranışın elastik olduğu, viskoelastik etkiler ise dinamik yer değiş tirmeler sırasında kendini gösterdiğinden, damara nonlineer elastik ve kuvasi- lineer viskoelastik bir model düşünülmüştür. Bu modele ait denklemler ve silindirik tüpün iç basınç ve eksenel germe altındaki gerilmesi ve şekil de ğiştirmesi Bölüm 2'de verilmiştir. Bu bölümde ayrıca büyük ön statik şekil değiştirmeler üzerine küçük dinamik yer değiştirmelerin süperpozisyonu sonucu oluşan artımsal denklemler elde edilmiştir. Üçüncü bölümde ise ön gerilmeye maruz ince tüplere ait artımsal hareket denklemlerinin türetimi verilmiştir. Viskoz akışkana ait hareket denklemleri, silindirik mambrana ait hareket denk lemleri ve sımr koşullan dördüncü bölümde verilmiştir. Beşinci bölümde kal bin uyguladığı peryodik basıncın harmonik karakteri dikkate alınarak alan denklemlerine harmonik tipten dalga çözümleri aranmış, sımr koşullan kul lanılarak dispersiyon bağıntısı elde edilmiştir. Damar yarıçapının dalga boyu na göre çok küçük olması nedeniyle uzun dalga boyu limit halinde dispersiyon bağıntısı, mümkün olan yerlerde analitik, diğer hallerde nümerik olarak ince lenmiştir. Sayısal incelemeden sonra sonuç ve öneriler kısmında elde edilen sonuçlar maddeler halinde sıralanmış, bu konuda yapılan diğer çalışmalarla karşılaştırıl mış ve geleceğe dair önerilerde bulunulmuştur.

Propagation of harmonic waves in initially stressed (or unstressed) cylin drical elastic (or viscoelastic) tubes filled with a viscous (or inviscid) fluid is a problem of interest since the time of Thomas Young who first studied the pulse wave speed in human arteries. Treating the artery as an elastic thin tube and the blood as an inviscid, incompressible fluid Moens-Korteweg (1809) studied the wave propagation in such a medium and obtained the wave mode, known in 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 viscosity. In 1954, Morgan and Kiely studied the same problem by assuming the artery as a lineer elastic tube and the blood as a Navier-Stokes fluid and obtained the dispersion relation. In 1955 Womersley treated the artery as a thick walled cylindrical shell and the blood as an incom pressible viscous fluid and obtained the dispersion equation. 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 all these works either the effects of initial stresses are neglected or the artery is treated to be as a membrane. Physiological studies indicate that, for a healthy human being the systolic pressure is about 120 mmHg and the diastolic pressure is around 80 mmHg. Furthermore the arteries are subjected to axial stretch, which is about 1.5. Thus, the large blood vessels are subjected to a mean static inner pressure, which is about 100 mmHg and the axial stretch ratio. In the course of blood flow a pressure increment ±20 mmHg is added by the left ventricle on this large initial static deformation. The initial stress of the arterial wall material had been taken into consideration first by Atabek and Lew, in 1966. Since in those years the nonlinear constitutive relations for arteries were not known in functional form, these initial stresses were not incorporated to initial de formations and to the incremental 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 de formation was properly taken into account by Rachev in 1980, but he simply treated the artery as a purely elastic material. However, many researchers (Fenn, Gow and Taylor, Moritz and Ogden) had pointed out that the blood vessels are nonlinear, viscoelastic and anisotropic. Therefore, the viscoelastic character of the arterial wall material should be taken into account. Rubinow and Keller studied the harmonic wave propagation in a thin viscoelastic tube filled with an incompressible Newtonian fluid, but neglected the effect of initial stresses. 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 propa gation in a prestressed viscoelastic tube filled with a viscous fluid. For that purpose, first, we reviewed the kinematics and the balans laws of continuous medium in Section 2. These are: (i) Conservation of mass: Po = P J (1) where po is the initial mass density, p is the mass density at time t and J is the Jacobian of the motion of viscoelastic body. (ii) Balance of linear and angular momenta: tki,k + p(fi - vi) = 0, tkı = tik. (2) where tki is the Cauchy stress tensor, // body force density and vi is the velocity vector. Here and throughout this work comma is used to denote the partial differentiation and the summation convention applies on repeated indices. (iii) Balance of energy: p i - tki dki + p h - qk,k (3) where e is the internal energy density, h is the heat source, qk is the heat flux vector and dki is the deformation rate tensor defined by 2 dM = Vk,ı + vi,k (4) Here, over-dot is used to denote the material time derivative of the correspond ing quantity. (iv) Second Law of thermodynamics: rv + ^ - Ç > o (5) where rj is the entropy density and 9 is the absolute temperature. vi Considering the physiological conditions that the arteries are subjected to and the inccompressibility of the material, we proposed a quasi-linear consti tutive equations for arterial wall material in the following form tki = P hi + exp[ a(Ji - 3)] i ckı [ Ş + / Xv(t-r) cmm(r) dr] J - oo + / //"(*- r) [ bkm(t,r) btm(t,T) + bkm(t,r) &{m(*,r) ] dr\ J -oo (6) where P is the hydrostatic pressure, I\ = ckk, oc and /? are two material con stants, Xv(t) and fiv(t) are two viscoelastic coefficient functions and cki(t) and bki(t, r) axe defined by cki(t) = FkK(t)FlK(t), bkl(t,r) = FkK(t)FiK(r) (7) Here we should note that, the material is assumed to be isotropic and over-dot under the integral sign denotes the differentiation with respect to parameter r. As is well-known the arterial wall material is subjected to a large static deformation, and in the course of blood flow a small displacement field is added on this static deformation (or stress). For simplicity in mathematical analysis, we shall treat the arterial wall material as a thin shell. The nonlinear and the linear equations of a thin shell with above described loading characteristics are given in Section 3. Assuming that the superposed dynamical motion is axially symmetric the governing equations may be given in cylindrical polar coordinates as follows N° Pi - - - = 0 Laplace Law,r0 iVju S« =, d2u NQ2u,2S+-^--- + Pn =Phw (8) / N* dT,z =,, d2w (-2- - Pi)u,s + -=f- + Pt = ph - r0 dz at2 Here Pi is the inner pressure, N§ and İV° are the initial stress resultants, u and w are the icremental displacement components in the radial and axial directions, So and S2 are the incremantal stress resultants and Pt and Pn stand for the fluid reaction and given by Pn = ( P ~ 2 f.1 drr )\r=r0, Pt = ~ ( 2 fl drz )\r=r0 (9) where p is the pressure, /j viscosity of the fluid and ro is the midradius of the tube after finite deformation. VII Using the definition of resultant membrane forces after final deformation, e.g., N'e = h' t'00, N'z = ti t'zz (10) where h' is the final thickness, t'9e and t'zz are the total stress components in the radial and axial directions, the incremental membrane forces may be given as follows. He - h [tg$ - t09 ( eeo + ezz ) ] (11) E* = h [ tZ2 - t°zz ( e9e + ezz)] Here h is the deformed thickness of the tube, tij is the incremental stress tensor referred to final configuration and ey is the incremental strain tensor. The incremental equations of the tube and the blood, which is assumed to be an incompressible Newtonian fluid, are given in the silindrical polar coor dinates and properly posed boundary conditions are given in Section 4. The incremental fluid equations may be given by dp. d2u 1 du u d2u _ du _ ~fo + ^( dr* + rd? ~T+dz~2)~Pdt = dp d2w 1 dw d2w _ dw (12) and the incompressibility condition dü ü dw _ dr r dz ~ ^ ' where p is the mass density of the fluid and ü, w are the incremental velocity components in the radial and axial directions, respectively. Introducing (11) into (8) and using the definition of incremental strain ten sor, the equations governing the incremental motion of the tube may be ex pressed as follows g w -h ft) "_»«,£_*/. aUt.T)*jpdT dz2 t% r0 dz r2J_00 ' dr h f* 0., d2w{r),. _ n -. d2u ~ To J- ""^ ~ ^ dzdr dT + İP ~ 2V drr )r=r0 = P h ~ğ^ VU1 (N° - N° + h fa) du 0 d2w h [* 0 d2u(r),, /' o, n d3w(r) J n,., d2w + hj_ «22(* ~ r) g22dT dT ~ 2V- ^«Ir-ro = ph - (14) where /3°,- are some coefficients depending on the initial deformation and «y(i) axe some known functions; the expressions of them are given in the main text. These equations are to be supplemented with the boundary conditions given by u(r0,z,t) = -jg, w(r0,z,t) = - (15) Solution to field equations: Considering the pulsatile motion of the left ventricle, it is suitable to seek a harmonic wave type of solution to field equations. For that purpose we set { u,w,p ; u,w } = { Ü(r),W(r),P(r) ; A,B } exp[ i(kz - cut) ] (16) where uj is the angular frequency, k is the complex wave number Ü(r), -P(r) are unknown amplitude constants. Introducing (16) into (12) and (13), the solution of Z7(r), W(r) and P(r) may be given by Ü = k [ Ch{kr) + DJx(sr) ], W = i[ kCI0(kr) + sDJQ(sr) } P = ipuCIo(kr), s2 = ^ - k2 A* (17) Here In and «7" are the modified and first kind of Bessel function of order n. Introducing (16) and (17) into (14) and (15), and ehminating the coefficients A and B among these equations the following homogeneous algebraic equations are obtained G -_7n G -_7n + ( ^- + 2iv\sü )£C }d = 0 { en m( & - Te + G ~27n ) - «^.n j + ^Jf^ + a,( a2 + 2iuae ) }c + {(C9[m(ü2 - Te + G "J" ) - 2iuX2ü ] *9 ix f c3fr,T - G + 721 s, 0.. 0 i, t ( A2 ~ 722e2 ) \~ 9 9 + { fc[ ^e2(r " ^2+ 721 ) + a,( n2 + 2ivne2 ) ] (is) ( fi2 - 722^2 ) 1 - + tTK- T } )D = 0 where we have defined the following non-dimensional quantities 7« = fi 7«. *** = W, İV? = hŞG k = 1_, s = i_, c2 = £ w = -5-n, m = t- -, // = pc0Rou (19) ilo P-*M) /(« = t4^t. «(C) = Msn) kr0I0(kr0) sr0I0(sr0) C = I0(kr0)C, D = J0(srQ)D. In order to have non-zero solution to these algebraic equations the determinant of the coefficient matrix must vanish. If this operation is carried out the following dispersion relation is obtained ( Axe + a2 )o4 + ( A3e + Ate4 )^2 + c a5£4 + a6£6 ) = o (20) where the coefficients Ai ( i = 1,2,....6 ) are defined by Ax = mX29 [ m(f - g) + fgX2g Xz ] A2 = A|AS ( m + gX"gXz ) A3 = - 2ivüX29Xz[ m(f - g) + g(f - 2)X29XZ ] + m2(f - g){G - 7ll) + mX29Xz[ fg(G - 711) + g(T - G + 712 + 721 ) - 722 ] (21) A4 = mA2[m(0 - f)(T + 722) - /<7TA2A, ] + 4(/ -2A2A2 A5 = (/ - flf){- 2iuQmX29Xz( T - G + 712 + 721 - 722 ) + m2[ (G - T)7i2 - G722 + 711722 - 712721 ] } A6 = m2X29T(f - (7)722. In genaral it is very difficult to analyse this relation by analytical means; a numerical approach is rather to be used. Before we study the general case numerically, some special cases are examined analytically and the results are compared with, previous works on the same subject. Then, the dispersion rela tion is studied by numerical means for various stretch ratios and inner pressure and the results are depicted on some figures. The numerical results reveal that: (i) the speed of the primary wave increases with Womersley parameter, inner pressure and the axial stretch ratio; (ii) the speed of the secondary wave increases with Womersley parameter and inner pressure but decreases with axial stretch ratio; (iii) the transmission coefficient of the primary wave first decreases with Womersley parameter then starts to increase with it. More over, this coefficient decreases with inner pressure but increases with axial stretch; (iv) the transmission coefficient of the secondary wave increases with Womersley parameter and inner pressure but decreases with axial stretch ra tio. These results are consistent with the results of previous works on the same subjects.