A MATHEMATICAL MODEL OF FLOW IN A LIQUID-FILLED VISCOELASTIC TUBE
Giuseppe Pontrelli Istituto per le Applicazioni del Calcolo - CNR Viale del Policlinico, 137 00161 Roma, Italy E-mail:
[email protected]
Abstract The propagation of waves in a distensible tube filled with a viscous fluid is studied numerically. The main application is in biofluid mechanics, where the fluid-solid interaction is of primary importance. Based on the assumption of long wavelength and small amplitude of pressure waves, a quasi-1D differential model was adopted. The model accounted for vessel wall viscoelasticity and included the wall deformations along both radial and axial directions. The nonlinear problem was solved in nondimensional form by a finite difference method on a staggered grid. The boundary conditions were for two relevant cases: natural oscillations in a deformable tube fixed at the ends and persistent oscillations due to a periodical forcing pressure. The natural frequency t was found to be nearly equal to the square root of the elasticity coefficient , with 0 6000 and is not affected by the viscosity. These results were strongly influenced by both wall viscoelasticity and blood viscosity: the damping time was found to be inversely proportional to the wall viscosity coefficient, while the fluid viscosity provides an even larger damping factor.
K
K
S
Key words: Wave propagation, wall-fluid interaction, viscoelasticity, hemodynamics.
1 Introduction Flows of viscous fluids in deformable tubes are quite common in many applications, for example in modelling blood flow problems. Experimental evidence shows that, when an unsteady forcing perturbs a steady flow in a distensible tube, damped waves are formed and propagated downstream. In spite of many theories having been developed to explain this phenomenon, the propagation mechanism is not yet fully understood, because of the complexity of the system and of the nonlinear fluid-structure interaction. Although physiological flows are better described in multi-dimensional models, simpler 1D models give some useful hints on the wave propagation and offer a easy tool in understanding the basic features of the damping effect allowing a systematic analysis on a wide range of parameters. As a matter of fact, the unidirectional nature of blood flow justifies the attempt to apply the 1D approximation to long arterial conduits. Several studies of pulse propagation in arteries have been carried out in order to gain some insight into the mechanical interaction between blood and vessel wall. Many of these models are concerned with an incompressible Newtonian fluid contained in a compliant tube. Moreover, most of them are based on the linearized equations of motion and linearized relations between stress and strains (L IGHTHILL, 1978; P EDLEY, 1980). 1
Under the simplistic assumption that the vessel wall is purely elastic, such linear models are known to produce shocklike flow pattern in the propagating pulses which are not observed under physiological conditions (A NLIKER et al., 1971; O LSEN, 1967). On the other hand, many authors have pointed out that the blood vessel walls are nonlinear, viscoelastic and anisotropic (ROCKWELL et. al., 1974). By including an appropriate mathematical model for the viscoelastic properties of the wall, the applicability of the one-dimensional theory can be extended. The damping resulting from viscoelasticity inhibits the sharp peaks of the pressure and flow pulses and flatten the abrupt rise of wave fronts. Thus, such models lead to more realistic results when a comparison with experimental data is carried out (H ORSTEN et al., 1989). In most of these papers, however, the distensibility of the arterial wall is modelled by nonlinear algebraic relations between the cross-section of the tube and the transmural pressure (M ORGAN and PARKER, 1989) and sometimes is a function of the frequency (R EUDERINK et. al., 1989): in some cases the integration is possible along the characteristics and the wave celerity is expressed in explicit form (ROCKWELL et al., 1974). In contrast with this approach, which lacks a reliable mechanical justification, some constitutive strainstress equation modelling the mechanical properties of the arterial wall has been proposed (H UMPHREY, 1995). A recent study has been carried out on a two-dimensional flow in a rigid channel, where a part of the wall is replaced by a viscoelastic membrane subject to deformation (P EDRIZZETTI, 1998). In this paper, a similar but simpler 1D model is demonstrated to be sufficient to describe propagative phenomena. The wall-fluid interaction in arterial flow problems and the role played by the constitutive equation of the vessel, which includes a viscoelastic term, is examined. The motivation of this study lies in the possibility to understand the evolution of small flow disturbances induced by a local vessel insertion or generated by a pathological state. The analysis of 1D models has revealed to provide much insight on the mechanism and the alteration of pulse propagation (R EUDERINK et al., 1989). The model used is based on the quasi1D flow equations coupled with the massless membrane equilibrium equation. A strain-stress constitutive equation for the wall is also provided, allowing wall deformations in both longitudinal and circumferential directions. Although blood flow in large arteries is inertia dominated, the limiting cases of very small and very large Reynolds number are also considered, in order to understand the important role of the dissipative mechanism induced by viscous friction and giving rise to damped waves when the system evolves to the steady state. The critical parameters involved in the mechanics and responsible for the possible numerical instability are pointed out.
2 Mathematical formulation Let us consider the flow of a homogeneous, incompressible newtonian fluid in a cylindrical distensible tube of circular cross section of radius R0 and length L. Being interested in the flow pulse propagation, the assumption of one-dimensional flow is a valid approximation under the hypothesis that the wave amplitude is small and the wave length is long compared with the tube radius, so that the slope of the deformed wall remains small at all times. The fluid equations By assuming that the flow is axisymmetric and indicating with x and r the longitudinal and the radial coordinates respectively, let us consider the x-component of the momentum equation:
@w + w @w = , 1 @p + @ r @w @t @x @x r @r @r
(2.1) 2
w is the axial component of the velocity, the kinematic viscosity and p the transmural pressure. @ 2 w , as well as the other velocity components, are comparatively small and have been The diffusive term @x2 neglected (P EDLEY, 1980). Let us define the flow rate and the averaged velocity over a cross section A = R2 respectively as: Z R Q = 2 w rdr u= Q A 0 where
and let us introduce a set of nondimensional variables:
x ! Rx 0 u u! U 0
r ! Rr 0 p p ! U 2 0
t ! tRU0 0
(2.2)
where U0 is a characteristic velocity and R0 is a reference radius. By integrating (2.1) over the cross section A, we obtain the quasi-1D nondimensional form of the momentum equation:
@u + u @u = , @p + f @t @x @x where f is the friction term.
(2.3)
If we locally approximate the friction term in (2.3) by that corresponding to the steady Poiseuille flow in a tube of radius R, then:
f ' , Re8uR2 U0R0 is the Reynolds number. Consequently the wall shear stress is approximated by: where Re = 4u ' , = du dr Re R R
(2.4)
(2.5)
Strictly speaking, the expressions (2.4) and (2.5) hold for a steady flow in a rigid tube but, since we are in-
@R
1
terested in the average effect of pulse propagation, they are reasonable when @x 0 is a nondimensional elasticity coefficient, with E the Young modulus 0 0 and h the undeformed arterial thickness, C > 0 is a wall viscosity coefficient and the dot denotes time In equations (2.9), K
derivative (P EDRIZZETTI, 1998). The former relations (2.9) hold in the case of an incompressible and isotropic material, wherein principal directions of strain and stress coincide and express the property that the instantaneous Young’s modulus increases with the strain, though of a different amount in the two directions. Although the inertia of the membrane is neglected and a general theoretical framework is still lacking, in the 1D case studied here the simple functional dependence strain-stress in equations (2.9) takes into account the viscous effects of a material in time dependent motions and model the response of the arterial wall to the deformation and to the rate of deformation. In other words, equations (2.9) mean that the membrane does not respond instantaneously to forces, as a purely elastic body, but with a dissipative mechanism as a viscoelastic material. Note that (2.3) and (2.7) depend explicitly on time, while the membrane equilibrium equations (2.8) depend on time through the dependence of the stresses T1 and T2 on the strain rates in equations (2.9). The boundary conditions are imposed by considering the wall fixed at the two ends, that is:
R (0; t) = 1 R (L; t) = 1
S (0; t) = 0 S (L; t) = Lu
(2.10)
where Lu is the length of the undeformed membrane. Following the experimental set up of a distensible tube fixed at the two extrema, only the values of p ; t and p L; t are assigned at both boundaries. This approach is different from that in (P EDRIZZETTI , 1998), where the value of flow rate is assigned at one of the boundaries together with the value of the pressure. In particular, in order to study the transient to the equilibrium configuration, the same value of the pressure is assigned at both ends:
(0 )
( )
p (0; t) = pref
p (L; t) = pref
(2.11)
As a second test case, an oscillating forcing pressure is given at the outlet:
p(L; t) = pref + sin(2 St t) (2.12) R0 are the nondimensional amplitude and the Strouhal number (with T the where < pref and St = U0 T period) of the excitation, respectively. The use of boundary conditions based on much physiological considerations are being developed (P ONTRELLI , 2002). The initial condition is chosen by considering a finite perturbed configuration of the steady flow, corresponding to a purely elastic wall and a constant pressure gradient. Then the system is left to evolve towards its equilibrium configuration (see (2.11)) or forced by an oscillating pressure (see (2.12)).
3 Numerical method and parameters The equations of evolution of the fluid (2.3) and (2.7), the equation of the equilibrium of membrane (2.8) together with the constitutive equations (2.9) are discretized by centered second order finite differences in space. 5
+1
=0
=
Let us consider a sequence of n equispaced grid points with x0 and xn L. The spatial discretization is obtained by evaluating membrane stresses, strains, and their time derivatives (see eqns.
= xi +2xi+1
(2.9)) at n inner points i of a staggered grid by considering averaged neighbouring quantities. On the other hand, equilibrium equations (2.8) and fluid equations (2.3)-(2.7) are computed at the n , inner points xi . The time discretization is based on the trapezoidal formula, in such a way the global scheme is of second order in space and time (F LETCHER , 1988). The resulting nonlinear system is solved by a globally convergent Newton type method. Nonlinear models turn out to be very sensitive to the choice of the material parameters which charactercm=s and R0 : cm. ize the specific flow problem. The reference values used in the (2.2) are U0 and the other parameters have been chosen around The nondimensional pref has been fixed as pref some typical values to obtain results of physiological interest and varied in a typical range to test the sensitivity of the system to their perturbation. The value of the elasticity parameter K must be larger than p and p=K is approximatively proportional
1
= 50
= 10
to the radial deformation of the membrane. In a physiological context,
200
=05
h 0:1 (N ICHOLS and ROURKE, R0
. Nevertheless, to better investigate the capabilities of the model and the 1990), and consequently K functional dependence of the solution, the problem is studied for K ranging in the wider interval : it turns out that for a larger value of K the stiffness of the wall becomes extremely high. On K the system undergoes an unrealistic large deformation and the present model the other hand, for K < is not representative of the physics (see section 4). ,2 and t ,3 . These values guarantee the , x In all experiments we selected L numerical stability of the system for the set of parameters considered. The accuracy of the solution was validated by considering a finer grid which did not reveal any different solution structure or unresolved patterns. For all the experiments, the same initial data set was chosen. Although the Strouhal number of the forcing in the physiological flow is extremely small (St : , in this model study we will consider it comparable with the natural frequency St of the membrane (see next section). This is computed by spectral analysis when the system, subject to an initial deformation, is left evolving toward the equilibrium configuration obtained by imposing the same value of the pressure at the two extrema.
6000
200
200
= 2 = 10
= 10
0 01
4 Results and discussion In this section the results of several numerical test are reported to reproduce realistic flow fields close to the biomechanical applications. When the system is left evolving from an initial perturbation toward the equilibrium, all the variables after an initial transient, asymptotically reach a steady state value with damped oscillations having an exponential decay and with a K -dependent frequency St (natural frequency). Due to the elasticity of the wall, a positive deformation and a positive flow rate at the final state at both ends is found equal. If not otherwise stated, a convection dominated fluid is considered (f in (2.3) and in (2.8.1)). Since no quantitative data for the dissipation of the wall in dynamical experiments are available, the influence of the membrane viscosity is investigated by varying the value of C . It turns out that the attenuation factor increases with C but is independent of K (see fig. 1). For C ! the solution is not damped and tends to that of a system with a purely elastic wall. To better understand the effect of the reference pressure, the value of pref is varied in the range :; , for two assigned values of K and C . Instability develops for larger pressures as a consequence
0
0
[0 5 200]
6
=0
of too large deformations. In such a case the 1D model is not adequate to describe the phenomenon. The value of u does not change with pref , while the deformation is proportional to it, and a slight increase of the natural frequency is observed (see fig. 2). However, the convergence of the system in the transient critically depends on the initial data. On the other hand, the steady state values of the deformation decrease with K , and are independent the natural frequency St increases with K (at the first instance, a of C and of the initial data. Similarly, p functional dependence St / K is assumed (L IGHTHILL , 1978)), but stays unchanged with C (see fig. 3) and with the initial data. The effect of the viscosity coefficient C is to dissipate energy, to damp the high frequency oscillations (but not to modify their frequency) and is observed only in the transient regime. The longitudinal displacement S turns out to be be very small (three order of magnitude lower) compared with R and is not displayed. However, the influence of such variable becomes more relevant when the fluid friction at the wall is considered. Oscillating forcing. The boundary condition (2.12) reflects the basic physiological waveform at outlet. Results from simulations of pref show the flow no longer decays to with several Strouhal numbers and amplitude a steady solution, but the persistence of steady oscillations of sinusoidal type occurs at the same frequency of the input forcing1 . The oscillations are about the same value as that corresponding to steady state with pref and their amplitudes attain a maximum when St St (resonance phenomenon) (figs. 4 and 5). The analysis of the radial velocity of the wall proves that the propagation features correspond to transverse waves which do not propagate along the tube and are due to the boundary conditions that generate reflections and spurious effects (fig. 6). The boundary conditions (2.10) of Dirichlet type may be unrealistic in physiological flows, but well reproduce a physical experiment. The phenomenon is similar to that of waves propagating in a stretched string of a finite length. The results do not differ if the excitation at the right boundary is replaced by one at the left boundary.
= 1(10%
= 10
= 10)
=
Influence of the fluid viscosity. In order to understand the role of viscous and inertial terms in the fluid motion and its interaction with the in wall, results of a inertia dominated flow (i.e. with Re ! 1) and of a very viscous flow (i.e. with Re (2.4) and (2.5)) (and with all the other parameters fixed) have been compared. Such a comparison shows that the mean velocities first exhibit a stronger damping to zero in the transient, and then a rising to the steady state values (see fig. 7, where a non-monotonic envelope is visible). The steady state deformation value and the frequency of oscillations in the two cases are the same, even with an oscillating forcing, and the wave celerity changes with Re only when K is small. Therefore it appears that the frequency of oscillations is an intrinsic characteristic of the wall and is not greatly influenced by the fluid properties. Nevertheless, the presence of fluid viscosity allows for a larger value of the amplitude of the forcing pressure, without blowing up. As reported elsewhere, the attenuation of the wall viscoelasticity turns out to be a dominant effect in larger vessels, while the damping induced by the fluid viscosity becomes more important in small vessels (P ONTRELLI , 1998).
=1
5 Conclusions A coupled wall-fluid model for studying the unsteady flow of a viscous fluid in a viscolastic tube has been presented, with application in the arterial flows. The unidirectional nature of blood flow suggests to model 1
A blow up of the solution is obtained with larger values of depending on the value of K .
7
the propagating flow pulses with a quasi-1D mathematical approximation which is well supported by measurements. However, this formulation yields realistic results provided that the wavelength is long and the wave amplitude small compared with the mean radius of the tube. A linear constitutive relation for the vessel wall that depends on both the strain and the strain rate is presented. The model includes the combined effects of the viscoelasticity of the solid tube as well as of the viscosity of the blood. The effect of the elasticity parameter is related to the frequency of oscillations in the transient, while the influence of viscosity parameter is to attenuate the high frequency oscillations, to reduce the tendency of shock formation as present in a purely elastic wall model, and to counterbalance possible instability phenomena. Though some simplifying hypotheses are necessary to assess the theory, the model study presented here is able to describe the basic physical mechanism of the 1D pulse propagation through the nonlinear interaction between the fluid and the wall. However, the numerical value of the elastic and viscous coefficients appearing in the constitutive equation as well the boundary conditions are critical and need to be carefully identified by comparing numerical results with experimental measurements.
Acknowledgments: The author wishes to thank prof. Gianni Pedrizzetti for many useful and valuable discussions. The work has been partially funded by the CNR Project Agenzia2000: CNRC00A3F1-002.
8
References [1] A NLIKER M., ROCKWELL R.L., O GDEN E. (1971) Nonlinear analysis of flow pulses and shock waves in arteries, p. I, J. Appl. Math. Phys. (ZAMP), 22, 217-246. [2] F LETCHER C.A. (1988) Computational Techniques for Fluid Dynamics I, Springer-Verlag. [3] H ORSTEN J.B., VAN S TEENHOVEN A.M., VAN D ONGEN A.A. (1989) Linear propagation of pulsatile waves in viscoelastic tubes, J. Biomech., 22, 477-484. [4] H UMPHREY J.D. (1995) Mechanics of the arterial wall: review and directions, Crit. Review in Biomed. Engineering, 23 (1/2), 1-162. [5] K YRIACOU S.K and H UMPHREY J.D. (1996) Influence of size, shape and properties on the mechanics of axisymmetric saccular aneurysms, J. Biomech., 29, 1015-1022. [6] L IGHTHILL J. (1978) Waves in fluids, Cambridge University Press. [7] M ORGAN P. and PARKER K.H. (1989) A mathematical model of flow through a collapsible tube - I Model and steady flow results, J. Biomech., 22 (11/12), 1263-1270. [8] N ICHOLS W.W. and ROURKE M. F. (1990) Mc Donald’s Blood flow in arteries, Edward Arnold. [9] O LSEN J.H. and S HAPIRO A.H. (1967) Large-amplitude unsteady flow in liquid-filled elastic tubes, J. Fluid Mech., 29, 513-538. [10] P EDLEY T.J. (1980) The fluid mechanics of large blood vessels, Cambridge University Press. [11] P EDRIZZETTI G. (1998) Fluid flow in a tube with an elastic membrane insertion, J. Fluid Mech., 375, 39-64. [12] P ONTRELLI G. (1998) Pulsatile blood flow in a pipe, Computers and Fluids, 27, 3, 367-380. [13] P ONTRELLI G. (2002) A multiscale approach for modelling blood flow in an arterial segment, Quad. IAC/CNR 14/02, 2002, submitted. [14] P ORENTA G., YOUNG D.F., ROGGE T.R. (1986) A finite element method of blood flow in arteries including taper, branches, and obstructions, J. Biomech. Eng., 108, 161-167. [15] R EUDERINK P.J., H OOGSTRATEN H.W., S IPKEMA P. , H ILLEN B., W ESTERHOF N. (1989) Linear and nonlinear one-dimensional model of pulse wave transmission at high Womersley numbers, J. Biomech., 22 (8/9), 819-827. [16] ROCKWELL R.L., A NLIKER M., E LSNER J. (1974) Model studies of the pressure and flow pulses in a viscoelastic arterial conduit, J. Franklin Inst., 297, 405-427. [17] T IMOSHENKO S. (1940) Theory of plates and shells, McGraw-Hill.
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Captions
(1; ) for 4 values of K . Thin line corresponds to C = 2, thick line to C = 8. 2 - Dependence of St on pref in the case K = 1000 C = 2. Starred points are results from
Fig 1. - Evolution of R
Fig. simulations, continuous curve is obtained by a cubic spline interpolation.
(1 )
Fig. 3 - Dependence of St and R ; % on the parameter K . Both are independent of C . Starred points are results from simulations, continuous curve is obtained by a cubic spline interpolation.
(1 )
Fig. 4 - Persistence of oscillations of R ; at the center of the tube when a forcing oscillatory pressure ; St St in eqn (2.12)) (Resonance case). is assigned at the right end (
=1
=
Fig. 5 - Ratio of forcing and natural amplitudes versus St =St - (? upstream mean velocity, pressure at St (Resonance the center of tube, + deformation at the center of tube). The ratio is maximum when St case).
=
Fig. 6 - Space-time evolution of the membrane radial velocity in the case of resonance ( = 1; St = St). The thick line indicates the zero level, continuous line positive levels, dashed line negative levels. Level
step is 0.01 in both plots.
(1 )
=1
) (thick line) and of a inertia domiFig. 7 - Evolution of R ; in the cases of a viscous fluid (Re nated (Re ! 1) fluid (thin line). Steady state case (above) and oscillating forcing (below) are compared.
10
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