J
Bmnechnnirs.
1977. Vol.
IO. pp
607409
Pergamon
Pras.
Prmted
in Great Braam
TECHNICAL THE EFFECT
NOTE
OF OSCILLATORY WAVY CONDUIT
FREQUENCY FLOWS*
FLOW
INTRODUIXION
Information relating blood flow characteristics to the development and progression of arterial disease has been sought diligently during the past decade. Theories on the initiation of atheroma have attributed it to both regions of high and low shear stress. Mechanical damage (that is, tearing of the endothelium by high shear stress) causes preferential sites for lipid deposits. In low shear regions. deposition of particles detrimental to the endothelium can occur when there is not sufficient transport of particles away from the wall. Precise flow patterns need to be calculated for unsteady, non-uniform flow situations to help find ways of relating flow characteristics to concomitant physiological events. This present study was initiated to help achieve a better understanding of these types of flows and to help find ways to solve the pressing problem of atherosclerosis. Most previous work in non-uniform hemodynamics has been done either by eliminating from the analysis the unsteadiness of the flow (Chow et al., 1971; Forrester and Young, 1970; Manton, 1971) or by adopting numerical techniques to handle the governing Navier-Stokes equations (Cheng, Clark and Robertson, 1972,1974,1976; Daly, 1976; Hung, 1970). The steady flow assumption represents a serious simplification to the actual arterial flow situation and numerical methods require excessive amounts of computer time to faithfully describe the timewise development and steady-state response to the imposed forcing functions. Analytical solutions, where available, are of more general applicability and many parametric studies can be made without a great amount of additional work. A recent perturbation analysis for describing plane oscillatory flow in wavy channels (Cheng et al., 1974) is used in this paper to delineate the manner in which flow characteristics are affected by variations in the frequency parameter of the oscillatory flows.
* Received 11 January 1977. t Department of Mechanical Engineering. WSU, Wichita, KS 67208. $ Combustion Engineering Company, Windsor. CT 06095.
IN PLANE
MODEL
To simplify the analysis, the assumptions imposed in this study are: (a) blood is an incompressible, homogeneous, Newtonian fluid, (b) the boundary of the rigid. stationary vessel is plane and sinusoidal, (c) the oscillating flow is laminar and fully-developed. At any conduit cross section, the volume flow rate is given by q = q. cos nt or in dimensionless form by Q = Q,, cos /IT (where II is angular velocity, t is the real time, Q. = qdv, /? = D%/a, T= tv/D2, D is half the mean separation distance of the channel, and v is kinematic viscosity). With the above assumptions, the system under study can be described by the following dimensionless fourth order differential equation in terms of the stream function $
where
and ? V4=64dX4+262m+a7
a4
d4
and the boundary conditions are $X=Jl,.=O
and
J,=F,cosfiT
* = lclx = Jlrr =0
at
at
Y=q
Y= 0
The variables appearing above are defined as (see Fig. 1)
(Wall profile)
Four dimensionless parameters are required to characterize the dynamics of the flow field: S and E are geometric parameters; F, and /? are flow parameters.
Fig. 1. The effect of oscillation frequency on the Row field at maximum flow rate. 607
608
Technical Note RESULTS AND
DISCUSSION
The manner in which the flow characteristics are affected by variations in the frequency parameter (/I) will be delineated in this note. The other governing parameters are held constant (F, = IO, 6 = 0.1, E = 0.5) as B is given values of 5n, 10x, and 1%. In Fig. 1, velocity profiles at different locations along the wavy conduit are compared for flows with different values of p. For the time in the oscillatory cycle shown (T= 0). flow rate is maximum. The velocity fields are seen to be affected by the flow frequency: the lower the frequency, the faster near the center region and the slower near the boundaries will be the flow. How’ ever, the total effect on the velocity field decreases with an increase in frequency. In Fig. 2, the centerline pressure drop per wavelength (Ap) is plotted against time for flows with the three different oscillation frequencies. The pressure drop per wavelength increases with the decrease in /I. The phase lag (Ae) between the pressure drop (Ap) and the flow rate (Q) increases with an increase of /I. Since the energy loss per wavelength of the flow field is proportional to the area enclosed by the pressure drop curves, it follows that the mechanical energy dissipated per cycle by the viscous effects is greater for flows with lower frequencies. It should be remembered, however, since the period of flow is inversely proportional to its frequency, Fig. 2. Temporal variation in centerline pressure drop per that while the flow with /I = SR completes one cycle, the wall wavelength at different oscillation frequencies. flows with /I = 1On and 15~ will complete two and three cycles, respectively. Therefore the total energy dissipated will still be greater for the higher frequency flows during an equal length of time. The solution of the basic equation throughout the region In Fig. 3, the temporal and spatial variation in shear of concern is obtained by expanding I+G in series in terms stress (T) at the throat station are shown for the various of the geometric parameter S; the asymptotic solution of frequencies. The definition of the dimensionless shear stress the stream function is then sought in the limit as d is given as approaches zero.
r=aV+,aV
ax'
ay
where Substituting the above expression into equation (1) and then collecting the terms with equal powers of 6 yields a set of perturbed equations. The solution of these equations has only been obtained up to and including the second order (i.e. JI = $e + a$, + S’$,). Further details of the method are given in an earlier paper (Cheng er al., 1974).
and
au av
Lr =It
( >
-.----
@r=o er
0.125
0.25
&fax.
/3=5r IOr 15t
0.375
Fig. 3. Temporal variation in shear stress at the throat station at different oscillation frequencies.
Technical Note The maximum shear stress in the flow field occurs near the wall and its value is proportional to the instantaneous flow rate. During most of the flow period, the higher frequency flows will exhibit higher instantaneous maximum shear stresses. Department Q Theoretical and Applied Mechanics, UIUC, Urbana IL 61801, U.S.A.
L. C.
CHENGt
M. E. W. C.
CLARK F%NG$
REFERENCES
Clark, M. E., Robertson, J. M. and Cheng, L. C. (1976) Numerical analysis of unsteady viscous flow in non-uniform channels Proc. 29th ACEMB. 18,334. Cheng, L. C., Clark, M. E. and Robertson, J. M. (1972) Numerical calculations of oscillating flow in the vicinity of square wall obstacles in plane conduits. J. Biomechanics 5, 467-484.
609
Cheng, L. C., Robertson, J. M. and Clark, M. E. (1974) Calculation of plane pulsatile flow past wall obstacles. Int. J. Comput. Fluids 2, 363-380. Cheng, L. C., Peng, W. C. and Clark, M. E. (1974) Analysis of oscillating tlbw in plane wavy conduits. Proc. 27th ACEMB. 16. 202. Chow, J. C. F., Soda, K. and Dean, C. (1971) On laminar fldw in wavy channels. Proc. 12th Midwest Mech. Conf 6, 247-253. Daly, B. J. (1976) A numerical study of pulsatile flow through stenosed canine femoral arteries. J. Biomechanics 9, 465-475. Forrester, J. H. and Young, D. F. (1970) Flow through a converging-diverging tube and its implications in occlusive vascular disease. J. Biomechanics 3, 297-305. Hung, T. K. (1970) Vortices in pulsatile flow. Proc. 1st Int. Co@ Rheology 2, 115-127. Manton, M. J. (1971) Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech. 49, 451-459.
Bmnechnnirs.
1977. Vol.
IO. pp
607409
Pergamon
Pras.
Prmted
in Great Braam
TECHNICAL THE EFFECT
NOTE
OF OSCILLATORY WAVY CONDUIT
FREQUENCY FLOWS*
FLOW
INTRODUIXION
Information relating blood flow characteristics to the development and progression of arterial disease has been sought diligently during the past decade. Theories on the initiation of atheroma have attributed it to both regions of high and low shear stress. Mechanical damage (that is, tearing of the endothelium by high shear stress) causes preferential sites for lipid deposits. In low shear regions. deposition of particles detrimental to the endothelium can occur when there is not sufficient transport of particles away from the wall. Precise flow patterns need to be calculated for unsteady, non-uniform flow situations to help find ways of relating flow characteristics to concomitant physiological events. This present study was initiated to help achieve a better understanding of these types of flows and to help find ways to solve the pressing problem of atherosclerosis. Most previous work in non-uniform hemodynamics has been done either by eliminating from the analysis the unsteadiness of the flow (Chow et al., 1971; Forrester and Young, 1970; Manton, 1971) or by adopting numerical techniques to handle the governing Navier-Stokes equations (Cheng, Clark and Robertson, 1972,1974,1976; Daly, 1976; Hung, 1970). The steady flow assumption represents a serious simplification to the actual arterial flow situation and numerical methods require excessive amounts of computer time to faithfully describe the timewise development and steady-state response to the imposed forcing functions. Analytical solutions, where available, are of more general applicability and many parametric studies can be made without a great amount of additional work. A recent perturbation analysis for describing plane oscillatory flow in wavy channels (Cheng et al., 1974) is used in this paper to delineate the manner in which flow characteristics are affected by variations in the frequency parameter of the oscillatory flows.
* Received 11 January 1977. t Department of Mechanical Engineering. WSU, Wichita, KS 67208. $ Combustion Engineering Company, Windsor. CT 06095.
IN PLANE
MODEL
To simplify the analysis, the assumptions imposed in this study are: (a) blood is an incompressible, homogeneous, Newtonian fluid, (b) the boundary of the rigid. stationary vessel is plane and sinusoidal, (c) the oscillating flow is laminar and fully-developed. At any conduit cross section, the volume flow rate is given by q = q. cos nt or in dimensionless form by Q = Q,, cos /IT (where II is angular velocity, t is the real time, Q. = qdv, /? = D%/a, T= tv/D2, D is half the mean separation distance of the channel, and v is kinematic viscosity). With the above assumptions, the system under study can be described by the following dimensionless fourth order differential equation in terms of the stream function $
where
and ? V4=64dX4+262m+a7
a4
d4
and the boundary conditions are $X=Jl,.=O
and
J,=F,cosfiT
* = lclx = Jlrr =0
at
at
Y=q
Y= 0
The variables appearing above are defined as (see Fig. 1)
(Wall profile)
Four dimensionless parameters are required to characterize the dynamics of the flow field: S and E are geometric parameters; F, and /? are flow parameters.
Fig. 1. The effect of oscillation frequency on the Row field at maximum flow rate. 607
608
Technical Note RESULTS AND
DISCUSSION
The manner in which the flow characteristics are affected by variations in the frequency parameter (/I) will be delineated in this note. The other governing parameters are held constant (F, = IO, 6 = 0.1, E = 0.5) as B is given values of 5n, 10x, and 1%. In Fig. 1, velocity profiles at different locations along the wavy conduit are compared for flows with different values of p. For the time in the oscillatory cycle shown (T= 0). flow rate is maximum. The velocity fields are seen to be affected by the flow frequency: the lower the frequency, the faster near the center region and the slower near the boundaries will be the flow. How’ ever, the total effect on the velocity field decreases with an increase in frequency. In Fig. 2, the centerline pressure drop per wavelength (Ap) is plotted against time for flows with the three different oscillation frequencies. The pressure drop per wavelength increases with the decrease in /I. The phase lag (Ae) between the pressure drop (Ap) and the flow rate (Q) increases with an increase of /I. Since the energy loss per wavelength of the flow field is proportional to the area enclosed by the pressure drop curves, it follows that the mechanical energy dissipated per cycle by the viscous effects is greater for flows with lower frequencies. It should be remembered, however, since the period of flow is inversely proportional to its frequency, Fig. 2. Temporal variation in centerline pressure drop per that while the flow with /I = SR completes one cycle, the wall wavelength at different oscillation frequencies. flows with /I = 1On and 15~ will complete two and three cycles, respectively. Therefore the total energy dissipated will still be greater for the higher frequency flows during an equal length of time. The solution of the basic equation throughout the region In Fig. 3, the temporal and spatial variation in shear of concern is obtained by expanding I+G in series in terms stress (T) at the throat station are shown for the various of the geometric parameter S; the asymptotic solution of frequencies. The definition of the dimensionless shear stress the stream function is then sought in the limit as d is given as approaches zero.
r=aV+,aV
ax'
ay
where Substituting the above expression into equation (1) and then collecting the terms with equal powers of 6 yields a set of perturbed equations. The solution of these equations has only been obtained up to and including the second order (i.e. JI = $e + a$, + S’$,). Further details of the method are given in an earlier paper (Cheng er al., 1974).
and
au av
Lr =It
( >
-.----
@r=o er
0.125
0.25
&fax.
/3=5r IOr 15t
0.375
Fig. 3. Temporal variation in shear stress at the throat station at different oscillation frequencies.
Technical Note The maximum shear stress in the flow field occurs near the wall and its value is proportional to the instantaneous flow rate. During most of the flow period, the higher frequency flows will exhibit higher instantaneous maximum shear stresses. Department Q Theoretical and Applied Mechanics, UIUC, Urbana IL 61801, U.S.A.
L. C.
CHENGt
M. E. W. C.
CLARK F%NG$
REFERENCES
Clark, M. E., Robertson, J. M. and Cheng, L. C. (1976) Numerical analysis of unsteady viscous flow in non-uniform channels Proc. 29th ACEMB. 18,334. Cheng, L. C., Clark, M. E. and Robertson, J. M. (1972) Numerical calculations of oscillating flow in the vicinity of square wall obstacles in plane conduits. J. Biomechanics 5, 467-484.
609
Cheng, L. C., Robertson, J. M. and Clark, M. E. (1974) Calculation of plane pulsatile flow past wall obstacles. Int. J. Comput. Fluids 2, 363-380. Cheng, L. C., Peng, W. C. and Clark, M. E. (1974) Analysis of oscillating tlbw in plane wavy conduits. Proc. 27th ACEMB. 16. 202. Chow, J. C. F., Soda, K. and Dean, C. (1971) On laminar fldw in wavy channels. Proc. 12th Midwest Mech. Conf 6, 247-253. Daly, B. J. (1976) A numerical study of pulsatile flow through stenosed canine femoral arteries. J. Biomechanics 9, 465-475. Forrester, J. H. and Young, D. F. (1970) Flow through a converging-diverging tube and its implications in occlusive vascular disease. J. Biomechanics 3, 297-305. Hung, T. K. (1970) Vortices in pulsatile flow. Proc. 1st Int. Co@ Rheology 2, 115-127. Manton, M. J. (1971) Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech. 49, 451-459.
