The boundary layer approximation and nonlinear waves in elastic tubes

Transcription

1 The boundary layer approximation and nonlinear waves in elastic tubes N.Antar Istanbul Technical University, Faculty of Sciences and Letters, Department of Mathematics, 80626, Maslak-Istanbul, Turkey Abstract In this paper, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and approximat equations of an incompressible viscous fluid, the propagation of weakly nonlinear waves is examined. In order to include the geometrical and structuraldispersion into analysis, the wall s inertial and shear deformation are taken into account in determining the inner pressure-inner cross sectional area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves, in the long-wave approximation, are shown to be governed by the Korteweg-de Vries (KdV) and the Korteweg-de Vries-Burgers (KdVB), depending on the balance between the nonlinearity, dispersion and/or dissipation. In the case of small viscosity (or large Reynolds number), the behaviour of viscous fluid is quite close to thatideal fluid and viscous effects are confined to a very thin layer near the boundary. In this case, using the boundary layer approximationwe obtain theviscous-korteweg-de Vries and viscous-burgers equations.

2 706 Advmces irl Fluid Mechunks W 1 Introduction The propagation of pressure pulses in fluid-filled distensible tubes has been studied by several researchers in the current literature. Such problems have been investigated, especially, in view of their applications to physiological problems involving pulse propagation in large blood vessels,pedley [l]and Fung [2]. Most of the works on wave propagation in compliant tubes have considered small amplitude waves ignoring the nonlinear effects and focused on the dispersive character of the waves (see, Atabek and Lew [3], Rachev [4]and Demiray [5]). However, when the nonlinear effects arising from the convective terms of fluids and/or the constitutive relations of tube materials are introduced, one has to consider either finite amplitude, or small-but-finite amplitude waves, depending on the order of nonlinearity. The propagation of finite amplitude waves in fluid-filled elastic or viscoelastic tubes has been examined, for instance, by Rudinger [6] and Tait and Moodie [7],by using the method of characteristics, in studying the shock formation. On the other hand, the propagationof small-but-finite amplitude waves in distensible tubes has been investigated by Johnson [8], Hashizume [g],yomosa [lo],erbay et a1 [ll]and Demiray [12]by employing various asymptotic methods. As is well-known, a far field (or long time) evolution of weakly nonlinear waves in dispersive or dissipative media can be described by some nonlinear evolution equations. For instance, in dissipative media the Burgers equation and in dispersive media the Korteweg-deVries equation are the simplest representative equations, Karpman [13], exhibiting a balance between the nonlinearity which causes the steepening and dissipation which causes the attenuation of waves, and nonlinearity and dispersion which causes the broadening of waves, respectively. Under certain conditions, such a balance between the nonlinearity and dispersion leads to the occurence of stable nonlinear structures such as solitary waves. On the other hand, when a balance exists between the nonlinearity, dispersion and dissipation, the simplest resulting evolution equation is the Korteweg-devries-Burgers (KdVB) equation which represents the combination of KdV and Burgers equations. Employing various asymptotic methods, the propagation of small-but-finite amplitude waves in fluid-filled distensible tubes has been investigated by several researchers in the current literature. For instance, Johnson [8] considered the laminar elastic jumps in an elastic tube containing a viscous fluid, and showed that the elastic jumps were governed by the KdVB equation. Hashizume [g] and Yomosa [lo] studied the propogation of weakly nonlinear waves in a thin nonlinear elastic tube filled with an incompressible inviscid fluid and showed that the propagation is governed by the KdV equation. Erbay et a1 [11]examined the propagation of weakly nonlinear waves in a fluid filled viscoelastic thin tube and

3 A d m c c ~ sill Fluid Mdxznics IV 707 obtained the governing equations as KdV, Burgers and KdVB, depending on the order of certain parameters. Demiray[l2] studied the propagation of weakly nonlinear waves in thin elastic and viscoelastic tubes filled with an incompressible inviscid fluid and obtained the KdV and KdVB equations, respectively, as the governing equations. In all these works the fluid body was considered to be inviscid and the axial motion of the tube is neglected. As far as the biological applications are concerned, in reality, the blood is incompressible and viscous fluid. The present study is undertaken with such an objective. In the present work, employing the nonlinear equations of a elastic thin tube containing an incompressible viscous fluid, the propagation of weakly nonlinear waves is studied. Considering the physiological conditions that the arteries experience, in the analysis, the tube is assumed to be subjected to a uniform inner pressure Po and a constant axial stretch ratio A,. In the course of blood flow in arteries, a pressure increment P, which depends both on time and theaxial coordinate is added by the left ventricle. As results of this, a large time dependent radial ( U") and axial (W*) displacement components, which also depends on the axial coordinate, are superimposed on this staticfield. The nonlinear equations of such a motion of an elastic tube are obtained both in the radial and axial directions. Treating the blood as an incompressible Newtonian fluid and utilizing the reductive perturbation method the propagation of weakly nonlinear waves, in the longwave approximation, is investigated. Depending on the balance between the nonlinearity and dissipation and/or dispersion, the evolution equations are obtained as the KdV, the KdVB and the nonlinear differential equations respectively. Using boundary layer approximation, we obtain viskosity-kdv (V-KdV) and viscosity-burgers equations (V-B). The solution of KdVB under some initial conditions is also investigated numerically by employing the finite-difference method. A finite-difference method is again used to obtain the solution of V-KdV, except for the integral which is evaluated by use of the discrete Fourier transformation. 2 Basic equations and theoretical preliminaries Equation of tube : We consider a circularly cylindirical long, straight and homogeneous elastic tube filled with an incompressible viscous fluid. The nonlinear equations of motion of an elastic tube can be given both in the radial and axial directions in non-dimensional form as follows

4 708 Advmces irl Fluid Mechmks W where E(R,, AS) is strain energy density function of tube material, p is the fluid pressure function, U and W are radial and axial displacements components, R, and Re are the stretches in the axial and circumferential directions, given by The components of the acceleration vector a, and a, defined as follows in eqn. (1)-(2) are In general, the strain energy density C is a function of R, and Re. For future purposes, we shall assume that C is analytic in h, and he and can be expanded into a power series around U = 0, duldz = 0 and dw/dz = 0. If these expansions are substituted into equations (1)-(2) we obtain the nonlinear equations of motion of an elastic tube in terms of radial and axial displacements. Equations of Fluid Treating blood as an incompressible Newtonian fluid, the governing equations for fluid are given by - dv du dp 1 d2v at + v dz dz R6z2 - + R( (7) +U) Here U is the averaged axial velocity, V, is the fluid velocity components in the axial direction, R is the Reynolds number and p is the averaged fluid pressure. The equations of tube, (1)-(2) and equations of fluid,(6)-(7)give sufficient relations to determine the unknowns U,v,p and W. The term av,/& appearing in (7)will be determined later by use of the boundary layer approximation. 3 Long-wave approximation In this section we shall examine the propagation of small-but-finite amplitude waves in a fluid-filled nonlinear thin elastic tube whose dimensionless governing equations are givenin eqn. (1)-(2) and (6)-(7). For this, we adopt the long-waveapproximation and employ the reductive

5 Admcc.s ill Fluid Mdxznics IV 709 perturbation method,jeffrey and Kawahara [14], and introduce the following stretched coordinate, < = P(z-gt), r = P+lgt, where E is a small parameter measuring the weakness of dispersion, nonlinearity and dissipation, a is a positive constant and g is a scale parameter which will be determined later. Assuming that all the field variables have the following series solutions expanded in terms of the small parameter E as m m U = C E~u,(<, 71, W = C E~-'/'w~(<,r), P = ~E'"P&:T), n=l n=l n=l m 'U = C En+1/2vrn((, T;x), v, = FEnvzn(<,r;x). (9) n=l Now substituting the expansion (9) together with the coordinate stretching (8) into the field equations (1)-(2), (6)-(7)and equating the terms with the same powers of E, we obtain a hierarchy of perturbation equations; since the elements of the hierarchy are complicated, only the results obtainedfor each order will be presented here. By obtaining the hierarchy we also assumed that R-' = ED i? and used the stretched coordinate y = (1- T + U). For O(E),we find the following results : n=l (8) Po-a 1 r= y1 - W'. where U((,r)is an unknown function whose governing equation will be obtained later. In order to have a non-zero solution for U we must have %l4 + g2b4po- Dl)- 2Yll + [T(Pl- Po)- (W- Po)2] = 0. (11) This equation makes it possible to determine the scale parameter g, which corresponds to the phase velocity in the long-wave limit. As is seen from equation (11) there are two waves propagating in the medium and interact with each other. Throughout this work, the waves associated with these roots of equation (11)will be called the first and second waves. For O(c2), using the results obtained for O(E) we obtain the following master equation, + p4&-a+p+1(f)= 0,?4=0

6 where the coefficients pi (i = 1,..., 4) are defined by PI = [log2+ 2(p2- p1 + po)+ 2r(-3~~~/2 + - p1 + p. - g2) From this master equation various evolution equations may be obtained by investigating some special cases. (i) a =,L3 = 1/2, i7 = 0(1) : In this case neglecting E and higher order terms in E, from (12) we obtain the Korteweg-de Vries Burgers equation as ar.+p1ug+p2del+p3 -= 0. &- at2 (14) (ii) a = l/2 and /? = 3/2, V = 0(1): In this case, the governing equation reduced to the well-known Korteweg-de Vries equation as (iii) a = = 3/2, i-i = 0(1) : In this case, the governing equation reduced to the nonlinear differential equation as XJ - + p d + W Boundary layer approximation In the case of small viscosity (or large Reynolds number) the behaviour of viscous fluid is quite close to that of ideal fluid and the viscous effects are confined to a very thin layer near the boundary. Now, we assume that = (E~+'R)-~/~and introduce stretched a coordinate y = g1/2('i7&)-1(l- T + U). Introducing this stretched coordinate and (8) into eqn. (7) we obtain

7 Admcc.sill Fluid Mdxznics IV Substituting the expansion (9) into equation (17) and using obtained for O(E)we get the following master equation the results E+/4U E+p2 E p(f)= 2a-1 E+p3i7&2'r+l d2u I p 4 0, dl- x at3 x 2 v=0 (18) In order to complete the solution we have to determine (dvzl/dy),=o. This term will be determined from the boundary layer ap roximation. Introducing the stretched coordinate transformation y = glf(%)-'(l - T + U), rii = (E~+~B)-~/~ into Naiver-Stokes equations, the boundary layer equations and theassociated boundary conditions can be found for the Q(&) order approximation. Solving the boundary layer equations and boundary conditions by use of the Duhamel's theorem and introducing this solution into equation (18) the following master equation is obtained dc'+/llue+/l2e 2a-l d3u I p3i7e2a+l E dl- x x 3 x 2 As some special cases we if select a = l/2 and a = 3/2 the following viscosity-kdv (V-KdV) and viscosity Burgers (V-B) equations are obtained, Antar and Demiray [15] J;; (l+ f)tu((<+ q,r)q-1/2drl. 0 (20) It is not so easy to obtain a progressive wave type of solution to these equations. Therefore, in the following section we shall seek a numerical solution for these equations. 5 Numerical results and discussion In this section we shall study the variation of wave profile with the initial deformation and viscosity parameter. For the numerical evaluations of the analytical results we need the values of elastic constants appearing in the theoretical model. The analytical results of the present elastic tissue model was compared by Demiray [16]with the experimental measurements

8 7 12 Advmces irl Fluid Mechunks W by Simon et a1 [17] on canina abdominal artery and for the best fit of theoreticalmodel to the experiment, the value of a was found as Q! = Utilizing this numericalvalue of the material constantin KdVB andv-kdv equations, the numerical solution of these equations under the initial conditions U(<,O) = Uosech2[ is presented. The KdVB and the left side of V-KdV equations we shall use the finite difference method, whereas for the right side of V-KdV equation we shall employ the discrete Fourier transformation. It is observed that the amplitude of the first wave for V-KdV equation decays much faster than that of KdVB equation.however, for the second wave the decay of the wave amplitudes are of the same order. References [l]pedley,t.j. The fluid mechanics of large blood vessels, Cambridge University Press, Cambridge, 1980 [a] Fung,Y.C.Biodynamics :Circulation, New York, Springer Verlag, [3] Atabek, H.B. and Lew, H.S. Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube, J. Biophys 7, , [4] Rachew, A.I. Effects of transmural pressure and muscular activity on pulse waves in arteries, J. Biomechanical Engineering. ASME 102, , [5] Demiray, H. Wave propagation through a viscous fluid contained in a prestressed thin elastic tube, kt. J. Engng. Sci., 30, , [6] Rudinger, G.,Shock waves in mathematical models of the aorta, J. Appl. Mech., 37,34-37, [7] Tait,R.S. and Moodie,T.B. Waves in nonlinear fluid filled tubes, Wave motion, 6, , [8]Johnson, R.S. A nonlinear equation incorporating damping and dispersion, J. Fluid Mechanics 42,49-60, [g] Hashizume, Y.Nonlinear pressure waves in a fluid-filled elastic tube, J. Phys. Soc. Japan, 54, , [lo] Yomosa, S. Solitary waves in large blood vessels, J. Phys. Soc. Japan, 56, , [ll]erbay, H.A, Erbay, S. and Dost, S. Wave propagation in fluid filled nonlinear viscoelastic tubes, Acta Mechanica, 95, , [12] Demiray, H. Solitary waves in prestressed elastic tubes, Bdletetin of Mathematical Biology, 58, , [13] Karpman, V. I. Non-linear waves in dispersive media, New York, Pergamon Press (1975).

9 Admcc~sill Fluid Mdxznics IV [14] Jeffrey A. and Kawahara T. Asymptotic methods in nonlinear wave theory, Pitman, Boston (1981). [15] Antar, N. and Demiray, H. Boundary layer approximation and nonlinear waves in elastic tubes, Int. J. Eng. Sci., 38, , [16] Demiray, H. Large deformation analysis of some basic problems in biophysics, Bull. Math. Biology, 38, , [17] Simon, B.R., Kobayashi, AS., Stradness, D.E. and Wiederhielm, C.A. Re-evaluation of arterial constitutive laws,, Circulation Research, 30, , 1972.