The Multilayer Impedance Pump Model

Transcription

1 12 Chapter 2 The Multilayer Impedance Pump Model 2.1 Phyical model The MIP wa a luid-illed elatic tube with an excitation zone located aymmetrically with repect to the length o the pump. The pump had an apect and a layered wall tructure imilar to the embryonic heart. The luid domain account or only 35% o the total volume occupied by the pump. The layered wall o the elatic tube were made o a thick gelatin layer or about 80% o the elatic tube volume, and a thin tier layer or the remaining 20%, ollowing the heart tube geometry (igure 5). Each layer contituting the tube wall wa made o an iotropic linear elatic material. The material propertie o each layer have been choen o that a large enough tine ratio between the elatic layer enable the combined eect o wave ampliication through the gelatin and the prevention o outward motion at the external layer. In addition, ollowing the embryonic heart tructure, the gelatin-like layer ha been given ome compreibility ( ν gel =0.3), while the tier layer wa relatively incompreible ( ν l =0.49) (table 1). The periodic excitation conited o impoed radial diplacement y ( t, z) on a ection o the outer urace o the tube (1). The pump wa actuated or 20% o the period time T. The tube external radiu wa maintained to original poition during the remaining 80% o the period time. During actuation, the elatic tube wa compreed ollowing a

2 13 inuoidal time unction g (t) that depended on the requency o excitation (2). The amplitude A o the compreion wa et to 10% o the pump external radiu, o that to model the diplacement reulting rom the myocite contraction. The patial repartition o the compreion zone ollowed a quadratic patial unction (z) to imulate a phyical pincher (3): y ( t, z) = g( t) * ( z), t, z) [ 0, T ]*[ a, a + a ], (1) T g( t) A*in(5 t) * Heaviide( t) 5 ( l l w = π, t ) [ 0,T ] 1 14 z ( ) 1 ( ) (, (2) z a a + a. (3) 4 = z, ( ) [ l, l w ] The impedance mimatch wa achieved by ixing the tube extremitie, enuring total relection o the elatic wave. The luid illing the tube wa water. Figure 5. (Top) 3D view and (Bottom) 2D view in longitudinal cro ection o the phyical model o the MIP.

3 14 Table 1. Phyical parameter o the MIP. Phyical parameter Symbol Value Length o the pump L 15.2 cm External radiu o the pump R ext 1.03 cm Fluid domain radiu R 0.55 cm Gelatin thickne h gel 0.405cm Stier layer thickne Actuator location with repect to the tube nearet extremity Actuator width h l a l a w cm 1.2 cm 1.8 cm Gelatin tine Stier layer tine Gelatin Poion ratio Stier layer Poion ratio Gelatin denity Stier layer denity E gel 5 e+ 4 dyn/cm 2 E l 1 e+ 7 dyn/cm 2 ν gel 0.3 ν l 0.49 gel 1 g/cm 3 l 1 g/cm 3 Fluid vicoity μ 0.01 g/cm Fluid denity 1 g/cm 3 Excitation amplitude A 0.1 cm Frequency 7 Hz to12.2 Hz

4 Mathematical model The luid motion wa derived by the conervative Navier-Stoke equation uing the Arbitrary Lagrange Eulerian ormulation: v = 0, (4) ( v v ) v + = 0 v + g τ, t (5) where τ i the tre tenor, v i the low velocity vector and velocity vector, i the denity o the luid and t i the time. v g i the local coordinate The luid i Newtonian, incompreible and vicou, and it tate o tre τ ollow: τ T = P I + μ ( v + v ), (6) where P i the tatic preure and μ i the dynamic vicoity. The dynamic o each layer o the lexible wall were calculated uing the balance o momentum equation in Lagrangian orm (7) and the contitutive relation or a linear iotropic elatic material (8): τ + b = u&&, (7) τ = λtr (ε )I + 2μ ε, (8) where τ i the Cauchy tre tenor, ε the train tenor, b the body orce vector per unit volume, u& & the acceleration vector the denity, and λ and μ the Lamé contant o the conidered tructural domain.

5 16 At the luid-tructure interace the luid i ully coupled to the gelatin. The undamental condition applied to the luid-tructure interace are diplacement compatibility and traction equilibrium between the two urace: d = u, (9) n τ = n, (10) τ where d and u are the luid and olid diplacement vector repectively, and n i the unit normal. To enure total wave relection, ixed end in both layer are modeled by impoing zero diplacement in all direction and at all time at the two tube extremitie: u =0 at z=0 and z=l. (11) The no-lip condition ( n u& = n v ) i applied at the luid-tructure interace, and the tube lie in a tre-ree and preure-ree environment (igure 3): n τ = 0 on the lateral urace o the tube, (12) P = 0 at z = 0 and z = L. (13) Initial condition are reting tate: zero preure and zero velocity in the luid, no tre or train in the tructure. Figure 6. 2D axiymmetric longitudinal outline o the MIP model with excitation and boundary condition (the haded region repreent the luid domain).

6 2.3 Numerical model 17 The inite element method wa ued to dicretize both the luid and tructure domain, and the ully coupled problem wa olved uing the commercial package ADINA (ADINA R&D, MA). The luid and the olid domain were mehed uing 4-noded axiymetric element. The olid meh wa reined at the pinching zone. A total o 10,500 element were ued, 6,000 or the luid and 4,500 or the olid (igure 7). An embedded actuation pincher wa modeled by impoing radial diplacement on a erie o node correponding to the pincher location, at the outer urace o the tube. The olid part i olved uing the mall train, mall deormation hypothei, and the low i aumed to be laminar. A contant number o 1,000 time tep per pinching cycle are ued to march throughout the tranient imulation. The time integration cheme i implicit Euler backward (α=1), which i irt-order accurate in time. The equation o motion are integrated by uing the implicit damped Newmark cheme (δ=0.5, α =0.25), and the ull Newton Method wa ued or the non linear equation. The luid and olid are 2-way direct ully coupled, and the luid meh i updated at each time tep uing Arbitrary Lagrange Eulerian ormulation. All computation are tarting rom reting tate and are carried on until periodicity in the luid motion i achieved (mean exit low i contant within 1% or at leat 5 period). Figure 7. 2D axiymmetric longitudinal view o the meh.