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07 O 01 Hydrodynamic torques and forces acting on nonspherical particles in a shear flow near a wall
d A e t o w d S c i . Vol. 24~ Suppl. I, pp. $ 4 5 - $ 4 6 , 1993 Printed in Great Britain.
0021-8502/93 $6.00 + 0.00 Pergamon Press Ltd
07 O 01 H Y D R O D Y N A M I C TORQUES AND FORCES ACTING ON N O N S P H E R I C A L PARTICLES IN A SHEAR FLOW NEAR A WALL
E. Gavze 1 and M. Shapiro 2 lIsrael Institute for Biological Research, P.O. Box 19, Ness Ziona 70450, Israel 2Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa 32000, Israel
KEYWORDS spheroidal particles, hydrodynamic resistance tensor, orientation, wall effect OBJECTIVES Dynamics and transport of aerosol particles are governed by its shape. The latter factor affects various processes, including aerosol deposition (Shapiro and Goldenberg, 1993), hydrodynamic dispersion, sampling, etc. Accounting for particles' geometric fo~xn when calculating the trajectory of its motion may be done on the basis of several simple models, according to which the particle's interactions are calculated as in the unbounded flow domain. Presence of solid walls may significantly influence the flow field around the particle, as well as its viscous interactions. The goal of this study is to calculate the hydrodynamic forces and torques acting on a nonspherical particle moving in a shear flow near a solid wall. METHOD Motion of a nonspherical particle in a viscous fluid is governed by the following system of equations m
du =F-K.u-C.o~ dt
d --(I-m) = L-Ct.udt
f~.ol
where u, {1}are the particle's respective translational and rotational velocities, m and I are its mass and inertia tensor, respectively, F and L are the force and the torques exerted on the particle by the fluid (Fig. 1), K, ~ and C are the particle's translational, rotational and coupling resistance tensors. This study deals with the case of negligible particle inertia, in which case the terms appearing in the left-hand side of the above equations are zeros. The forces and torques were calculated numerically by the boundary integral method (Gavze, 1990) for various particle orientations 0 and distances z from the wall for the case of a simple shear flow v(z)=voz/c, where c is the longer axis of the spheroid. These interactions were compared with their limiting values F~,, Loo prevailing far from the wall.
~
a
/
Uz/Vo
-O.OI~A~\
./~
-oo2L \ "F -0.031 0
/ I
I
0.1
0.2
/
11 <-z'°-"'l I
z/°-l41
L "--zic 6J I
I
0.3
0.4
I 0.5 -
0/rt Fig. 1 Spheroidal particle moving in a she,uflow near a solid wall
Fig. 2 Dependence of the normal velocity component of the spheroidal particle with a/c=0.2 upon its orientation $45
RESULTS The calculated results revealed the following effects: (i) the difference Fx-Fxoo of the horizontal force (parallel to the wall) is almost independent of the distance z, which trend prevails for all particle orientations. (ii) the character of the z-dependence of the torque component Ly was 12)und to be strongly affected by the particle's orientation. For 0=re/2 Ly exhibits a maximum in the wall's vicinity, however, for/9=0, rJ4 Ly monotonically increases with z. (iii) for 0 different from 0, td2 there exists a nonzero z-component of the viscous force, which monotonically decreases with z. Using the symmetry of the problem, the velocity components were calculated from the relationship
K Ko cq,q= q'x Cyz
*yJLcoJLLIJ
The angular velocity, co was found to be insignificantly affected by the presence of the wall. The effect of the wall was found to create a nonzero z-component of the velocity, Uz,, Which vanishes in the absence of the wall (Brenner, 1973). A typical O-distribution of Uz is shown in Fig. 2. One can see that the particle moves towards the wall (Uz <0) with the velocity which reaches a maximum at about 0.157r. The maximum of the absolute value of uz depends upon the particle's shape, with the maximum-maximorum lying between the values of the aspect ratio 0.35
o)P(z,
O)dO.
Due to the above symmetry properties of Uz and P(z, 0), one easily obtains that tT~= 0. This is, however no longer true in cases, where P(z, O) is not symmetric. In spite of the above conclusion, the data shown in Fig. 2 indicates that at any surface z=const there exist a positive total (orientation-averaged) particle's fluxjz, i.e., directed upward from the wall, even in case where spatial particles' distribution is uniform and their angular distribution is symmetric. This flux stems from the observation that the absolute value of Uz decays with z, and, hence, on average the amount of particles crossing a surface z=const upward from below is larger than the comparable amount of particles, crossing this surface downward, from above. This effect is further quantified by a physical model, which is being developed by us now. The results obtained by the boundary element method serve as a necessary input data for calculating the trajectories of nonspherical particles in viscous shear flows. In particular, such calculations are important in verification of applicability of simplified models developed for comparable trajectories' calculations (Gallily and Cohen, 1979). ACKNOWLEDGEMENT This work was partially supported by the Ministry of Science and Technology, State of Israel. REFERENCES Gallily, I. and Cohen, A-H. J. Colloidlnterface Sci. 68, (2) 338 (1979). Gavze, E. Int. J. Multiphase Flows 16 (3) 529-543 (1990) Happel, H. and Brenner, H. Low Reynolds Number Hydrodynamics with Special Application to Particulate Media, Noordhoff, Leyden (1973) Shapiro, M. and Goldenberg, M. Deposition of glass fiber particles from turbulent air flow in a pipe. J. Aerosol Science, in press (1993).
0021-8502/93 $6.00 + 0.00 Pergamon Press Ltd
07 O 01 H Y D R O D Y N A M I C TORQUES AND FORCES ACTING ON N O N S P H E R I C A L PARTICLES IN A SHEAR FLOW NEAR A WALL
E. Gavze 1 and M. Shapiro 2 lIsrael Institute for Biological Research, P.O. Box 19, Ness Ziona 70450, Israel 2Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa 32000, Israel
KEYWORDS spheroidal particles, hydrodynamic resistance tensor, orientation, wall effect OBJECTIVES Dynamics and transport of aerosol particles are governed by its shape. The latter factor affects various processes, including aerosol deposition (Shapiro and Goldenberg, 1993), hydrodynamic dispersion, sampling, etc. Accounting for particles' geometric fo~xn when calculating the trajectory of its motion may be done on the basis of several simple models, according to which the particle's interactions are calculated as in the unbounded flow domain. Presence of solid walls may significantly influence the flow field around the particle, as well as its viscous interactions. The goal of this study is to calculate the hydrodynamic forces and torques acting on a nonspherical particle moving in a shear flow near a solid wall. METHOD Motion of a nonspherical particle in a viscous fluid is governed by the following system of equations m
du =F-K.u-C.o~ dt
d --(I-m) = L-Ct.udt
f~.ol
where u, {1}are the particle's respective translational and rotational velocities, m and I are its mass and inertia tensor, respectively, F and L are the force and the torques exerted on the particle by the fluid (Fig. 1), K, ~ and C are the particle's translational, rotational and coupling resistance tensors. This study deals with the case of negligible particle inertia, in which case the terms appearing in the left-hand side of the above equations are zeros. The forces and torques were calculated numerically by the boundary integral method (Gavze, 1990) for various particle orientations 0 and distances z from the wall for the case of a simple shear flow v(z)=voz/c, where c is the longer axis of the spheroid. These interactions were compared with their limiting values F~,, Loo prevailing far from the wall.
~
a
/
Uz/Vo
-O.OI~A~\
./~
-oo2L \ "F -0.031 0
/ I
I
0.1
0.2
/
11 <-z'°-"'l I
z/°-l41
L "--zic 6J I
I
0.3
0.4
I 0.5 -
0/rt Fig. 1 Spheroidal particle moving in a she,uflow near a solid wall
Fig. 2 Dependence of the normal velocity component of the spheroidal particle with a/c=0.2 upon its orientation $45
RESULTS The calculated results revealed the following effects: (i) the difference Fx-Fxoo of the horizontal force (parallel to the wall) is almost independent of the distance z, which trend prevails for all particle orientations. (ii) the character of the z-dependence of the torque component Ly was 12)und to be strongly affected by the particle's orientation. For 0=re/2 Ly exhibits a maximum in the wall's vicinity, however, for/9=0, rJ4 Ly monotonically increases with z. (iii) for 0 different from 0, td2 there exists a nonzero z-component of the viscous force, which monotonically decreases with z. Using the symmetry of the problem, the velocity components were calculated from the relationship
K Ko cq,q= q'x Cyz
*yJLcoJLLIJ
The angular velocity, co was found to be insignificantly affected by the presence of the wall. The effect of the wall was found to create a nonzero z-component of the velocity, Uz,, Which vanishes in the absence of the wall (Brenner, 1973). A typical O-distribution of Uz is shown in Fig. 2. One can see that the particle moves towards the wall (Uz <0) with the velocity which reaches a maximum at about 0.157r. The maximum of the absolute value of uz depends upon the particle's shape, with the maximum-maximorum lying between the values of the aspect ratio 0.35
o)P(z,
O)dO.
Due to the above symmetry properties of Uz and P(z, 0), one easily obtains that tT~= 0. This is, however no longer true in cases, where P(z, O) is not symmetric. In spite of the above conclusion, the data shown in Fig. 2 indicates that at any surface z=const there exist a positive total (orientation-averaged) particle's fluxjz, i.e., directed upward from the wall, even in case where spatial particles' distribution is uniform and their angular distribution is symmetric. This flux stems from the observation that the absolute value of Uz decays with z, and, hence, on average the amount of particles crossing a surface z=const upward from below is larger than the comparable amount of particles, crossing this surface downward, from above. This effect is further quantified by a physical model, which is being developed by us now. The results obtained by the boundary element method serve as a necessary input data for calculating the trajectories of nonspherical particles in viscous shear flows. In particular, such calculations are important in verification of applicability of simplified models developed for comparable trajectories' calculations (Gallily and Cohen, 1979). ACKNOWLEDGEMENT This work was partially supported by the Ministry of Science and Technology, State of Israel. REFERENCES Gallily, I. and Cohen, A-H. J. Colloidlnterface Sci. 68, (2) 338 (1979). Gavze, E. Int. J. Multiphase Flows 16 (3) 529-543 (1990) Happel, H. and Brenner, H. Low Reynolds Number Hydrodynamics with Special Application to Particulate Media, Noordhoff, Leyden (1973) Shapiro, M. and Goldenberg, M. Deposition of glass fiber particles from turbulent air flow in a pipe. J. Aerosol Science, in press (1993).