Journal of Fluid Science and Technology

Transcription

1 Bulletin of the JSME Vol.12, No.2, 2017 Journal of Fluid Science and Technology Two-dimensional numerical simulation of the behavior of a circular capsule subject to an inclined centrifugal force near a plate in a fluid Suguru MIYAUCHI*, Toshiyuki HAYASE*, Arash Alizad BANAEI**, Jean-Christophe LOISEAU**, Luca BRANDT** and Fredrik LUNDELL** * Institute of Fluid Science, Tohoku University Katahira, Aoba-ku, Sendai , Japan miyauchi@reynolds.ifs.tohoku.ac.jp ** Linne Flow Centre, KTH Mechanics S Stockholm, Sweden Received: 26 March 2017; Revised: 5 July 2017; Accepted: 10 August 2017 Abstract In order to examine mechanical interactions between erythrocytes and a blood vessel surface, the frictional characteristics between erythrocytes and plates in plasma have been measured by an inclined centrifuge microscope. The frictional characteristics have been properly reproduced by a numerical simulation of a rigid erythrocyte model assuming a flat bottom surface. However, validity of the assumption has not been confirmed. The purpose of this fundamental study, therefore, was to clarify the behavior of a two-dimensional circular capsule subjected to inclined centrifugal force near a plate in a fluid. An unsteady simulation was performed for various values of the angles of the inclined centrifugal force and membrane elasticity. In equilibrium states, a lubrication domain with high pressure and a large shear stress is formed between the capsule and the base plate, and the bottom surface of the capsule becomes flat with a positive attack angle. The gap distance and translational and rotational velocities increase with decreasing membrane elasticity or increasing centrifugal force angle. The attack angle increases with increasing membrane elasticity or centrifugal force angle. The results in this study qualitatively justified the assumption of the former numerical study that erythrocytes in an inclined centrifuge microscope have a flat bottom surface and its result that they have a positive attack angle in equilibrium state. Key words : Inclined centrifuge microscope, Frictional characteristics, Erythrocyte, Elastic capsule, Numerical simulation, Fluid-membrane interaction 1. Introduction Blood flow in microcirculation plays an important role in the supply of nutrients and collection of waste from cells. Especially in blood capillaries, strong interaction between erythrocytes and endothelial cells occurs since the diameter of the blood vessel may be as large as or smaller than the size of the erythrocyte. Fiber-like structures called glycocalyx exist on the endothelial cell surface (Weinbaum, et al., 2011), and it is considered that the interaction between erythrocytes and endothelial cells becomes complex. Most cardiovascular diseases are related to the blood flow condition, and elucidation of the mechanical interaction between the erythrocytes and endothelial cells is an important issue that may lead to clarification of the mechanisms of cardiovascular diseases and the development of new treatments (Weinbaum, et al., 2011). In order to elucidate the interaction between erythrocytes and endothelial cells, an inclined centrifuge microscope was developed (Fig. 1). In Fig. 1(a), a medium including erythrocytes is set in an inclined sample container, and centrifugal force is applied by rotating the sample container. Erythrocytes are pushed onto a base plate by the normal component F N of the centrifugal force F and move with a constant velocity U on the plate by tangential component F T. Paper No

2 (a) Schematic of erythrocytes in a sample container. (b) Erythrocytes moving on a glass plate. Fig. 1 An inclined centrifuge microscope. It is possible to give arbitrary values of F N and F T by controlling the setting angle and rotational angular velocity of the sample container. The friction force acting on the erythrocytes moving with a constant velocity U is evaluated based on F T. Figure 1(b) shows an experimental result in the case of erythrocytes on a glass plate in plasma (Kandori, et al., 2008). Figure 2 shows experimental and analysis results for the frictional characteristics of erythrocytes moving on a plain glass plate (Hayase, et al., 2005), an endothelia-cultured plate (Hayase, et al., 2013), and diamond-like carbon (DLC) and 2-methacryloyloxyethyl phosphorylcholine (MPC) coated plates (Kandori, et al., 2008) in plasma. In the experimental results, the frictional characteristics of the endothelia-cultured plate are larger than those of the plain and material-coated plates. In order to understand the mechanism of the frictional characteristics, numerical simulations were performed. Oshibe et al. (2014) performed three-dimensional numerical simulation of a flow around a rigid erythrocyte model with an attack angle and flat bottom surface, and reproduced the frictional characteristics in cases of plain and material-coated plates as the equilibrium state where flow and centrifugal forces and their moments are balanced ( 3-D simulation ). However, the validity of the assumption that the erythrocyte model has a flat bottom surface has not been confirmed. The behavior of erythrocytes in the inclined centrifuge microscope should be clarified in consideration of their deformability. Fig. 2 Nondimensional frictional characteristics of erythrocytes. 2

3 Fig. 3 Computational setup for capsule behavior in an inclined centrifuge microscope. The purpose of this fundamental study, therefore, was to clarify the behavior of a two-dimensional circular capsule subject to inclined centrifugal force near a plate in a fluid. A 2-D unsteady simulation was performed to obtain a transient behavior and a quasi-steady equilibrium state of a capsule subjected to inclined centrifugal force near a plate in a fluid. The existence of the equilibrium state and parameters in the equilibrium state, such as capsule shape, attack angle, gap distance, moving velocity, and tank-treading motion were investigated for several values of the membrane elasticity and the angle of the inclined centrifugal force. The results were compared with those of the former study. 2. Methods In this study, a 2-D numerical simulation was performed to clarify the behavior of a single circular capsule in an inclined centrifuge microscope. The computational domain for the fluid is a square domain in the vicinity of the capsule, bottom surface of which corresponds to the plate (solid boundary) (Fig. 3). The x- and y-axes are, respectively, set in the direction along the plate and its vertical direction, and the origin is located at the bottom-left corner of the computational domain. A capsule with circular shape is set in the domain. The inclined centrifugal force with the angle against the vertical direction of the plate is applied to the fluid inside the membrane due to the density difference of the inner and outer fluids of the membrane. In the rotating coordinate system with a constant rotational speed, the Coriolis force also acts to the capsule. However, the Coriolis force is sufficiently smaller than the inclined centrifugal force because of the small translational velocity of the capsule, and the Coriolis force is ignored in this study. The computation was unsteady analysis and continued until the motion of the capsule became quasi-steady state. The fluid was assumed to be an incompressible Newtonian fluid, and the fluid motion is governed by the following continuity equation and Navier-Stokes equation: u 0, (1) u p u u p 2 u f c f e, (2) t where u is fluid velocity, p is pressure, p is density of the outer fluid of the membrane, is viscosity, f c is centrifugal force applied to the inner fluid, and f e is interaction force from the membrane applied to the fluid near the membrane. It is noted that the centrifugal force acting on the whole domain and the static pressure generated by the centrifugal force 23

4 Density of inner fluid r 1087 [kg/m 3 ] Density of outer fluid p 1025 [kg/m 3 ] Viscosity [Pa s] Diameter of a capsule D 8.0 [m] Rotational speed 423 [rad/s] Radius of a rotor r [m] Number of grid points N x N y for fluid Number of grid points N s 800 for membrane Time increment -6 [s] t [s](for T 0 = [N/m]) are neglected in the formulation while the force caused by the difference in the centrifugal forces between the inner and outer fluids, which mainly drives the capsule motion, is considered. The static pressure due to the centrifugal force does not affect the velocity field as well as the capsule behavior because the fluid force acting on the capsule is related to the stress difference on both sides of the membrane (Le, et al., 2006) and the distribution of the static pressure is continuous in the whole domain. Although the density and viscosity are different between the fluids inside and outside the membrane in reality, the same density and viscosity of the outer fluid were used for the inner fluid when Eq. (2) was solved except for the evaluation of the centrifugal force f c applied inside the membrane for the sake of simplicity. Centrifugal force f c was calculated by the following equation: 2 r p 0 c f c r e, (3) where r is the density of the inner fluid of the membrane, r 0 is the centrifugal radius, is the rotational angular velocity of the rotor of the inclined centrifugal microscope, and e c is the unit vector pointing in the direction of the inclined centrifugal force. The centrifugal radius is treated as a fixed value because the migration distance of the capsule (see Fig.4) is relatively small compared to the radius of a rotor. The force acting on the membrane F e was given as follows: X F e T0 1τ, (4) s s where s is arc length, T 0 is tension coefficient to represent membrane elasticity, X is position vector of the membrane, and is the unit tangential vector against the membrane surface. In this study, the membrane has zero-thickness as often assumed in fluid-membrane interaction problems because the membrane thickness of the real erythrocyte is very small compared to the size of the capsule. The immersed boundary method (IBM) (Peskin, 1977, Peskin, 2002) was used for the model of interaction between the fluid and the membrane. The interpolation of the fluid velocity onto the material points of the membrane and the distribution of the elastic force of the membrane to flow field were performed by the following equations: DX Dt u x X dx, (5) Table 1 Physical and computational parameters. f e F e x X ds, (6) where x is the position vector of the fluid, and is the approximate delta function. In this study, the following equation 24

5 was used to approximate the delta function (Peskin, 1977,Peskin, 2002) : where 1 x y ( x, y), (7) h h h 2 1 r 1 cos r 2 r 4 2 (8) 0 r 2, and h is mesh width. For the boundary condition, a no-slip condition was imposed on the upper and lower boundaries and a periodic boundary condition was imposed on the left and right boundaries. The no-slip condition was imposed on the interface between the fluid and membrane according to Eq. (5). As the initial condition, the fluid velocity was set to 0. The SMAC method (Amsden and Harlow, 1970) was used for the governing equation of fluid (1), (2). For the discretization in the time direction, the Adams-Bashforth method for the convective term and the Crank-Nicholson method for the viscous term were applied. For the discretization in the space direction, the central difference method with 2nd order accuracy was applied to the governing equations for both the fluid and membrane. The computational parameters are summarized in Table 1. The density of inner fluid r corresponds to that of an erythrocyte, and p and to those of plasma. The diameter of capsule D was set based on the major axis diameter of the erythrocyte. and r 0 were referenced from the literature (Hayase, et al., 2013). It was confirmed that the number of grid points and time increment used in this study were sufficient by performing preliminary computations. Numerical simulation was performed for five tension coefficients, T 0 = , , , , [N/m], with the angle of the inclined centrifugal force = 30 for the effect of the membrane elasticity, and for three angles of the inclined centrifugal force, = 20, 30, 40, with the tension coefficient T 0 = [N/m] for the effect of the angle of the inclined centrifugal force. 3. Result Figure 4 shows the trajectory of the capsule in the case of 0 = 30 and T 0 = [N/m]. In Fig. 4(a), the capsule moves toward the lower wall and then along the wall with deformation due to the inclined centrifugal force. In the figure, the red circles show specific points on the membranes. The broken lines are the neighbor capsules in the periodic boundary condition. In Fig. 4(b) for the result after the time elapsed sufficiently, the membrane shape reaches a quasisteady state with almost the same capsule shape. In the figure, the specific point moves in a clockwise direction showing a tank-treading motion. Figure 5 shows the velocity and pressure fields in the quasi-steady state at t = 0.25[s] in the same condition as described above. These results are viewed in the moving coordinate with the translational velocity of the capsule. In the fluid velocity vector field in Fig. 5(a) and streamlines in Fig. 5(b), two vortices appear inside and above the membrane due to the rotational motion of the membrane. In the pressure distribution in Fig. 5(c), high pressure appears above the lower front surface of the capsule normal to centrifugal force and in the fluid domain in the clearance between the membrane and the wall. The results for the effect of the membrane elasticity are shown below. Figure 6 shows the capsule shapes in quasisteady state for different tension coefficients. In all cases, the capsules achieve the equilibrium state where their bottom surfaces become flat with positive attack angles against the moving direction. The attack angle was defined as the angle consisting of the line tangential to the capsule s concave bottom surface and the plate. The front and back sides of the capsule, respectively, become more rounded and sharper as the tension coefficient decreases. Figure 7 shows the gap distance h, attack angle, and translational and rotational velocities U t, U r with the tension coefficient. The translational and rotational velocities are defined as the moving velocity of the capsule in x-direction and the velocity on the membrane in the moving coordinate system with the translational velocity of the capsule, respectively. 25

6 (a) t [s] (b) [m] [m] t 0.005[s] t [s] [m] t [s] t 0.200[s] t [s] t [s] [m] Fig. 4 Time variation of the capsule ( T 0 = [N/m], = 30 ). The gap distance decreases, the attack angle increases, and the translational and rotational velocities decrease with increasing tension coefficient. The rotational velocity is much smaller than the translational velocity. Next, the results for the effect of the angle of the inclined centrifugal force 0 are shown. Figure 8 shows the capsule shapes in the quasi-steady state for different angles of the inclined centrifugal force. The capsule shape, gap distance, and attack angle differ depending on the angle of the inclined centrifugal force. Figure 9 shows the gap distance, attack angle, and translational and rotational velocities with the angle of the inclined centrifugal force. The result of the study performed by Oshibe et al. using the rigid erythrocyte model is also shown in the figure for comparison. The gap distance, attack angle, and translational and rotational velocities increase with increasing angle of the inclined centrifugal force. Although the present results in Fig. 9(a) are larger than the results of Oshibe et al., the tendency is consistent with their results. In Fig. 9(b), both the computational results are in agreement. In Fig. 9(c), the translational velocities of this study are larger than those of Oshibe et al., but the tendency is consistent. The nondimensional frictional characteristics obtained from the present results are plotted in Fig. 2. Although the nondimensional friction force F T/F N obtained in the present 2-D study and those of the 2-D lubrication analysis (Yatsuyanagi, et al., 2016) are much smaller than those of the experiment for plain, DLC-coated plates and 3-D simulation, the inclinations of the curves are almost the same as those of the experiment and 3-D simulation. 4. Discussion As a fundamental study to elucidate the behavior of the erythrocytes in the inclined centrifuge microscope, unsteady numerical analysis was performed to clarify the behavior of a 2-D circular capsule subject to inclined centrifugal force near a plate in a fluid. The transient behavior of the capsule toward the quasi-steady state was clarified (Fig. 4). In a quasi-steady state, a lubrication domain is formed between the capsule and the plate, and high pressure occurs in that domain (Fig. 5(c)). Besides, the tension of the capsule increases due to a large shear stress from the flow in the lubrication domain and the large pressure inside the capsule, and, therefore, the bottom surface of the capsule becomes flat. A quasisteady state of the capsule was obtained for various values for the tension coefficient and the angle of inclined centrifugal force. In the all quasi-steady states, the capsule has a flat bottom surface and a positive attack angle (Figs. 6 and 8). This result supports the assumption of the former study by Oshibe et al. (2014) that erythrocytes in an inclined centrifuge microscope have flat bottom surface and its result that they have a positive attack angle in equilibrium state. The pressure distribution between the capsule and the plate is similar to that of Oshibe et al., i.e. the pressure distribution where the pressure becomes large in the lubrication domain and small in the rear side of the capsule. The effects of the tension 26

7 (a) velocity vectors. (b) streamlines. (c) pressure. Fig. 5 Velocity and pressure fields in the vicinity of the capsule in the quasi-steady state on a moving coordinate ( T 0 = [N/m], 0= 30 ). 27

8 Fig. 6 Capsule shapes in the quasi-steady state for various membrane elasticities (t = 0.25[s], 0 = 30 ). (a) gap distance. (b) attack angle. (c) translational and rotational velocities. Fig. 7 Parameters for the capsule in the quasi-steady state with the membrane elasticity. 28

9 Fig. 8 Capsule shapes in the quasi-steady state for various angles of inclined centrifugal force ( t = 0.25 [s], T 0 = [N/m]). (a) gap distance. (b) attack angle. (c) translational and rotational velocities. Fig. 9 Parameters for the capsule in the quasi-steady state with the angle of inclined centrifugal force. coefficient, or membrane elasticity, on gap distance, attack angle, and translational and rotational velocities in the quasisteady state were clarified (Fig. 7). The variation of parameters for the tension coefficient is explained as follows. To be an equilibrium state where the capsule moves above the plate with a constant distance, the inclined centrifugal force and 29

10 flow force need to be in balance. The capsules with different tension coefficient receive the same value of the inclined centrifugal force because the volume inside the capsule is constant. The capsule shape is stretched in x-direction by the shear stress for small tension coefficient T 0. The stretch becomes small as the tension coefficient increases, and the area of the flat bottom surface becomes small. Therefore, the gap distance h decreases with increasing the tension coefficient to make a balance between the drag force and tangential force F T. On the other hand, the attack angle 0 increases with increasing the tension coefficient to make a balance between the lift force and normal force F N. The translational and rotational velocities decrease with increasing the tension coefficient probably due to increase of the projected area of the capsule. The effect of the angle of the inclined centrifugal force on the above-mensioned parameters in the quasi-steady state were also clarified (Fig. 9). The gap distance, attack angle and translational velocity increase with increasing angle since the normal force F N increases and tangential force F T decreases with increasing angle. Regarding the gap distance, the present result is much larger than that of Oshibe et al. This is caused by overestimation of the lift force due to the lack of span-wise flow in present 2-D analysis and the difference of the shape between the 3-D erythrocyte model and circular capsule. Regarding the attack angle, the present result is in good agreement with that of Oshibe et al., but this agreement is not significant since the attack angle varied according to the membrane elasticity. It is considered that the larger translational velocity of this study compared to that of Oshibe et al. is caused by overestimation of the lift force by the 2-D analysis because the shear stress acting on the bottom side of the capsule becomes small in the case of the large gap distance. The present capsule in quasi-steady state has positive rotational velocity, implying occurrence of the tanktreading motion. Although the rotational velocity of the tank-treading motion is much smaller than that of the translational velocity, the effective tank-treading motion U r/u t increases with decreasing angle. Regarding the nondimensional frictional characteristics (Fig. 2), result of the present study and that of the lubrication analysis (Yatsuyanagi, et al., 2016) are in good agreement although the range of nondimensional cell velocity DU/F N are different. Although the nondimensional friction forces F T/F N of these 2-D analysis results are much smaller than those of the experiment for plain and DLC-coated plates and 3-D simulation probably because of above-mentioned over estimation of the lift force in 2-D analysis, inclinations of the curves in these 2-D analyses are similar to those of the experiment and 3-D simulation. It is, therefore, considered that the present result qualitatively reproduces the phenomenon. The limitations of this study are its 2-D analysis, capsule shape, and membrane properties. The 2-D circular shape is far different from 3-D biconcave shape of a real erythrocyte. The present membrane model considering only tension is too simple and it does not express the resistance of the area dilation. More appropriate membrane model such as Skalak model (Skalak, et al., 1973) should be considered in the next step. 3-D analysis of the erythrocyte with a proper shape and mechanical properties should be necessary for quantitative evaluation of the behavior of the erythrocytes under the inclined centrifugal force field in future work. 5. Conclusion As a fundamental study for the elucidation of the behavior of erythrocytes in an inclined centrifuge microscope, this study performed 2-D unsteady simulation for the behavior of a circular capsule subject to inclined centrifugal force near the plate in a fluid. In quasi-steady state, the capsule shows a flat bottom surface and a positive attack angle. The effects of membrane elasticity and the angle of the inclined centrifugal force on the capsule in quasi-steady state were clarified for the capsule shape, gap distance, attack angle, and translational and rotational velocities. The present results qualitatively justifies the assumption of the former study that erythrocytes in an inclined centrifugal microscope have a flat bottom surface and its result that they have a positive attack angle in equilibrium state. Acknowledgment This study was partly supported by the JSPS Core-to-Core Program, A. Advanced Research Network, International research core on smart layered materials and structures for energy saving,jsps KAKENHI Grant Number 15H06026, 15H03914 and a collaborative research project of the Institute of Fluid Science, Tohoku University. 10 2

11 References Amsden, A. A. and Harlow, F. H., A simplified MAC technique for incompressible fluid flow calculations, Journal of Computational Physics Vol.6, No.2, (1970), pp Hayase, T., Sugiyama, H., Yamagata, T., Inoue, K., Shirai, A. and Takeda, M. Inclined centrifuge microscope for measuring frictional characteristics of red blood cells moving on glass plate in plasma, The 2005 Summer Bioengineering Conference, (2005), pp.1-2. Hayase, T., Inoue, K., Funamoto, T. and Shirai, A. Frictional Charcteristics of Erythrocytes on Endothelia-Cultured or Material -Coated Glass Plates Subject to Inclined Cnetrifugal Forces, 8th International Conference on Multiphase Flow ICMF, (2013), pp Kandori, T., Hayase, T., Inoue, K., Funamoto, K., Takeno, T., Ohta, M., Takeda, M. and Shirai, A., Frictional characteristics of erythrocytes on coated glass plates subject to inclined centrifugal forces, Journal of biomechanical engineering Vol.130, No.5, (2008), pp Le, D. V., Khoo, B. C. and Peraire, J., An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries, Journal of Computational Physics Vol.220, No.1, (2006), pp Oshibe, T., Hayase, T., Funamoto, K. and Shirai, A., Numerical Analysis for Elucidation of Nonlinear Frictional Characteristics of a Deformed Erythrocyte Moving on a Plate in Medium Subject to Inclined Centrifugal Force, Journal of biomechanical engineering Vol.136, No.12, (2014), pp Peskin, C. S., Numerical analysis of blood flow in the heart, Journal of computational physics Vol.25, No.3, (1977), pp Peskin, C. S., The immersed boundary method, Acta numerica Vol.11, (2002), pp Skalak, R., Tozeren, A., Zarda, R. and Chien, S., Strain energy function of red blood cell membranes, Biophysical Journal Vol.13, No.3, (1973), pp Weinbaum, S., Duan, Y., Thi, M. M. and You, L., An integrative review of mechanotransduction in endothelial, epithelial (renal) and dendritic cells (osteocytes), Cellular and molecular bioengineering Vol.4, No.4, (2011), pp Yatsuyanagi, A., Hayase, T., Miyauchi, S., Funamoto, K., Inoue, K., Shirai, A. and Brandt, L., Numerical analysis for elucidation of mechanical interaction between an erythrocyte moving in medium subject to inclined centrifugal force and endothelial cells on a plate, Journal of Fluid Science and Technology Vol.11, No.4, (2016), pp. JFST