CC BY
32«ШУУШШШУМ-ЛШТМак» 2©21 / PHYSICS and mathematics
19
PHYSICS AND MATHEMATICS
УДК 532.5.031
Koroleva Yulia,
Department of Higher Mathematics, Russian Gubkin State University of Oil and Gas
Moscow, 119991, Russia https://orcid.org/0000-0003-0290-5236 Korolev Alexander,
Department of Higher Mathematics, Russian Gubkin State University of Oil and Gas
Moscow, 119991, Russia https://orcid.org/0000-0001-9339-4974 Amit Kumar Verma Department of Mathematics, Indian Institute of Technology
Patna, 801106, India https://orcid.org/0000-0001-8768-094x DOI: 10.24412/2520-2480-2021-188-19-26 ANALYSIS OF HERSCHEL-BULKLEY MODEL OF BLOOD FLOW THROUGH ROUGH VESSELS
Abstract.
The paper deals with Herschel-Bulkley system describing a two-phase flow of blood trough vessels with rough walls. We analyze a stationary case with a variable viscosity depending on the boundary roughness. We derive the asymptotic formula for the blood velocity behavior in the limit when both vessel's radius and the roughness period are small. Theoretical results are approved by numerical simulations. It turned out that the high boundary oscillations of vessel's wall cause rapid oscillations of velocity.
Keywords: hemodynamics, Herschel-Bulkley system, small parameter, flow of variable viscosity, stress tensor, non-newtonian fluid
1. INTRODUCTION
The object of the presented study is a problem on two-phase liquid flow through thin tube with roughness on the boundary. More precisely, we discuss the Her-schel-Bulkley model (see [1]) for viscous flows.
Our focus is done on the application of this theory to the blood flow in thin vessels. We use Herschel-Bulkley model to describe mathematically the behavior of blood passing through vessels with rough walls.
The research of hemodynamics [2] shows that blood is a Newtonian fluid when it passes through arteries of wide diameters at high shear rates. However, this flow becomes a non-Newtonian one in the case of a path through small diameter arteries at low shear rates. In addition, there is a region in the center of arteria where the concentration of erythrocytes is high. The flow in this zone is subject to another rheological properties characteristic for non-Newtonian fluid. The best representative of blood flow is a two-phase liquid model (see [3,4]) because blood vessels can be described by combined two-layers: the one where the flow consists of the plasma with erythrocytes, and the other one near-wall layer is the pure plasma.
The rough structure of vessels walls can be caused by a certain disease. For example, it is important for medicine to understand possible issue of the stenosed arteria and vessels. This topic of research was analyzed by mathematicians well enough. The case of simply stenosed arteria was studied in [5] under the assumption that flow is governed only by Newtonian model. Similar approach based on just Newtonian blood representation one can find in [6]. Here the author treated the case of blood flow through an axially asymmetric stenosed cylinder. More complicated geometry of stenosis
having a dilation was considered in [7]. A perturbation method was used to study the influence of varying ar-teria's geometry in [8,9]. Papers [10,11] deal with flows through tapered stenosed arteria where power-law blood model was assumed. This model is known as Herschel-Bulkley system and it has two parameters: flow index and yield stress. The effects of pulsality and stenosis were considered in [12] where the blood flow was represented through power-law Herschel-Bulkley model. A similar approach was considered in [13,14] for the case of catheterized arteria.
There exists another non-Newtonian fluid model named after Casson [15] which could be used to blood representative. In that approach blood is a non-Newtonian fluid which has the only yield stress as a parameter. This model was studied extensively in the literature, see e.g. [16-18].
There exist a number of works dealing with computational research in hemodynamics. In paper [19] the authors developed an anatomically detailed computational model of the arterial vasculature. They have used computational fluid dynamics to simulate blood flow and wave propagation phenomena in such arterial network. Some results of numerical study of two-phase blood flow are given in [20].
A new research around Herschel-Bulkley blood model is an analysis of a visosity depending on vessel radius. In paper [21] we have derived some apriori estimates for the solution to both Herschel-Bulkley and Casson's blood models. In particular, it was justified rigorously that the velocity of flow reduces when the viscosity increases. Authors have developed methods of estimations for parameters of variable viscosity flows in [21,23] and applied it to the present problem.
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Let us mention that we discuss here a two-phase model of a laminar blood flow which occurs in thin vessels. The paper [24] studies a question on the possibility blood flow to be turbulent. The experimental investigations in [24] show that this case may occur only for flow in aorta which is out of our study. In general the turbulence of flow does not occur in the human circulatory system except in the aorta. This effect might be caused by a reduce in the viscosity or by an increase in the velocity as was the case in anemia disease. There are two facts important for possibility of the turbulence: a Reynolds number of 2000 and a peak (not mean) velocity. Experimental measurments lead to the conclusion that the turbulence occurs when Reynolds number was below 2000. In the present paper we consider only the
case of laminar flow what excludes the case of turbulence.
The novelty of this manuscript is a theoretical analysis of the Herschel-Bulkley model for all possible types of rough vessel walls and the derivation of asymptotic formulas for the solution. These knowledge would help to understand the flow behavior. We have chosen the vessel's radius and the wavelength as small parameters. Our investigation considers different types of roughness effecting on the flow. We justified rigorously how asymptotics of both velocity and blood resistance depend on the ratio between vessel radius and the wavelength of vessel walls. It was discovered that high boundary oscillations lead to a significant velocity variation.
2. STATEMENT OF THE PROBLEM AND PRELIMINARIES
Let us consider blood flow as a viscous flow of variable viscosity p,(r) through the thin cylindrical vessel Qe of the length L and radius R (z), where Oe = {(z, r), 0 < z < L, 0 < r < R (z)}. Here (r, z) are the
cylindrical coordinates. Vessels' boundary r = R (z) is assumed to be an oscillating function depending on a small parameter £ which describes the wall's roughness.
Fig. 1: two-phase geometry of vessel
We shall analyze the stationary symmetric flow caused by constant pressure gradient. We assume that only one component of velocity u(r, z) = (ur, u , uz) is non-zero: ur = u = 0, uz = uz(r). Let us denote further
Uz (r) by u(r) for simplicity. Mathematical model describing this flow is the following Herschel-Bulkley system (see e.g. [12-14]):
t = T„ +
du
(Kr))" , if T > Ty O Rp (Ty) < r < R(z)
Equation Section (Next)(1.1)
du
— = 0, if t < ty o 0 < r < Rp (ty)
(1.2)
Here u is the unknown flow velocity, p is the pressure, t is the stress tensor, t is the yield stress, p,(r)
is the viscosity, n > 1 is a flow index. We assume that all functions and variables are dimensionless.
Herschel-Bulkley system characterizes flow as a two -phases motion governed by (1.1) and (1.2) correspondingly. Indeed, there is a domain Q,Newt = {(z,r), 0 < z < L, 0 < r < R (t )}, where the concentration of eretrocitus is comparatively high. The flow is Newtonian one in this area and velocity does not depend on vessel's radius (see (1.2)). The boundary r (t ) between Newtonian and non-Newtonian flow regimes depends on limit
stress t thorough the formula t(R^(Ty),z) = Ty.
In a common case the pressure of flow is related to velocity via momentum equation
du dp 1 d
- = dp +1 -( n ) dt dz r dr
(1.3)
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Moreover, we assume natural boundary conditions: a given pressure on the lateral surface of the vessel, no-slip conditions on the vessel's walls and a finite stress in the center of vessel:
t < œ as r = 0
u = 0 as r = R( z) P(°) = Po, P(L) = Pl ■
(1.4)
Apriori estimates
Due to zero boundary condition on vessel walls one can derive an a-priori estimate for velocity in the domain with non-Newtonian behavior of blood. Lemma (Friedrich's inequality)
The velocity satisfies to the estimate
R ( z )
R(z ^ du 2
J u 2dr < C(Rp, R(z)) {j^ I dr, C = max{2Rp (R(z) - Rp ),(R(z) - Rp )2}
(1.5)
Inequality (1.5) and formula (1.1) lead to estimation of velocity via stress and viscosity of blood: Corollary
" " < C
llL2( rp ,r ( z ))
( T - T» )"
(1.6)
lj( rp ,r( z))
The detailed derivation of Lemma and Corollary are given in [21].
Estimate (1.6) proves that velocity of flow reduces when the viscosity increases. In addition, the closer stress approaches to value x , the smaller values can take the velocity. One more important observation is that velocity
norm depends on size of the non-Newtonian layer since constant C in (1.6) is proportional to max(R(z) — R ), see (1.5). Hence, the thinner the layer nearby vessels wall the lower values velocity achieves.
3. STATIONARY FLOW
In stationary case we obtain analytical formulas for velocity function and some other flow characteristics. Indeed, the momentum equation (1.3) reduces to the following system:
dP = o,
dr
dP +1 rT ) = 0 .
dz r dr
Equation Section (Next)
Theorem
The stationary solution to problem (1.1)-(1.4) is represented as
R(z) i ir \ Y
r (z ) y u(r, z) = i "Tx
r
r ( z )
r (z ) I
u(z) = i -n
w ( f
-rp21-t >
dr as R < r < R(z)
y
y
-ps
-rK V t y
W 2 ) y y
dr as 0 < r < R .
The boundary R between flow phases satisfies
2T y
Rp = —L
p ps
(2.1)
(2.2)
(2.3)
(2.4)
Proof.
Using the boundary condition u(R(z), z) = 0 and the Newton-Leibnitz formula, we derive for
RP < r < R(z)
R( z )
-u(r, z) = u(R(z), z) - u(r, z) = [ —dr.
j pjr
du
—I
dr
(2.5)
r
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Hence,
R (z) du
u(r, z) = j--dr for R < r < R(z). (2.6)
r Or
Equation (1.1) implies that
Ou 1 / \n „ .
- — = — ) for R <r <R(z). (2.7)
Or ^(r y '
Combining (2.6) and (2.7), we derive the formula
R(z) i
u(r, z) = j —— (r-Ty)"dr for R < r < R(z). (2.8)
Multiplying equation (2.1) by r, integrating it over [0, r] and taking into the account the known pressure
Op
gradient — = ps, one obtains
Oz
r f d ^ ^2
0 = j^rpx + —(rr)Jdr = + rr(r, z) ^ r(r, z) = -pLr.
Thus,
p 2t Ty ="Rpp « RP =---. (2-9)
2 P P
Substituting (2.9) into(2.8), we obtain the velocity function in region R < r < R(z) :
R(,z) 1 in \
z ) 1 ( p Y u(r,z) = I---r dr for R < r < R(z).
r Mr) V 2 y p
(2.10)
V 2 J
The condition (1.2) implies that u does not depend on r as 0 < r < R . Therefore,
u(r, z) = u(z) as 0 < r < Rp. (2.11)
Since function u is smooth, formulas (2.10) and (2.11) must coincide at r = R . Therefore,
R (z) i f
u =
r) 1 ( p Y
[-rfJ dr a'12)
in the layer {0 < r < R }. The proof is complete.
(2.13)
Other important flow characteristics are the volumetric flow rate
R(z) (Rp R(z) 1 ( p Yn R(r) R(? 1 ( p Yn ^
Q = 2n rudr = 2n r -1 -r^ - t I drdr + r -1 -r^- - t I drdr
Q 0 tJ0 Rp ^(r)V 2 y) J J ^(r)I 2 yJ j
which shows the volume of blood passing through the vessel's segment, and blood resistance:
x = p - Pl (2.14)
Q
4. ROUGHNESS EFFECTS
Consider a small parameter 8, 0 < £ <Si 1. Assume that the vessel radius as small as the order of 8 : R (z) = O(s). Introduce another parameter v(s) which is the length of the wall's wavelength (see Fig. 2). We suppose that v(s) ^ 0 as s ^ 0. The goal now is to analyze the blood flow behavior for all possible relations between vessel radius and its wavelength. For this reason we define the parameter k = lim-. Different
v(s)
types of roughness are possible depending on this value:
• the case 0 < k < œ corresponds to «middle oscillations» which means that the roughness period as small as vessel radius;
• the value k = 0 corresponds to «small oscillations» which means that the vessel radius is much smaller than the wavelength;
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• if k = x> then one deals with the «high frequency regime» which describes the case when the roughness period is much smaller than the vessel radius.
Assume that rough vessels wall is modeled as an oscillating function
Re (Z) = 8h)(z)+8he
f z ^
V v(8) J
where h describes the global geometry of wall while hs notes periodically oscillations of wavelength v(8)
Ps
. Denote further p = —— . Taking into the account the derived analytical formula for blood velocity, one can observe the dependence of limits of the integration on roughness:
sho( z )+8he
u(r, z) = [ i rp — t ) dr as R < r < R(z),
i (rp - ty )"dr as Rp
s\ (z )—8h£
Equation Section (Next)(3.1)
u(z) = i (rp - t ) dr as 0 < r < R .
i ^(r r p
Below we derive the asymptotics with respect to £ —> 0 for the velocity and the blood resistance.
Fig. 2: geometry of vessel
Asymptotic formula
Lemma. The velocity and resistance of blood flow satisfy to the following asymptotics as 8 ^ 0 :
u - k max
(rp - t y )"
x - k3
re[ rp ,r( z)] ^(r )
r (rp - t y)"V
max -
rs[Rp ,R(z)] jx(r )
(3.2)
(3.3)
J
Proof.
Applying the Lagrange formula to integrals in (3.1), we obtain an estimate
u < max
(rp - t y r d
r ^(r) dz
8h0 (z) + 8h8
f X\
z
vv(8)jj
< max
(rp - t v )H
Kr)
8 max h ——— max h
( 8 )
(3.4)
Due to regularity of functions h, h and compactness of the domain there exist constants K, K2 such that
(3.5)
8max h0 - 8Kl, 8 max he - 8 K2.
z V(8) z V(8)
Passing to the limit in (3.4) and having in mind (3.5), we derive asymptotics (3.2) for velocity as 8 ^ 0. In a similar way, applying Lagrange theorem and (3.2) to (2.13) and (2.14), one justifies (3.3):
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Q « k3 max
(rp -
t )n
Mr )
X ^ F
f
max
(rp-
T y )
> Y
^(r )
v r~v 7 J
5. NUMERICAL VALIDATION OF THE RESULTS AND CONCLUSIONS
In this section we prove our theoretical results on asymptotics by numerical computations for a particular
i
choice ofthe viscosity function. We model our problem with rough boundary Re (z) = 3s + ssin
A
v(e).
where
v(s) = s, v(s) = -\[s, v(s) = s2 which could correspond to one of the situation: 0 < k < ro, k = 0, k = ro. The viscosity is assumed to have the polynomial growth: p,(r) = ra.
Figures 3 and 4 demonstrate plots for velocity: the solution u(z) is plotted on Fig. 3 and u(r, z = 0.5) is shown on Fig. 4. Analyzing the roughness effect, one can observe that the velocity behavior is similar to the behavior of oscillations: small boundary oscillations (v(e) = 4e) give the slow growth of velocity. The variance of velocity is more rapid in case of middle oscillations (v(s) = s) and one can observe the change in monotonicity regime. Finally, the high oscillations (v(e) = s2) effect more significantly: the graph for velocity is a rapidly oscillating function as well. Numerical validations show that the bigger the value of yield stress x the smaller the
blood velocity what is agreed with the estimate (1.6). Moreover, one can see from Fig. 3 that bigger value of flow index n give greater velocity values (compare n = 3 and n = 5). Figure 4 demonstrates the velocity behavior depending on radius in the layer near the walls (in non-Newtonian layer) for different roughness types. The bigger the radius (i.e. the wider the non-Newtonian layer) the smaller the velocity. One can observe also the roughness effects on how fast the velocity decreases.
Analogous conclusions are valid for dependence of the volumetric flow rate (Fig. 6) on roughness, since Q is strictly proportional to the velocity. However, the behavior of blood resistance (Fig.5) is the opposite one. The more intensively roughness oscillates the greater variations of the resistance. Small and middle roughness regime give a significant reduce of the resistance. The viscosity effect is also important: high viscosity leads to small resistance while more viscous flow lias greater resistance.
Fig. 3: Blood velocity regimes for different oscillation regimes for n = 3 and 5, e = 0.1, a= 4.1, r = 0.1, p=10
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Fig. 4: Blood velocity depending on radius in the non-Newtonian layer: a = 2.1, a = 4.1, e = 0.5, n=3, ^ = 0.1, p=10
Fig. 5: Blood resistance for a = —1.1, a = 4.1, > e = 0.1, n=5, x = 0.1, p=10
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PHYSICS AND MATHEMATICS / «©OLlOMUM-JOUrMaL» #1(88), 2©21
a=4.1,n=5
flow rate Q
—1-1-1-■-1-1-■-■-1-1-'-1-1 ■-1-1-1— variable z
0.2 0.4 0 6 0 8 1.0
Fig. 6: Volumetric flow rate
Conflict of Interest
The authors declared no potential conflicts of interest with respect to the research, authorship and publication of this article.
Funding
This work was supported by RFBR (project 18-31-00311).
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