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A mathematical model of pulsatile flows of microstretch fluids in circular tubes
International Journal of Engineering Science 41 (2003) 231–247 www.elsevier.com/locate/ijengsci
A mathematical model of pulsatile flows of microstretch fluids in circular tubes M.N.L. Narasimhan
*
10455 S.W. 152nd Avenue, Beaverton, OR 97007, USA Received 24 April 2002; accepted 22 July 2002
Abstract Pulsatile flows of micropolar fluids with stretch whose microelements can undergo expansions and contractions besides translations and rotations in straight circular tubes are considered. The governing field equations for such flows of linear microstretch fluids turn out to be a nonlinear coupled partial differential system. Solutions are sought for this system starting with a reasonable initial approximation for microinertia and the consequent linearization of the field equations. One of the coupled equations governing the microstretch and microinertia is solved approximately by the method of Laplace transforms taken with respect to the time variable. Making use of this approximate solution, the other coupled equation is solved leading to explicit higher order approximation solutions for microinertia, microstretch and micropressure. Next, the coupled equations governing the velocity and the microrotation fields are solved by employing the finite Hankel transform operators on a space variable and their inversions, and higher order approximation solutions are determined. All the above-mentioned explicit solutions are obtained in computationally suitable forms. These solutions have the promise of application to many practically important physical situations such as flows of polymeric fluids with deformable springy suspensions and flows of biological fluids including blood with deformable cell suspensions in small arteries. Ó 2002 Published by Elsevier Science Ltd. Keywords: Microstretch continuum theory; Microstretch fluids; Suspension rheology; Arterial blood flow
1. Introduction The theory of micropolar fluids with stretch (or microstretch fluids) was introduced by Eringen [1] in 1969 as a subclass of the more general type of fluids known as simple microfluids, also known as Eringen fluids (Eringen [2]). The microstretch fluids possess a local substructure whose *
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0020-7225/03/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 0 2 0 - 7 2 2 5 ( 0 2 ) 0 0 2 0 4 - 5
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material particles or microelements are endowed with seven degrees of freedom, three of translation, three of rotation and one of stretch or axial motion, consisting of expansions and contractions (breathing motion), as opposed to twelve degrees of freedom possessed by their parent class of simple microfluids, namely, three of translation, three of rotation, and six of stretch and shears. These fluids differ from the micropolar fluids, another subclass of the simple microfluids, in having one extra degree of freedom, namely, that of stretch, while the micropolar fluids whose particles are rigid, possess only six degrees of freedom, three of translation and three of rotation. Also, the microstretch fluids like micropolar fluids can support couple stresses as well as body couples and possess microinertia. The stress tensor of the microstretch fluid theory is asymmetric as in the case of micropolar fluids. It is important, however, to note that the special theory mentioned above can also be applied, in the limiting case, to cover the classical Navier–Stokes fluids as well. Eringen [3] extended his microstretch fluid theory to that of thermomicrostretch fluids incorporating thermal effects and applied the theory to the problem of propagation of acoustic waves in bubbly fluids obtaining agreement with available experimental results. Eringen also developed the theory of microstretch liquid crystals [4] and the electrodynamics of microstretch liquid crystal polymers [5]. Ariman [6] applied the microstretch fluid theory to the problem of steady-state blood flow in small arteries. Noting that micropressure is missing from ArimanÕs analysis, Eringen ([7], pp. 250–252) introduced a new investigation of the same problem but with the incorporation of micropressure obtaining a gratifying agreement with the experimental results of Goldsmith and Marlow [8]. Questions of existence, uniqueness and stability of solutions for microstretch fluid flows were examined by Rao and Raju [9,10], and by Iesßan [11]. Aydemir and Venart [12] performed numerical computations to obtain the solutions for the problem of nonsteady flow of a thermomicrostretch fluid between two parallel and heated plates, stationary with respect to the flow. While there exists a vast literature on the theory and applications of micropolar fluids, a relatively scant amount of work has been done on the dynamics of microstretch fluids. To the authorÕs knowledge, no work, analytical or numerical, has been done yet either on nonsteady flows or on pulsatile flows of microstretch fluids in tubes with circular cylindrical geometry. In the present investigation, the problem of pulsatile flows of microstretch fluids due to a sinusoidally varying pressure gradient in circular tubes is analyzed. Starting with a reasonable initial approximation for the microinertia and the consequent linearization of the field equations, the coupled partial differential system incorporating micropressure is solved. The mathematical analysis basically consists of two parts, one involving the determination of higher order approximation solutions for microstretch, microinertia, and the micropressure fields, and the other involving the determination of solutions for the velocity and the microrotation fields. Explicit solutions of the former set of equations are obtained by employing the Laplace transform with respect to time of the relevant equations, solving them and then taking the inverse transform. The second set is solved by employing the finite Hankel transform of the appropriate equations with respect to a space variable, solving them and then taking the inverse Hankel transforms. In both cases, explicit solutions are obtained in computationally suitable forms. This work may find application to many biophysical and technological fields. To mention a few, pulsatile blood flow in small arteries under the pumping action of the heart can be modeled after this work since blood consists of cell suspensions which can undergo expansions and contractions in addition to translations and rotations. Similarly, some of the industrial applications of
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this work include pulsatile flows of polymeric fluids with deformable springy suspensions in circular tubes. 2. Formulation of the problem The following set of pertinent field equations governing the flow of isothermal microstretch fluids in vectorial form are given by (cf. Eringen [7]) q_ þ q$ v ¼ 0;
ð2:1Þ
Dj 2jm ¼ 0; Dt
ð2:2Þ
$p þ k0 $m þ ðk þ 2l þ jÞ$$ v ðl þ jÞ$ $ v þ j$ m þ qðf v_ Þ ¼ 0;
ð2:3Þ
ða þ b þ cÞ$$ m c$ $ m þ j$ v 2jm þ qðl rÞ ¼ 0;
ð2:4Þ
a0 r2 m þ p0 k1 m k0 $ v þ qðl rÞ ¼ 0;
ð2:5Þ
where q ¼ qðx; tÞ, v ¼ vðx; tÞ, m ¼ mðx; tÞ, j ¼ jðx; tÞ, and m ¼ mðx; tÞ are, respectively, the mass density, velocity, microrotation, microinertia density, and the microstretch fields that are unknowns to be determined under appropriate boundary and initial conditions. The microstretch rotatory inertia r and the microstretch scalar inertia r are given by r¼
D ðjmÞ; Dt
1 r ¼ jðm_ þ m2 Þ jm m; 2
ð2:6Þ
where j ¼ jkk is the scalar microinertia density and m is the microrotation vector and the microstretch m is defined by mkl ¼ mdkl klr mr ;
ð2:7Þ
mkl , mr , and klr being, respectively, the microgyration tensor, the microgyration vector (microrotation vector), and the permutation tensor, and here and in the sequel, summation being understood over repeated indices. The quantities f, l and l, respectively, denote the body force density, the body couple density and the trace of the microstretch body moment density tensor. The quantities p and p0 , respectively, denote the thermodynamic pressure and the micropressure. A superposed dot on a quantity denotes the material time derivative of that quantity. The quantities k0 , k1 , a0 , k, l, and j are the viscosity coefficients for the translatory motions, while a, b, and c are those for the rotatory motions. In this paper, we consider the linear microstretch theory in which the fields jvj, jmj and jmj are small in comparison with other terms in (2.6). As a consequence, (2.6) reduces to r ¼ j_m;
1 r ¼ jm_ : 2
ð2:8Þ
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In general, all the viscosity coefficients depend on j, q1 , and the temperture. But, in the linear approximation, they are considered as constants. The other basic assumptions made in the present study are (1) the fluid is incompressible and p is replaced by p, the fluid pressure, (2) the flow is isothermal, (3) external body force, body couple and microstretch body moment are absent, (4) the flow is microisotropic in the sense that the microinertia tensor jkl ¼ ð1=3Þjdkl , and (5) the flow in the circular tube is axisymmetric. We now introduce a cylindrical coordinate system ðr; h; zÞ with z-axis taken along the axis of the tube and r and h are, respectively, the radial and azimuthal coordinates taken in cross-sectional planes perpendicular to the tube axis. On account of axial symmetry and for a long tube all field quantities are independent of both h and z-coordinates with the exception of the macropressure p ¼ pðr; z; tÞ. The unknown fields are now represented by v ¼ ½0; 0; v ;
m ¼ ½0; mh ; 0 ;
m ¼ mðr; tÞ;
j ¼ jðr; tÞ;
ð2:9Þ
where v ¼ vðr; tÞ and mh ¼ mh ðr; tÞ. The field equations (2.2)–(2.5) now reduce to oj ¼ 2jm; ot k0
ð2:10Þ
om op ¼ ; or or
ð2:11Þ
1 o ov op ov r þ ju ¼ q ; ðl þ jÞ r or or oz ot 1 ou ov omh c j qj mh ¼ ; 2j or or ot 1 o om 1 om r k1 m þ p0 ¼ qj ; a0 r or or 2 ot
ð2:12Þ ð2:13Þ ð2:14Þ
where u ¼ uðr; tÞ ¼
1 o ðrmh Þ: r or
ð2:15Þ
These equations are to be solved under appropriate boundary and initial conditions for the unknowns v, mh , m; j, and p. We assume in the present study that the nonsteady axial flow is of a suddenly generated and accelerated type produced by an impressed sinusoidal axial pressure gradient of the form op ¼ oz
0 for t < 0 A0 þ A1 sin xt ¼ A0 ð1 þ sin xtÞ for t P 0;
ð2:16Þ
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where ¼ A1 =A0 , A0 is the mean axial pressure gradient, A1 is the amplitude of the axial pressure gradient and x is the circular frequency. The five equations (2.10)–(2.14) for the five unknowns are nonlinear and coupled partial differential equations. They are difficult to solve analytically, if not intractable. Hence, no progress is possible unless we adopt some reasonable approximation procedure. To this end, it is necessary to generate a suitable initial approximation for j in order to perform linearization of the equations. Bearing this in mind, we proceed as follows. To begin with, we note that m is uncoupled from v and mh in equations (2.10) and (2.14). Noting that at the tube wall r ¼ a, a being the radius of the tube, mh ¼ nv;r ;
m ¼ n$ v;
ð2:17Þ
where n is a constant, and a comma followed by an index indicates the partial derivative with respect to that index, is suggested as a possible general boundary condition by Kirwan [13]. We exploit this situation as follows. Substituting (2.17) into (2.12) and (2.13), one obtains after simplification, 1 ov ; j ¼ K c þ ð1 2nÞj ng or
ð2:18Þ
where 1
K ¼ fl þ jð1 nÞg ;
o g¼ or
1 o ov r : r or or
ð2:19Þ
This expression for j is nonlinear and is unsuitable for purposes of performing the desired linearization of the equations. It is clear that by setting n ¼ 1=2 in (2.18) and (2.19), we obtain a suitable initial or zeroth order approximate value for j denoted by jð0Þ where j¼j
ð0Þ
1 1 ¼c lþ j : 2
ð2:20Þ
We remark here that when n ¼ 1=2 in (2.17), it amounts to equating the microrotation velocity m to the local vorticity, and the stress tensor of the micropolar stretch theory becomes symmetric and reduces to that of the Navier–Stokes fluid. This condition is rather a severe restriction to impose on the microstretch fluid flow. As pointed out by Turk et al. [14], in some applied problems, however, such as those involving blood flow in small arteries, the experimental blood flow data of Bugliarello and Sevilla [15] show that the relation (2.17) with n ¼ 1=2 is a reasonably good approximation for the actual situation involving flow of blood with any hematocrit value (volumetric percentage of red cells in blood) in small arteries. But, in our analysis here, we employ j ¼ jð0Þ as an initial (or zeroth order) approximation for purposes of linearizing the field equations. Later, we obtain higher order approximation for jðr; tÞ by a successive iteration procedure.
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3. Method of solution Microstretch, microinertia and micropressure: Before we begin the solution process, we note that by direct integration of (2.10) between t ¼ 0 and t, we can express the formal solution of Eq. (2.10) as Z t jðr; tÞ ¼ jðr; 0Þ exp 2 mðr; sÞ ds ; 0
where jðr; 0Þ denotes the initial value of j at t ¼ 0. It is clear from the equation above that jðr; 0Þ 6¼ 0 since jðr; tÞ does not vanish. To begin with the solution process, we first treat equations (2.10) and (2.14). As the first step, we linearize (2.10) and (2.14) by substituting jð0Þ for j on the right-hand sides of these equations obtaining oj ¼ 2jð0Þ m; ot 1 o om 1 om r k1 m þ p0 ¼ qjð0Þ : a0 r or or 2 ot
ð3:1Þ ð3:2Þ
Initial condition: For the initial condition on the microstretch field, we have mðr; 0Þ ¼ m0 ðrÞ;
0 6 r < a:
ð3:3Þ
Boundary condition: The boundary condition on the microstretch field is mða; tÞ ¼ m1 ðtÞ;
t > 0:
ð3:4Þ
We will discuss the nature and the physical implications of these initial and boundary conditions later. We note that the linearization of the field equations (2.10) and (2.14) has been achieved by replacing j in the product terms on their right-hand sides with the constant quantity jð0Þ given by (2.20). As is clear from the work of Turk et al. [14], the observation that this constant value jð0Þ is a close approximation for j is well supported by the experimental results of [15] on blood flow. Hence, it is reasonable to assume that at t ¼ 0; jðr; tÞ possesses this constant value jð0Þ . Thus, jðr; 0Þ ¼ jð0Þ
ð3:5Þ
can serve as the initial condition for jðr; tÞ. Now, integration of (3.1) with respect to t between t ¼ 0 and t, gives Z t ð0Þ mðr; sÞ ds ; jðr; tÞ ¼ j 1 þ 2
ð3:6Þ
0
which is clearly consistent with the above-obtained formal solution for jðr; tÞ, applied to the linear approximation case with (3.5) taken into account.
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We now turn our attention to the nature of the micropressure field. The concept of micropressure was introduced by Eringen (cf. [7]) through a constitutive law derived from the free energy expression. That is, p0 ¼ 2q
ow j; oj
w ¼ Wðq1 ; h; jÞ:
ð3:7Þ
For the case of small microinertia j, the free energy w can be expanded into a power series in j, retaining terms up to the first degree in j. Then, the first of the equations in (3.7) reduces, in the case of incompressible, isotropic, and isothermal fluids to p0 ¼ d0 j;
ð3:8Þ
where d0 is a constant. Furthermore, when j is small, it is physically reasonable to assume, in the zeroth order approximation, that the influence of microinertia on the free energy is negligibly small. Hence, we may neglect the micropressure p0 in our zeroth order approximation, but retain it in the first-order approximation. As a consequence, (3.2) takes the form 1 o om om r ð3:9Þ m2 m ¼ k 2 ; r or or ot where m2 ¼
k1 ; a0
k2 ¼
qjð0Þ ; 2a0
ð3:10Þ
and jð0Þ is given by (2.20). We now take the Laplace transform of (3.9) with respect to time t where an overbar on a quantity denotes its Laplace transform, s being the transform parameter. We obtain after employing (3.3), o2 m 1 o m þ kH2 m ¼ k 2 m0 ðrÞ; 2 or r or
ð3:11Þ
where kH ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ k 2 s :
ð3:12Þ
The general solution of (3.11) is mðr; sÞ ¼ A0 ðsÞI0 ðrkH Þ þ A1 ðsÞK0 ðrkH Þ þ mp ðr; sÞ; where mp ðr; sÞ is a particular solution of (3.11) which can be obtained by the method of variation of parameters. The functions y1 ðr; sÞ ¼ I0 ðrkH Þ and y2 ðr; sÞ ¼ K0 ðrkH Þ are two linearly independent solutions of the corresponding homogeneous equation. Since the integrals involved in the
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particular solutions contain W 1 ; where W is the Wronskian of the solutions y1 and y2 , all we need here is W 1 for which we can employ the result in [18, p. 375] namely, 1
W 1 ¼ fW ½I0 ðrkH Þ; K0 ðrkH Þ g
¼ rkH :
Thus, the general solution of (3.11) is mð0Þ ðr; sÞ ¼ A0 ðsÞI0 ðrkH Þ þ A1 ðsÞK0 ðrkH Þ k 2 I0 ðrkH Þ Z r H 2 kH ym0 ðyÞI0 ðykH Þ dy; þ k K0 ðrk Þ
Z
r
kH ym0 ðyÞK0 ðykH Þ dy ð3:13Þ
where mð0Þ ðr; sÞ denotes the zeroth order solution in the transform space. The regularity condition at r ¼ 0 leads to A1 ðsÞ ¼ 0. Furthermore, mð0Þ ðr; sÞ becomes unbounded at r ¼ 0, because of the presence of K0 ðrkH Þ in the last term on the right-hand side of (3.13). This means that the only solution possible occurs when m0 ðrÞ is zero. Moreover, it is physically reasonable to assume that initially at t ¼ 0, microstretch is zero for all r 2 ½0; aÞ. In this case, we obtain mð0Þ ðr; sÞ ¼ A0 ðsÞI0 ðrkH Þ:
ð3:14Þ
Applying the boundary condition (3.4) and denoting mð0Þ ða; sÞ by m1 ðsÞ; we obtain the solution (3.14) in the form mð0Þ ðr; sÞ ¼
m1 ðsÞI0 ðrkH Þ : I0 ðakH Þ
It is possible in the case of some fluids such as, for instance, certain polymer melts while being in contact with a solid boundary, that they could possess a uniform distribution of their microelements. This will allow the fluids to have the microstretch remain constant at the boundary and yet satisfy the incompressibility condition. This is analogous in some ways to the case of a uniform distribution of bubble-like suspensions of microelements in the case of bubbly fluids. But, here we have the case of incompressible polymer melts or other polymeric fluids. In such a situation, it is physically reasonable to assume the microstretch to be a constant at the solid boundary. Hence, we can take here m1 ¼ constant at the boundary r ¼ a and hence its Laplace transform becomes m1 ðsÞ ¼ m1 =s. In this situation, taking (3.12) into account, we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I0 ðrkH Þ m1 I0 ðr m2 þ k 2 sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ m ðr; sÞ ¼ s I0 ðakH Þ s I0 ða m2 þ k 2 sÞ ð0Þ
ð3:15Þ
We now follow the numerical procedure of Krylov and Skoblaya [19] for evaluating the inverse transform of (3.15). The procedure consists in expanding the inverse transform in a series of shifted Legendre polynomials defined by Qn ðxÞ ¼ Pn ð2x 1Þ;
n ¼ 0; 1; 2; 3; . . . ;
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which form a set of orthogonal functions in the interval [0, 1] instead of the usual interval of orthogonality [)1, 1] for the Legendre polynomials. The Laplace transform (3.15) is defined by Z 1 est mð0Þ ðr; tÞ dt: ð3:16Þ mð0Þ ðr; sÞ ¼ 0
In order to prepare for the numerical inversion, we substitute et ¼ s; s 2 ½0; 1 , in the Laplace transform obtaining mð0Þ ðr; sÞ ¼
Z
1
ss1 mð0Þ ðr; sÞ ds:
ð3:17Þ
0
The values of this integral at the points s ¼ 1; 2; 3; . . . or, equivalently, sn ¼ n þ 1, for n ¼ 0; 1; 2; . . . can be written as ð0Þ
m ðr; sn Þ ¼ fn ðrÞ
Z
1
sn mð0Þ ðr; sÞ ds;
n ¼ 0; 1; 2; . . .
0
Now we expand the inverse transform in a series of shifted Legendre polynomials Qn ðsÞ: mð0Þ ðr; sÞ ¼
1 X ð2n þ 1Þcn ðrÞQn ðsÞ;
ð3:18Þ
n¼0
where for each fixed value of r, in 0 6 r 6 a, the coefficients cn are given by cn ðrÞ ¼
Z
1
mð0Þ ðr; sÞQn ðsÞ ds;
n ¼ 0; 1; 2; . . . ;
ð3:19Þ
0
and hence are clearly functions of r. Furthermore, it is shown by Krylov and Skoblaya [19] that the series expansion in (3.18) in terms of the orthogonal polynomials Qn is uniformly convergent and that the coefficients cn ðrÞ for each fixed value of r can be expressed in terms of the values of the shifted Legendre polynomials in the form cn ¼ Qn ðfÞ;
n ¼ 0; 1; 2; 3; . . . ;
ð3:20Þ
where the powers fn present in the shifted Legendre polynomials are replaced with fn the latter being the values of the Laplace transform at the points sn ¼ n þ 1, for n ¼ 0; 1; 2; . . . Hence, we can write fn ðrÞ ¼ m ðr; sn Þ
ð0Þ
sn ¼nþ1
¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I0 ðr m2 þ k 2 sn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : sn I0 ða m2 þ k 2 sn Þ sn ¼nþ1
ð3:21Þ
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For n ¼ 0, that is, for s0 ¼ 1, we obtain from (3.21) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I0 ðr m2 þ k 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : f0 ðrÞ ¼ I0 ða m2 þ k 2 Þ For n ¼ 1, that is, for s1 ¼ 2, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I0 ðr m2 þ 2k 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; f1 ðrÞ ¼ 2I0 ða m2 þ 2k 2 Þ
ð3:22Þ
ð3:23Þ
etc. ð1Þ Next, to obtain the first-order approximation solutions jð1Þ ; mð1Þ , and p0 , we proceed as follows. ð0Þ We first substitute m for m on the right-hand side of (3.1) and write the resulting equation in the form ojð1Þ ðr; sÞ ¼ 2jð0Þ mð0Þ ðr; sÞ: os
ð3:24Þ
Next, substitute the solution (3.18) for mð0Þ ðr; sÞ into (3.24), and integrate the result with respect to s between s ¼ 1 and s obtaining the first-order approximation jð1Þ for j. Thus, the final solution can be expressed in the form " # Z s 1 X ð1Þ ð0Þ ð2n þ 1Þcn ðrÞ Qn ðyÞ dy ; ð3:25Þ j ðr; sÞ ¼ j 1 þ 2 n¼0
1
where we have made use of (3.5) for the initial condition jð1Þ ðr; tÞ t¼0 ¼ jð1Þ ðr; sÞ s¼1 ¼ jð1Þ ðr; 0Þ ¼ jð0Þ ; since s ¼ et . In order to obtain the first-order approximation mð1Þ , we first rewrite (2.10) in the form mð1Þ ðr; sÞ ¼
1 ojð1Þ ðr; sÞ ; 2jð1Þ os
ð3:26Þ
and then substitute for jð1Þ from (3.25) into (3.26). We thus obtain mð1Þ ðr; sÞ ¼
1 jð0Þ X jð0Þ ð0Þ ð2n þ 1Þc ðrÞQ ðsÞ ¼ m ðr; sÞ: n n jð1Þ n¼0 jð1Þ
ð3:27Þ
ð1Þ
The expression for p0 , can be obtained by first rewriting (3.2) in the form ð1Þ p0 ðr; sÞ
ð1Þ
¼ k1 m
2 ð1Þ 1 ð0Þ omð1Þ om 1 omð1Þ þ þ qj a0 ; 2 2 or os or2
ð3:28Þ
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241
and then substituting (3.25) for jð1Þ and (3.27) for mð1Þ into (3.28). We then obtain ð1Þ p0 ðr; sÞ
1 1 k1 jð0Þ X qjð0Þ X o jð0Þ ¼ ð1Þ ð2n þ 1Þcn ðrÞQn ðsÞ þ ð2n þ 1Þcn ðrÞ Qn ðsÞ os jð1Þ j 2 n¼0 n¼0 2 ð1Þ om 1 omð1Þ a0 þ : r or or2
ð3:29Þ
Further simplification of the expressions in (3.29) is possible. But, we avoid the tedious and lengthy algebra caused by the presence of jð1Þ in the denominators of the expressions involved. However, the solutions obtained above, namely, (3.18), (3.25), (3.27), and (3.29) can all be numerically evaluated once the coefficients cn ðrÞ; n ¼ 0; 1; 2; 3; . . . are numerically computed. Velocity and microrotation fields: Before we proceed further to obtain the solutions for the velocity and microrotation fields, we present here, for later use, the finite Hankel transform of order zero which is defined by (cf. Sneddon [16], pp. 82–91): H0 ff ðrÞg ¼ f~ðgi Þ ¼
Z
r
rf ðrÞJ0 ðrgi Þ dr;
ð3:30Þ
0
where H0 and the tilde placed on a quantity denote the finite Hankel transform operator of order zero, and where gi ; i ¼ 1; 2; 3; . . . are the positive roots of the transcendental equation J0 ðagi Þ ¼ 0;
i ¼ 1; 2; 3; . . . ;
ð3:31Þ
J0 being the Bessel function of the first kind and of order zero. The inverse Hankel transform is given by 2 X~ J0 ðrgi Þ f ðrÞ ¼ H01 ff~ðgi Þg ¼ 2 ; f ðgi Þ a i fJ1 ðagi Þg2
ð3:32Þ
where the summation extends over all the positive roots of (3.31). Sneddon [17] has proved that the following property of the finite Hankel transform holds: H0
1 o of r ¼ g2i f~ðgi Þ þ agi f ðaÞJ1 ðagi Þ: r or or
ð3:33Þ
We also record here a frequently used result in this paper (cf. [16], p. 513, Eq. (11)), Z 0
a
rJ0 ðrgi Þ dr ¼
a J1 ðagi Þ: gi
ð3:34Þ
Now, we consider the solutions of (2.12) and (2.13) for vðr; tÞ and mh ðr; tÞ. As before, replacing j by its initial approximate value jð0Þ given by (2.20), and denoting the corresponding v and mh by
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vð0Þ ðr; tÞ and mh ðr; tÞ, respectively, as our zeroth order approximations, we find that (2.12), (2.13) and (2.15) reduce to the equations of Turk et al. [14]: ovð0Þ l þ j 1 o ovð0Þ 1 r ¼ ð3:35Þ þ fjuð0Þ þ A0 ð1 þ sin xtÞg; q r or q ot or ( ) ð0Þ 1 ouð0Þ ovð0Þ om ð0Þ j ; ð3:36Þ mh ¼ c qjð0Þ h 2j or or ot and uð0Þ ¼ uð0Þ ðr; tÞ ¼
1 o ð0Þ ðrmh Þ: r or
These equations are subject to the boundary and initial conditions ð0Þ omh ð0Þ ¼ 0; vða; tÞ ¼ 0; vðr; 0Þ ¼ 0; mh ðr; 0Þ ¼ 0; or t¼0;r¼0 1 o ð0Þ uð0Þ ða; tÞ ¼ ðrm Þ ¼ 0: r or h r¼a
ð3:37Þ
ð3:38Þ
Following [14], upon integration of the last of these conditions on the boundary r ¼ a, one obtains C ; a
ð0Þ
mh ða; tÞ ¼
ð3:39Þ
with C, a constant of integration which can be determined once the full solution for mh is obtained. In order to utilize the solutions of [14] as our zeroth order approximation solutions in our analysis, we employ the same boundary and initial conditions. For future reference, we find it convenient to express the solutions of Turk et al. [14] in the following simplified forms which we employ in our analysis as our initial or zeroth order approximations for vðr; tÞ and mh ðr; tÞ. Thus, we have vð0Þ ðr; tÞ ¼ ð0Þ
mh ðr; tÞ ¼
2 X ð0Þ J0 ðrgi Þ v~ ðgi ; tÞ ; 2 2 a i fJ1 ðagi Þg
ð3:40Þ
2 X ð0Þ J1 ðrgi Þ ; u~ ðgi ; tÞ 2 a i gi fJ1 ðagi Þg2
ð3:41Þ
where v~ð0Þ ðgi ; tÞ ¼
aA0 J1 ðagi Þ ðI1 þ E1 er1 t E2 er2 t þ E3 Þ; gi q2 jð0Þ r1 r2
ð3:42Þ
M.N.L. Narasimhan / International Journal of Engineering Science 41 (2003) 231–247
u~ð0Þ ðgi ; tÞ ¼
jgi aA0 J1 ðagi Þ ð1 þ E4 er1 t E5 er2 t þ E6 Þ; q2 jð0Þ r1 r2
243
ð3:43Þ
with E1 ¼ E1 ðgi Þ ¼ E4 ðqjð0Þ r1 þ I1 Þ;
E2 ¼ E2 ðgi Þ ¼ E5 ðqjð0Þ r2 þ I1 Þ;
E3 ¼ E3 ðgi Þ ¼ I2 ða1 cos xt þ b1 sin xtÞ; E5 ¼ E5 ðgi Þ ¼
r1 ðr22 þ xr2 þ x2 Þ ; ðr1 r2 Þðr22 þ x2 Þ
I1 ¼ I1 ðgi Þ ¼ 2j þ cg2i ;
r2 ðr12 þ xr1 þ x2 Þ ; ðr1 r2 Þðr12 þ x2 Þ
E6 ¼ E6 ðgi Þ ¼ I2 ða2 cos xt þ b2 sin xtÞ;
I2 ¼ I2 ðgi Þ ¼
a1 ¼ qjð0Þ xðr1 r2 x2 Þ þ xðr1 þ r2 ÞI1 ; a2 ¼ xðr1 þ r2 Þ;
E4 ¼ E4 ðgi Þ ¼
ð3:44Þ
r1 r2 ; ðr12 þ x2 Þðr22 þ x2 Þ b1 ¼ ðr1 r2 x2 ÞI1 qjð0Þ x2 ðr1 þ r2 Þ;
b 2 ¼ r 1 r 2 x2 ;
r1 and r2 being the roots of the quadratic equation s2 þ M1 s þ M2 ¼ 0;
ð3:45Þ
and M1 and M2 are given by g2i 2j ð0Þ M1 ¼ M1 ðgi Þ ¼ ð0Þ ðl þ jÞj þ c þ 2 ; qj gi g2i cðl þ jÞ 2 jð2l þ jÞ gi þ M2 ¼ M2 ðgi Þ ¼ : qjð0Þ cðl þ jÞ Turk et al. [14] have shown that the roots of (3.45) are real, distinct, and negative, which is the desired result since both r1 and r2 appear as coefficients of time in the exponential terms in (3.42) and (3.43). First order approximation for mh ðr; tÞ: On comparing (2.13) and (3.36), it becomes clear that ð0Þ when one replaces u; v; mh and j, respectively, with uð0Þ ; vð0Þ ; mh and jð0Þ on the right-hand side of (2.13), there results on the left-hand side, as can be seen from (3.36), the zeroth order approxið0Þ mation for mh obtained in (3.41). If, instead, one replaces them with uð0Þ ; vð0Þ ; mh and jð1Þ , the firstorder approximation for j given by (3.25), on the right-hand side of (2.13), one obtains a higher ð0Þ ð1Þ approximation for mh than mh , which we designate as mh ðr; tÞ, the first-order approximation for m. Thus, ( ) ð0Þ ð0Þ ð0Þ 1 ou ov om ð1Þ j : ð3:46Þ c qjð1Þ h mh ðr; tÞ ¼ 2j or or ot
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We note that it will be computationally simpler to evaluate ouð0Þ =or in (3.46) directly from the expression uð0Þ ðgi ; tÞg ¼ uð0Þ ¼ uð0Þ ðr; tÞ ¼ H01 f~
2 X ð0Þ J0 ðrgi Þ ; u~ ðgi ; tÞ 2 a i fJ1 ðagi Þg2
ð3:47Þ
where u~ð0Þ ðgi ; tÞ is given by (3.43). The term ovð0Þ =or can be evaluated from the expression (3.40) by employing (3.42). Similarly, the last term in the parentheses on the right-hand side of (3.46) can be evaluated by employing (3.25) and (3.41). Thus, after some lengthy but straightforward algebra, we obtain ( ) X X 1 ð1Þ ð1Þ B1 ðgi ; tÞ qj B2 ðgi ; tÞ ; ð3:48Þ mh ðr; tÞ ¼ 2 aj i i where jð1Þ is known from (3.25) and where B1 ðgi ; tÞ ¼ B2 ðgi ; tÞ ¼
gi J1 ðrgi Þ 2
fJ1 ðagi Þg J1 ðrgi Þ
ðj~ vð0Þ c~ uð0Þ Þ;
gi fJ1 ðagi Þg2
ð3:49Þ /ðgi ; tÞ;
o~ uð0Þ ¼ Rðgi Þfr1 E4 er1 t r2 E5 er2 t þ xI2 ðb2 cos xt a2 sin xtÞg; ot aA0 jgi J1 ðagi Þ Rðgi Þ ¼ ; q2 jð0Þ r1 r2
/ðgi ; tÞ ¼
ð3:50Þ
and E4 ; E5 ; I2 ; a2 and b2 are given by (3.44), while A0 is given by (2.16). First-order approximation for vðr; tÞ: We now obtain the first-order approximation vð1Þ ðr; tÞ for ð1Þ vðr; tÞ. First, we replace in (2.15) mh with mh given by (3.48) and designate the result as uð1Þ obtaining uð1Þ ¼ uð1Þ ðr; tÞ ¼
1 o ð1Þ ðrmh Þ; r or
ð3:51Þ
which is a higher order approximation for u than uð0Þ given by (3.37). Next, replace on the left-side of (2.12), u with uð1Þ from (3.51) and op=oz with the expression from (2.16). This leads to a higher order approximation for v, which we designate as vð1Þ . Thus, (2.12) now takes the form ovð1Þ 1 o ovð1Þ r ¼ j0 þ vðr; tÞ; r or ot or
ð3:52Þ
M.N.L. Narasimhan / International Journal of Engineering Science 41 (2003) 231–247
245
where j0 ¼
lþj ; q
1 vðr; tÞ ¼ fjuð1Þ þ A0 ð1 þ sin xtÞg: q
ð3:53Þ
The boundary and initial conditions are vð1Þ ða; tÞ ¼ 0;
vð1Þ ðr; 0Þ ¼ 0:
ð3:54Þ
Now, we take the finite Hankel transforms of (3.52) and (3.54) with respect to r. Then, employing (3.33) and noting that vð1Þ ða; tÞ ¼ 0, we obtain o~ vð1Þ þ j0 g2i v~ð1Þ ¼ v~ðgi ; tÞ; ot v~ð1Þ ðgi ; 0Þ ¼ H0 fvð1Þ ðr; 0Þg ¼ 0;
ð3:55Þ
where j aA0 J1 ðagi Þ v~ðgi ; tÞ ¼ u~ð1Þ ðgi ; tÞ þ ð1 þ sin xtÞ; q qgi Z a 1 o ð1Þ ð1Þ ð1Þ ~ ðrm Þ ¼ rmh gi J1 ðrgi Þ dr; u ðgi ; tÞ ¼ H0 r or h 0
ð3:56Þ
and where we have applied the result (3.34). The solution of (3.55) is readily obtained as v~ð1Þ ðgi ; tÞ ¼
Z
t
v~ðgi ; sÞ exp½j0 g2i ðt sÞ ds:
0
By applying the inversion theorem (3.32) on the above equation, we obtain Z t 2 X J0 ðrgi Þ v~ðgi ; sÞ exp½j0 g2i ðt sÞ ds: v ðr; tÞ ¼ 2 a i fJ1 ðagi Þg2 0 ð1Þ
ð3:57Þ
The determination of the solution for the indeterminate macropressure field p is a matter of simple integration of (2.11) taken together with (2.16) and employing the known solution (3.27) for mðr; tÞ as well as utilizing a known condition on the pressure such as, for instance, its value at, say, r ¼ 0, z ¼ 0, and t ¼ 0. Thus, Eqs. (2.20) and (3.18) provide complete solutions for the zeroth order approximation for j and m, while (3.25), (3.27), (3.29), (3.48) and (3.57) provide those for the first-order approximations of the unknowns j, m, p0 , mh , and v.
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4. Discussion and conclusions The equations governing the pulsatile flow of microstretch fluids in circular tubes are formulated into a nonlinear, coupled partial differential system (2.10)–(2.14) involving micropressure. The introduction of a physically reasonable initial approximation for microinertia leads to the linearization of the governing equations. Explicit solutions for the microstretch, the microinertia and the micropressure fields have been obtained by taking the Laplace transforms of their governing equations, solving the resulting equations in the transform space under appropriate initial and boundary conditions and then performing a numerical inversion of the Laplace transforms. On the other hand, the equations governing the microrotation and the velocity fields have been solved by applying the finite Hankel transform on their governing equations, solving the resulting equations in the Hankel transform space under appropriate initial and boundary conditions and performing the Hankel inversion procedure. Next, successive iteration techniques have been applied leading to higher order approximation solutions in computationally suitable forms appropriate to the cylindrical geometry of the problem. It is worth noting that if the micropressure is set equal to zero in our solution (3.27) for mð1Þ , it is equivalent to replacing mð1Þ by the zeroth order approximation solution mð0Þ , which in turn implies that jð1Þ reduces to jð0Þ . This means that when the micropressure is dropped from our analysis, our solution reduces to the micropolar solution of Turk et al. [14]. Furthermore, our solution (3.27) for microstretch has no counterpart in the micropolar case. Again, it can be readily verified that when we set jð1Þ ¼ jð0Þ in our solutions for the microrotation and velocity fields, (3.48) and (3.57), we find after some lengthy but straightforward algebra, that the latter reduce to the micropolar solutions of Turk et al. [14]. For purposes of obtaining a graphical illustration of the solutions, a complete numerical computation is necessary. The coefficients cn ðrÞ present in the Legendre series expansion in (3.18), (3.25), (3.27) and (3.29) can be readily computed as shown in (3.20)–(3.23). Moreover, the Legendre polynomials are extensively tabulated in [18]. Thus, the solutions mð0Þ ðr; tÞ; ð1Þ mð1Þ ðr; tÞ; jð1Þ ðr; tÞ and p0 ðr; tÞ can all be numerically evaluated. Next, the positive roots gi of the equation J0 ðagi Þ ¼ 0 are furnished by the tables ([18], pp. 359 and 409). It is clear from [18] that the graphs of the functions J0 ðrgi Þ and J1 ðrgi Þ,regarded as functions of gi for each given value of r in 0 6 r 6 a, possess the character similar to that of damped oscillations, rapidly converging and thus facilitating the evaluation of the sums of the series involved in the above solutions. Furthermore, the integrals involved in the expressions (3.56) and (3.57) can be readily evaluated by the use of SimpsonÕs rule. Thus, the numerical computations of the solutions can be performed in a straightforward manner leading to the graphical illustration of the solutions. Acknowledgement I sincerely thank Professor A. Cemal Eringen for helpful discussions and suggestions. References [1] A.C. Eringen, Int. J. Engng. Sci. 7 (1) (1969) 115–127. [2] A.C. Eringen, Int. J. Engng. Sci. 2 (1964) 205–217. [3] A.C. Eringen, Int. J. Engng. Sci. 28 (1990) 133–143.
M.N.L. Narasimhan / International Journal of Engineering Science 41 (2003) 231–247 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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A.C. Eringen, J. Math. Phys. 33 (12) (1992) 4078–4086. A.C. Eringen, Int. J. Engng. Sci. 38 (2000) 959–987. T. Ariman, J. Biomech. 4 (1971) 185–191. A.C. Eringen, Microcontinum Field Theories, vol. II: Fluent Media, Springer-Verlag, New York, 2001. H.L. Goldsmith, J. Marlow, J. Colloid Interf. Sci. 71 (2) (1979) 383–407. S.K.L. Rao, K.V. Raju, Int. J. Engng. Sci. 17 (4) (1979) 465–473. S.K.L. Rao, K.V. Raju, Int. J. Engng. Sci. 18 (1980) 1411–1419. D. Iesßan, Int. J. Engng. Sci. 35 (7) (1997) 669–679. N.U. Aydemir, J.E.S. Venart, Int. J. Engng. Sci. 28 (1990) 1211–1222. A.D. Kirwan Jr., Lett. Appl. Engng. Sci. 24 (7) (1986) 1237–1242. M.A. Turk, N.D. Sylvester, T. Ariman, Trans. Soc. Rheol. 17 (1) (1973) 1–21. G. Bugliarello, J. Sevilla, Biorheology 7 (1970) 85. I.N. Sneddon, Fourier Transforms, McGraw-Hill Book Co., New York, 1951. I.N. Sneddon, Phil. Mag. 37 (7) (1946) 17–25. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1965. V.I. Krylov, N.S. Skoblaya, Handbook of Numerical Inversion of Laplace Transforms, Translated from the Russian by D. Louvish, Israel Program of Scientific Translations, Jerusalem, 1969.
A mathematical model of pulsatile flows of microstretch fluids in circular tubes M.N.L. Narasimhan
*
10455 S.W. 152nd Avenue, Beaverton, OR 97007, USA Received 24 April 2002; accepted 22 July 2002
Abstract Pulsatile flows of micropolar fluids with stretch whose microelements can undergo expansions and contractions besides translations and rotations in straight circular tubes are considered. The governing field equations for such flows of linear microstretch fluids turn out to be a nonlinear coupled partial differential system. Solutions are sought for this system starting with a reasonable initial approximation for microinertia and the consequent linearization of the field equations. One of the coupled equations governing the microstretch and microinertia is solved approximately by the method of Laplace transforms taken with respect to the time variable. Making use of this approximate solution, the other coupled equation is solved leading to explicit higher order approximation solutions for microinertia, microstretch and micropressure. Next, the coupled equations governing the velocity and the microrotation fields are solved by employing the finite Hankel transform operators on a space variable and their inversions, and higher order approximation solutions are determined. All the above-mentioned explicit solutions are obtained in computationally suitable forms. These solutions have the promise of application to many practically important physical situations such as flows of polymeric fluids with deformable springy suspensions and flows of biological fluids including blood with deformable cell suspensions in small arteries. Ó 2002 Published by Elsevier Science Ltd. Keywords: Microstretch continuum theory; Microstretch fluids; Suspension rheology; Arterial blood flow
1. Introduction The theory of micropolar fluids with stretch (or microstretch fluids) was introduced by Eringen [1] in 1969 as a subclass of the more general type of fluids known as simple microfluids, also known as Eringen fluids (Eringen [2]). The microstretch fluids possess a local substructure whose *
Tel.: +1-503-590-9437; fax: +1-503-590-2554.
0020-7225/03/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 0 2 0 - 7 2 2 5 ( 0 2 ) 0 0 2 0 4 - 5
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M.N.L. Narasimhan / International Journal of Engineering Science 41 (2003) 231–247
material particles or microelements are endowed with seven degrees of freedom, three of translation, three of rotation and one of stretch or axial motion, consisting of expansions and contractions (breathing motion), as opposed to twelve degrees of freedom possessed by their parent class of simple microfluids, namely, three of translation, three of rotation, and six of stretch and shears. These fluids differ from the micropolar fluids, another subclass of the simple microfluids, in having one extra degree of freedom, namely, that of stretch, while the micropolar fluids whose particles are rigid, possess only six degrees of freedom, three of translation and three of rotation. Also, the microstretch fluids like micropolar fluids can support couple stresses as well as body couples and possess microinertia. The stress tensor of the microstretch fluid theory is asymmetric as in the case of micropolar fluids. It is important, however, to note that the special theory mentioned above can also be applied, in the limiting case, to cover the classical Navier–Stokes fluids as well. Eringen [3] extended his microstretch fluid theory to that of thermomicrostretch fluids incorporating thermal effects and applied the theory to the problem of propagation of acoustic waves in bubbly fluids obtaining agreement with available experimental results. Eringen also developed the theory of microstretch liquid crystals [4] and the electrodynamics of microstretch liquid crystal polymers [5]. Ariman [6] applied the microstretch fluid theory to the problem of steady-state blood flow in small arteries. Noting that micropressure is missing from ArimanÕs analysis, Eringen ([7], pp. 250–252) introduced a new investigation of the same problem but with the incorporation of micropressure obtaining a gratifying agreement with the experimental results of Goldsmith and Marlow [8]. Questions of existence, uniqueness and stability of solutions for microstretch fluid flows were examined by Rao and Raju [9,10], and by Iesßan [11]. Aydemir and Venart [12] performed numerical computations to obtain the solutions for the problem of nonsteady flow of a thermomicrostretch fluid between two parallel and heated plates, stationary with respect to the flow. While there exists a vast literature on the theory and applications of micropolar fluids, a relatively scant amount of work has been done on the dynamics of microstretch fluids. To the authorÕs knowledge, no work, analytical or numerical, has been done yet either on nonsteady flows or on pulsatile flows of microstretch fluids in tubes with circular cylindrical geometry. In the present investigation, the problem of pulsatile flows of microstretch fluids due to a sinusoidally varying pressure gradient in circular tubes is analyzed. Starting with a reasonable initial approximation for the microinertia and the consequent linearization of the field equations, the coupled partial differential system incorporating micropressure is solved. The mathematical analysis basically consists of two parts, one involving the determination of higher order approximation solutions for microstretch, microinertia, and the micropressure fields, and the other involving the determination of solutions for the velocity and the microrotation fields. Explicit solutions of the former set of equations are obtained by employing the Laplace transform with respect to time of the relevant equations, solving them and then taking the inverse transform. The second set is solved by employing the finite Hankel transform of the appropriate equations with respect to a space variable, solving them and then taking the inverse Hankel transforms. In both cases, explicit solutions are obtained in computationally suitable forms. This work may find application to many biophysical and technological fields. To mention a few, pulsatile blood flow in small arteries under the pumping action of the heart can be modeled after this work since blood consists of cell suspensions which can undergo expansions and contractions in addition to translations and rotations. Similarly, some of the industrial applications of
M.N.L. Narasimhan / International Journal of Engineering Science 41 (2003) 231–247
233
this work include pulsatile flows of polymeric fluids with deformable springy suspensions in circular tubes. 2. Formulation of the problem The following set of pertinent field equations governing the flow of isothermal microstretch fluids in vectorial form are given by (cf. Eringen [7]) q_ þ q$ v ¼ 0;
ð2:1Þ
Dj 2jm ¼ 0; Dt
ð2:2Þ
$p þ k0 $m þ ðk þ 2l þ jÞ$$ v ðl þ jÞ$ $ v þ j$ m þ qðf v_ Þ ¼ 0;
ð2:3Þ
ða þ b þ cÞ$$ m c$ $ m þ j$ v 2jm þ qðl rÞ ¼ 0;
ð2:4Þ
a0 r2 m þ p0 k1 m k0 $ v þ qðl rÞ ¼ 0;
ð2:5Þ
where q ¼ qðx; tÞ, v ¼ vðx; tÞ, m ¼ mðx; tÞ, j ¼ jðx; tÞ, and m ¼ mðx; tÞ are, respectively, the mass density, velocity, microrotation, microinertia density, and the microstretch fields that are unknowns to be determined under appropriate boundary and initial conditions. The microstretch rotatory inertia r and the microstretch scalar inertia r are given by r¼
D ðjmÞ; Dt
1 r ¼ jðm_ þ m2 Þ jm m; 2
ð2:6Þ
where j ¼ jkk is the scalar microinertia density and m is the microrotation vector and the microstretch m is defined by mkl ¼ mdkl klr mr ;
ð2:7Þ
mkl , mr , and klr being, respectively, the microgyration tensor, the microgyration vector (microrotation vector), and the permutation tensor, and here and in the sequel, summation being understood over repeated indices. The quantities f, l and l, respectively, denote the body force density, the body couple density and the trace of the microstretch body moment density tensor. The quantities p and p0 , respectively, denote the thermodynamic pressure and the micropressure. A superposed dot on a quantity denotes the material time derivative of that quantity. The quantities k0 , k1 , a0 , k, l, and j are the viscosity coefficients for the translatory motions, while a, b, and c are those for the rotatory motions. In this paper, we consider the linear microstretch theory in which the fields jvj, jmj and jmj are small in comparison with other terms in (2.6). As a consequence, (2.6) reduces to r ¼ j_m;
1 r ¼ jm_ : 2
ð2:8Þ
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In general, all the viscosity coefficients depend on j, q1 , and the temperture. But, in the linear approximation, they are considered as constants. The other basic assumptions made in the present study are (1) the fluid is incompressible and p is replaced by p, the fluid pressure, (2) the flow is isothermal, (3) external body force, body couple and microstretch body moment are absent, (4) the flow is microisotropic in the sense that the microinertia tensor jkl ¼ ð1=3Þjdkl , and (5) the flow in the circular tube is axisymmetric. We now introduce a cylindrical coordinate system ðr; h; zÞ with z-axis taken along the axis of the tube and r and h are, respectively, the radial and azimuthal coordinates taken in cross-sectional planes perpendicular to the tube axis. On account of axial symmetry and for a long tube all field quantities are independent of both h and z-coordinates with the exception of the macropressure p ¼ pðr; z; tÞ. The unknown fields are now represented by v ¼ ½0; 0; v ;
m ¼ ½0; mh ; 0 ;
m ¼ mðr; tÞ;
j ¼ jðr; tÞ;
ð2:9Þ
where v ¼ vðr; tÞ and mh ¼ mh ðr; tÞ. The field equations (2.2)–(2.5) now reduce to oj ¼ 2jm; ot k0
ð2:10Þ
om op ¼ ; or or
ð2:11Þ
1 o ov op ov r þ ju ¼ q ; ðl þ jÞ r or or oz ot 1 ou ov omh c j qj mh ¼ ; 2j or or ot 1 o om 1 om r k1 m þ p0 ¼ qj ; a0 r or or 2 ot
ð2:12Þ ð2:13Þ ð2:14Þ
where u ¼ uðr; tÞ ¼
1 o ðrmh Þ: r or
ð2:15Þ
These equations are to be solved under appropriate boundary and initial conditions for the unknowns v, mh , m; j, and p. We assume in the present study that the nonsteady axial flow is of a suddenly generated and accelerated type produced by an impressed sinusoidal axial pressure gradient of the form op ¼ oz
0 for t < 0 A0 þ A1 sin xt ¼ A0 ð1 þ sin xtÞ for t P 0;
ð2:16Þ
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235
where ¼ A1 =A0 , A0 is the mean axial pressure gradient, A1 is the amplitude of the axial pressure gradient and x is the circular frequency. The five equations (2.10)–(2.14) for the five unknowns are nonlinear and coupled partial differential equations. They are difficult to solve analytically, if not intractable. Hence, no progress is possible unless we adopt some reasonable approximation procedure. To this end, it is necessary to generate a suitable initial approximation for j in order to perform linearization of the equations. Bearing this in mind, we proceed as follows. To begin with, we note that m is uncoupled from v and mh in equations (2.10) and (2.14). Noting that at the tube wall r ¼ a, a being the radius of the tube, mh ¼ nv;r ;
m ¼ n$ v;
ð2:17Þ
where n is a constant, and a comma followed by an index indicates the partial derivative with respect to that index, is suggested as a possible general boundary condition by Kirwan [13]. We exploit this situation as follows. Substituting (2.17) into (2.12) and (2.13), one obtains after simplification, 1 ov ; j ¼ K c þ ð1 2nÞj ng or
ð2:18Þ
where 1
K ¼ fl þ jð1 nÞg ;
o g¼ or
1 o ov r : r or or
ð2:19Þ
This expression for j is nonlinear and is unsuitable for purposes of performing the desired linearization of the equations. It is clear that by setting n ¼ 1=2 in (2.18) and (2.19), we obtain a suitable initial or zeroth order approximate value for j denoted by jð0Þ where j¼j
ð0Þ
1 1 ¼c lþ j : 2
ð2:20Þ
We remark here that when n ¼ 1=2 in (2.17), it amounts to equating the microrotation velocity m to the local vorticity, and the stress tensor of the micropolar stretch theory becomes symmetric and reduces to that of the Navier–Stokes fluid. This condition is rather a severe restriction to impose on the microstretch fluid flow. As pointed out by Turk et al. [14], in some applied problems, however, such as those involving blood flow in small arteries, the experimental blood flow data of Bugliarello and Sevilla [15] show that the relation (2.17) with n ¼ 1=2 is a reasonably good approximation for the actual situation involving flow of blood with any hematocrit value (volumetric percentage of red cells in blood) in small arteries. But, in our analysis here, we employ j ¼ jð0Þ as an initial (or zeroth order) approximation for purposes of linearizing the field equations. Later, we obtain higher order approximation for jðr; tÞ by a successive iteration procedure.
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3. Method of solution Microstretch, microinertia and micropressure: Before we begin the solution process, we note that by direct integration of (2.10) between t ¼ 0 and t, we can express the formal solution of Eq. (2.10) as Z t jðr; tÞ ¼ jðr; 0Þ exp 2 mðr; sÞ ds ; 0
where jðr; 0Þ denotes the initial value of j at t ¼ 0. It is clear from the equation above that jðr; 0Þ 6¼ 0 since jðr; tÞ does not vanish. To begin with the solution process, we first treat equations (2.10) and (2.14). As the first step, we linearize (2.10) and (2.14) by substituting jð0Þ for j on the right-hand sides of these equations obtaining oj ¼ 2jð0Þ m; ot 1 o om 1 om r k1 m þ p0 ¼ qjð0Þ : a0 r or or 2 ot
ð3:1Þ ð3:2Þ
Initial condition: For the initial condition on the microstretch field, we have mðr; 0Þ ¼ m0 ðrÞ;
0 6 r < a:
ð3:3Þ
Boundary condition: The boundary condition on the microstretch field is mða; tÞ ¼ m1 ðtÞ;
t > 0:
ð3:4Þ
We will discuss the nature and the physical implications of these initial and boundary conditions later. We note that the linearization of the field equations (2.10) and (2.14) has been achieved by replacing j in the product terms on their right-hand sides with the constant quantity jð0Þ given by (2.20). As is clear from the work of Turk et al. [14], the observation that this constant value jð0Þ is a close approximation for j is well supported by the experimental results of [15] on blood flow. Hence, it is reasonable to assume that at t ¼ 0; jðr; tÞ possesses this constant value jð0Þ . Thus, jðr; 0Þ ¼ jð0Þ
ð3:5Þ
can serve as the initial condition for jðr; tÞ. Now, integration of (3.1) with respect to t between t ¼ 0 and t, gives Z t ð0Þ mðr; sÞ ds ; jðr; tÞ ¼ j 1 þ 2
ð3:6Þ
0
which is clearly consistent with the above-obtained formal solution for jðr; tÞ, applied to the linear approximation case with (3.5) taken into account.
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We now turn our attention to the nature of the micropressure field. The concept of micropressure was introduced by Eringen (cf. [7]) through a constitutive law derived from the free energy expression. That is, p0 ¼ 2q
ow j; oj
w ¼ Wðq1 ; h; jÞ:
ð3:7Þ
For the case of small microinertia j, the free energy w can be expanded into a power series in j, retaining terms up to the first degree in j. Then, the first of the equations in (3.7) reduces, in the case of incompressible, isotropic, and isothermal fluids to p0 ¼ d0 j;
ð3:8Þ
where d0 is a constant. Furthermore, when j is small, it is physically reasonable to assume, in the zeroth order approximation, that the influence of microinertia on the free energy is negligibly small. Hence, we may neglect the micropressure p0 in our zeroth order approximation, but retain it in the first-order approximation. As a consequence, (3.2) takes the form 1 o om om r ð3:9Þ m2 m ¼ k 2 ; r or or ot where m2 ¼
k1 ; a0
k2 ¼
qjð0Þ ; 2a0
ð3:10Þ
and jð0Þ is given by (2.20). We now take the Laplace transform of (3.9) with respect to time t where an overbar on a quantity denotes its Laplace transform, s being the transform parameter. We obtain after employing (3.3), o2 m 1 o m þ kH2 m ¼ k 2 m0 ðrÞ; 2 or r or
ð3:11Þ
where kH ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ k 2 s :
ð3:12Þ
The general solution of (3.11) is mðr; sÞ ¼ A0 ðsÞI0 ðrkH Þ þ A1 ðsÞK0 ðrkH Þ þ mp ðr; sÞ; where mp ðr; sÞ is a particular solution of (3.11) which can be obtained by the method of variation of parameters. The functions y1 ðr; sÞ ¼ I0 ðrkH Þ and y2 ðr; sÞ ¼ K0 ðrkH Þ are two linearly independent solutions of the corresponding homogeneous equation. Since the integrals involved in the
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particular solutions contain W 1 ; where W is the Wronskian of the solutions y1 and y2 , all we need here is W 1 for which we can employ the result in [18, p. 375] namely, 1
W 1 ¼ fW ½I0 ðrkH Þ; K0 ðrkH Þ g
¼ rkH :
Thus, the general solution of (3.11) is mð0Þ ðr; sÞ ¼ A0 ðsÞI0 ðrkH Þ þ A1 ðsÞK0 ðrkH Þ k 2 I0 ðrkH Þ Z r H 2 kH ym0 ðyÞI0 ðykH Þ dy; þ k K0 ðrk Þ
Z
r
kH ym0 ðyÞK0 ðykH Þ dy ð3:13Þ
where mð0Þ ðr; sÞ denotes the zeroth order solution in the transform space. The regularity condition at r ¼ 0 leads to A1 ðsÞ ¼ 0. Furthermore, mð0Þ ðr; sÞ becomes unbounded at r ¼ 0, because of the presence of K0 ðrkH Þ in the last term on the right-hand side of (3.13). This means that the only solution possible occurs when m0 ðrÞ is zero. Moreover, it is physically reasonable to assume that initially at t ¼ 0, microstretch is zero for all r 2 ½0; aÞ. In this case, we obtain mð0Þ ðr; sÞ ¼ A0 ðsÞI0 ðrkH Þ:
ð3:14Þ
Applying the boundary condition (3.4) and denoting mð0Þ ða; sÞ by m1 ðsÞ; we obtain the solution (3.14) in the form mð0Þ ðr; sÞ ¼
m1 ðsÞI0 ðrkH Þ : I0 ðakH Þ
It is possible in the case of some fluids such as, for instance, certain polymer melts while being in contact with a solid boundary, that they could possess a uniform distribution of their microelements. This will allow the fluids to have the microstretch remain constant at the boundary and yet satisfy the incompressibility condition. This is analogous in some ways to the case of a uniform distribution of bubble-like suspensions of microelements in the case of bubbly fluids. But, here we have the case of incompressible polymer melts or other polymeric fluids. In such a situation, it is physically reasonable to assume the microstretch to be a constant at the solid boundary. Hence, we can take here m1 ¼ constant at the boundary r ¼ a and hence its Laplace transform becomes m1 ðsÞ ¼ m1 =s. In this situation, taking (3.12) into account, we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I0 ðrkH Þ m1 I0 ðr m2 þ k 2 sÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ m ðr; sÞ ¼ s I0 ðakH Þ s I0 ða m2 þ k 2 sÞ ð0Þ
ð3:15Þ
We now follow the numerical procedure of Krylov and Skoblaya [19] for evaluating the inverse transform of (3.15). The procedure consists in expanding the inverse transform in a series of shifted Legendre polynomials defined by Qn ðxÞ ¼ Pn ð2x 1Þ;
n ¼ 0; 1; 2; 3; . . . ;
M.N.L. Narasimhan / International Journal of Engineering Science 41 (2003) 231–247
239
which form a set of orthogonal functions in the interval [0, 1] instead of the usual interval of orthogonality [)1, 1] for the Legendre polynomials. The Laplace transform (3.15) is defined by Z 1 est mð0Þ ðr; tÞ dt: ð3:16Þ mð0Þ ðr; sÞ ¼ 0
In order to prepare for the numerical inversion, we substitute et ¼ s; s 2 ½0; 1 , in the Laplace transform obtaining mð0Þ ðr; sÞ ¼
Z
1
ss1 mð0Þ ðr; sÞ ds:
ð3:17Þ
0
The values of this integral at the points s ¼ 1; 2; 3; . . . or, equivalently, sn ¼ n þ 1, for n ¼ 0; 1; 2; . . . can be written as ð0Þ
m ðr; sn Þ ¼ fn ðrÞ
Z
1
sn mð0Þ ðr; sÞ ds;
n ¼ 0; 1; 2; . . .
0
Now we expand the inverse transform in a series of shifted Legendre polynomials Qn ðsÞ: mð0Þ ðr; sÞ ¼
1 X ð2n þ 1Þcn ðrÞQn ðsÞ;
ð3:18Þ
n¼0
where for each fixed value of r, in 0 6 r 6 a, the coefficients cn are given by cn ðrÞ ¼
Z
1
mð0Þ ðr; sÞQn ðsÞ ds;
n ¼ 0; 1; 2; . . . ;
ð3:19Þ
0
and hence are clearly functions of r. Furthermore, it is shown by Krylov and Skoblaya [19] that the series expansion in (3.18) in terms of the orthogonal polynomials Qn is uniformly convergent and that the coefficients cn ðrÞ for each fixed value of r can be expressed in terms of the values of the shifted Legendre polynomials in the form cn ¼ Qn ðfÞ;
n ¼ 0; 1; 2; 3; . . . ;
ð3:20Þ
where the powers fn present in the shifted Legendre polynomials are replaced with fn the latter being the values of the Laplace transform at the points sn ¼ n þ 1, for n ¼ 0; 1; 2; . . . Hence, we can write fn ðrÞ ¼ m ðr; sn Þ
ð0Þ
sn ¼nþ1
¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I0 ðr m2 þ k 2 sn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : sn I0 ða m2 þ k 2 sn Þ sn ¼nþ1
ð3:21Þ
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For n ¼ 0, that is, for s0 ¼ 1, we obtain from (3.21) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I0 ðr m2 þ k 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : f0 ðrÞ ¼ I0 ða m2 þ k 2 Þ For n ¼ 1, that is, for s1 ¼ 2, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 I0 ðr m2 þ 2k 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; f1 ðrÞ ¼ 2I0 ða m2 þ 2k 2 Þ
ð3:22Þ
ð3:23Þ
etc. ð1Þ Next, to obtain the first-order approximation solutions jð1Þ ; mð1Þ , and p0 , we proceed as follows. ð0Þ We first substitute m for m on the right-hand side of (3.1) and write the resulting equation in the form ojð1Þ ðr; sÞ ¼ 2jð0Þ mð0Þ ðr; sÞ: os
ð3:24Þ
Next, substitute the solution (3.18) for mð0Þ ðr; sÞ into (3.24), and integrate the result with respect to s between s ¼ 1 and s obtaining the first-order approximation jð1Þ for j. Thus, the final solution can be expressed in the form " # Z s 1 X ð1Þ ð0Þ ð2n þ 1Þcn ðrÞ Qn ðyÞ dy ; ð3:25Þ j ðr; sÞ ¼ j 1 þ 2 n¼0
1
where we have made use of (3.5) for the initial condition jð1Þ ðr; tÞ t¼0 ¼ jð1Þ ðr; sÞ s¼1 ¼ jð1Þ ðr; 0Þ ¼ jð0Þ ; since s ¼ et . In order to obtain the first-order approximation mð1Þ , we first rewrite (2.10) in the form mð1Þ ðr; sÞ ¼
1 ojð1Þ ðr; sÞ ; 2jð1Þ os
ð3:26Þ
and then substitute for jð1Þ from (3.25) into (3.26). We thus obtain mð1Þ ðr; sÞ ¼
1 jð0Þ X jð0Þ ð0Þ ð2n þ 1Þc ðrÞQ ðsÞ ¼ m ðr; sÞ: n n jð1Þ n¼0 jð1Þ
ð3:27Þ
ð1Þ
The expression for p0 , can be obtained by first rewriting (3.2) in the form ð1Þ p0 ðr; sÞ
ð1Þ
¼ k1 m
2 ð1Þ 1 ð0Þ omð1Þ om 1 omð1Þ þ þ qj a0 ; 2 2 or os or2
ð3:28Þ
M.N.L. Narasimhan / International Journal of Engineering Science 41 (2003) 231–247
241
and then substituting (3.25) for jð1Þ and (3.27) for mð1Þ into (3.28). We then obtain ð1Þ p0 ðr; sÞ
1 1 k1 jð0Þ X qjð0Þ X o jð0Þ ¼ ð1Þ ð2n þ 1Þcn ðrÞQn ðsÞ þ ð2n þ 1Þcn ðrÞ Qn ðsÞ os jð1Þ j 2 n¼0 n¼0 2 ð1Þ om 1 omð1Þ a0 þ : r or or2
ð3:29Þ
Further simplification of the expressions in (3.29) is possible. But, we avoid the tedious and lengthy algebra caused by the presence of jð1Þ in the denominators of the expressions involved. However, the solutions obtained above, namely, (3.18), (3.25), (3.27), and (3.29) can all be numerically evaluated once the coefficients cn ðrÞ; n ¼ 0; 1; 2; 3; . . . are numerically computed. Velocity and microrotation fields: Before we proceed further to obtain the solutions for the velocity and microrotation fields, we present here, for later use, the finite Hankel transform of order zero which is defined by (cf. Sneddon [16], pp. 82–91): H0 ff ðrÞg ¼ f~ðgi Þ ¼
Z
r
rf ðrÞJ0 ðrgi Þ dr;
ð3:30Þ
0
where H0 and the tilde placed on a quantity denote the finite Hankel transform operator of order zero, and where gi ; i ¼ 1; 2; 3; . . . are the positive roots of the transcendental equation J0 ðagi Þ ¼ 0;
i ¼ 1; 2; 3; . . . ;
ð3:31Þ
J0 being the Bessel function of the first kind and of order zero. The inverse Hankel transform is given by 2 X~ J0 ðrgi Þ f ðrÞ ¼ H01 ff~ðgi Þg ¼ 2 ; f ðgi Þ a i fJ1 ðagi Þg2
ð3:32Þ
where the summation extends over all the positive roots of (3.31). Sneddon [17] has proved that the following property of the finite Hankel transform holds: H0
1 o of r ¼ g2i f~ðgi Þ þ agi f ðaÞJ1 ðagi Þ: r or or
ð3:33Þ
We also record here a frequently used result in this paper (cf. [16], p. 513, Eq. (11)), Z 0
a
rJ0 ðrgi Þ dr ¼
a J1 ðagi Þ: gi
ð3:34Þ
Now, we consider the solutions of (2.12) and (2.13) for vðr; tÞ and mh ðr; tÞ. As before, replacing j by its initial approximate value jð0Þ given by (2.20), and denoting the corresponding v and mh by
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M.N.L. Narasimhan / International Journal of Engineering Science 41 (2003) 231–247 ð0Þ
vð0Þ ðr; tÞ and mh ðr; tÞ, respectively, as our zeroth order approximations, we find that (2.12), (2.13) and (2.15) reduce to the equations of Turk et al. [14]: ovð0Þ l þ j 1 o ovð0Þ 1 r ¼ ð3:35Þ þ fjuð0Þ þ A0 ð1 þ sin xtÞg; q r or q ot or ( ) ð0Þ 1 ouð0Þ ovð0Þ om ð0Þ j ; ð3:36Þ mh ¼ c qjð0Þ h 2j or or ot and uð0Þ ¼ uð0Þ ðr; tÞ ¼
1 o ð0Þ ðrmh Þ: r or
These equations are subject to the boundary and initial conditions ð0Þ omh ð0Þ ¼ 0; vða; tÞ ¼ 0; vðr; 0Þ ¼ 0; mh ðr; 0Þ ¼ 0; or t¼0;r¼0 1 o ð0Þ uð0Þ ða; tÞ ¼ ðrm Þ ¼ 0: r or h r¼a
ð3:37Þ
ð3:38Þ
Following [14], upon integration of the last of these conditions on the boundary r ¼ a, one obtains C ; a
ð0Þ
mh ða; tÞ ¼
ð3:39Þ
with C, a constant of integration which can be determined once the full solution for mh is obtained. In order to utilize the solutions of [14] as our zeroth order approximation solutions in our analysis, we employ the same boundary and initial conditions. For future reference, we find it convenient to express the solutions of Turk et al. [14] in the following simplified forms which we employ in our analysis as our initial or zeroth order approximations for vðr; tÞ and mh ðr; tÞ. Thus, we have vð0Þ ðr; tÞ ¼ ð0Þ
mh ðr; tÞ ¼
2 X ð0Þ J0 ðrgi Þ v~ ðgi ; tÞ ; 2 2 a i fJ1 ðagi Þg
ð3:40Þ
2 X ð0Þ J1 ðrgi Þ ; u~ ðgi ; tÞ 2 a i gi fJ1 ðagi Þg2
ð3:41Þ
where v~ð0Þ ðgi ; tÞ ¼
aA0 J1 ðagi Þ ðI1 þ E1 er1 t E2 er2 t þ E3 Þ; gi q2 jð0Þ r1 r2
ð3:42Þ
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u~ð0Þ ðgi ; tÞ ¼
jgi aA0 J1 ðagi Þ ð1 þ E4 er1 t E5 er2 t þ E6 Þ; q2 jð0Þ r1 r2
243
ð3:43Þ
with E1 ¼ E1 ðgi Þ ¼ E4 ðqjð0Þ r1 þ I1 Þ;
E2 ¼ E2 ðgi Þ ¼ E5 ðqjð0Þ r2 þ I1 Þ;
E3 ¼ E3 ðgi Þ ¼ I2 ða1 cos xt þ b1 sin xtÞ; E5 ¼ E5 ðgi Þ ¼
r1 ðr22 þ xr2 þ x2 Þ ; ðr1 r2 Þðr22 þ x2 Þ
I1 ¼ I1 ðgi Þ ¼ 2j þ cg2i ;
r2 ðr12 þ xr1 þ x2 Þ ; ðr1 r2 Þðr12 þ x2 Þ
E6 ¼ E6 ðgi Þ ¼ I2 ða2 cos xt þ b2 sin xtÞ;
I2 ¼ I2 ðgi Þ ¼
a1 ¼ qjð0Þ xðr1 r2 x2 Þ þ xðr1 þ r2 ÞI1 ; a2 ¼ xðr1 þ r2 Þ;
E4 ¼ E4 ðgi Þ ¼
ð3:44Þ
r1 r2 ; ðr12 þ x2 Þðr22 þ x2 Þ b1 ¼ ðr1 r2 x2 ÞI1 qjð0Þ x2 ðr1 þ r2 Þ;
b 2 ¼ r 1 r 2 x2 ;
r1 and r2 being the roots of the quadratic equation s2 þ M1 s þ M2 ¼ 0;
ð3:45Þ
and M1 and M2 are given by g2i 2j ð0Þ M1 ¼ M1 ðgi Þ ¼ ð0Þ ðl þ jÞj þ c þ 2 ; qj gi g2i cðl þ jÞ 2 jð2l þ jÞ gi þ M2 ¼ M2 ðgi Þ ¼ : qjð0Þ cðl þ jÞ Turk et al. [14] have shown that the roots of (3.45) are real, distinct, and negative, which is the desired result since both r1 and r2 appear as coefficients of time in the exponential terms in (3.42) and (3.43). First order approximation for mh ðr; tÞ: On comparing (2.13) and (3.36), it becomes clear that ð0Þ when one replaces u; v; mh and j, respectively, with uð0Þ ; vð0Þ ; mh and jð0Þ on the right-hand side of (2.13), there results on the left-hand side, as can be seen from (3.36), the zeroth order approxið0Þ mation for mh obtained in (3.41). If, instead, one replaces them with uð0Þ ; vð0Þ ; mh and jð1Þ , the firstorder approximation for j given by (3.25), on the right-hand side of (2.13), one obtains a higher ð0Þ ð1Þ approximation for mh than mh , which we designate as mh ðr; tÞ, the first-order approximation for m. Thus, ( ) ð0Þ ð0Þ ð0Þ 1 ou ov om ð1Þ j : ð3:46Þ c qjð1Þ h mh ðr; tÞ ¼ 2j or or ot
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We note that it will be computationally simpler to evaluate ouð0Þ =or in (3.46) directly from the expression uð0Þ ðgi ; tÞg ¼ uð0Þ ¼ uð0Þ ðr; tÞ ¼ H01 f~
2 X ð0Þ J0 ðrgi Þ ; u~ ðgi ; tÞ 2 a i fJ1 ðagi Þg2
ð3:47Þ
where u~ð0Þ ðgi ; tÞ is given by (3.43). The term ovð0Þ =or can be evaluated from the expression (3.40) by employing (3.42). Similarly, the last term in the parentheses on the right-hand side of (3.46) can be evaluated by employing (3.25) and (3.41). Thus, after some lengthy but straightforward algebra, we obtain ( ) X X 1 ð1Þ ð1Þ B1 ðgi ; tÞ qj B2 ðgi ; tÞ ; ð3:48Þ mh ðr; tÞ ¼ 2 aj i i where jð1Þ is known from (3.25) and where B1 ðgi ; tÞ ¼ B2 ðgi ; tÞ ¼
gi J1 ðrgi Þ 2
fJ1 ðagi Þg J1 ðrgi Þ
ðj~ vð0Þ c~ uð0Þ Þ;
gi fJ1 ðagi Þg2
ð3:49Þ /ðgi ; tÞ;
o~ uð0Þ ¼ Rðgi Þfr1 E4 er1 t r2 E5 er2 t þ xI2 ðb2 cos xt a2 sin xtÞg; ot aA0 jgi J1 ðagi Þ Rðgi Þ ¼ ; q2 jð0Þ r1 r2
/ðgi ; tÞ ¼
ð3:50Þ
and E4 ; E5 ; I2 ; a2 and b2 are given by (3.44), while A0 is given by (2.16). First-order approximation for vðr; tÞ: We now obtain the first-order approximation vð1Þ ðr; tÞ for ð1Þ vðr; tÞ. First, we replace in (2.15) mh with mh given by (3.48) and designate the result as uð1Þ obtaining uð1Þ ¼ uð1Þ ðr; tÞ ¼
1 o ð1Þ ðrmh Þ; r or
ð3:51Þ
which is a higher order approximation for u than uð0Þ given by (3.37). Next, replace on the left-side of (2.12), u with uð1Þ from (3.51) and op=oz with the expression from (2.16). This leads to a higher order approximation for v, which we designate as vð1Þ . Thus, (2.12) now takes the form ovð1Þ 1 o ovð1Þ r ¼ j0 þ vðr; tÞ; r or ot or
ð3:52Þ
M.N.L. Narasimhan / International Journal of Engineering Science 41 (2003) 231–247
245
where j0 ¼
lþj ; q
1 vðr; tÞ ¼ fjuð1Þ þ A0 ð1 þ sin xtÞg: q
ð3:53Þ
The boundary and initial conditions are vð1Þ ða; tÞ ¼ 0;
vð1Þ ðr; 0Þ ¼ 0:
ð3:54Þ
Now, we take the finite Hankel transforms of (3.52) and (3.54) with respect to r. Then, employing (3.33) and noting that vð1Þ ða; tÞ ¼ 0, we obtain o~ vð1Þ þ j0 g2i v~ð1Þ ¼ v~ðgi ; tÞ; ot v~ð1Þ ðgi ; 0Þ ¼ H0 fvð1Þ ðr; 0Þg ¼ 0;
ð3:55Þ
where j aA0 J1 ðagi Þ v~ðgi ; tÞ ¼ u~ð1Þ ðgi ; tÞ þ ð1 þ sin xtÞ; q qgi Z a 1 o ð1Þ ð1Þ ð1Þ ~ ðrm Þ ¼ rmh gi J1 ðrgi Þ dr; u ðgi ; tÞ ¼ H0 r or h 0
ð3:56Þ
and where we have applied the result (3.34). The solution of (3.55) is readily obtained as v~ð1Þ ðgi ; tÞ ¼
Z
t
v~ðgi ; sÞ exp½j0 g2i ðt sÞ ds:
0
By applying the inversion theorem (3.32) on the above equation, we obtain Z t 2 X J0 ðrgi Þ v~ðgi ; sÞ exp½j0 g2i ðt sÞ ds: v ðr; tÞ ¼ 2 a i fJ1 ðagi Þg2 0 ð1Þ
ð3:57Þ
The determination of the solution for the indeterminate macropressure field p is a matter of simple integration of (2.11) taken together with (2.16) and employing the known solution (3.27) for mðr; tÞ as well as utilizing a known condition on the pressure such as, for instance, its value at, say, r ¼ 0, z ¼ 0, and t ¼ 0. Thus, Eqs. (2.20) and (3.18) provide complete solutions for the zeroth order approximation for j and m, while (3.25), (3.27), (3.29), (3.48) and (3.57) provide those for the first-order approximations of the unknowns j, m, p0 , mh , and v.
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4. Discussion and conclusions The equations governing the pulsatile flow of microstretch fluids in circular tubes are formulated into a nonlinear, coupled partial differential system (2.10)–(2.14) involving micropressure. The introduction of a physically reasonable initial approximation for microinertia leads to the linearization of the governing equations. Explicit solutions for the microstretch, the microinertia and the micropressure fields have been obtained by taking the Laplace transforms of their governing equations, solving the resulting equations in the transform space under appropriate initial and boundary conditions and then performing a numerical inversion of the Laplace transforms. On the other hand, the equations governing the microrotation and the velocity fields have been solved by applying the finite Hankel transform on their governing equations, solving the resulting equations in the Hankel transform space under appropriate initial and boundary conditions and performing the Hankel inversion procedure. Next, successive iteration techniques have been applied leading to higher order approximation solutions in computationally suitable forms appropriate to the cylindrical geometry of the problem. It is worth noting that if the micropressure is set equal to zero in our solution (3.27) for mð1Þ , it is equivalent to replacing mð1Þ by the zeroth order approximation solution mð0Þ , which in turn implies that jð1Þ reduces to jð0Þ . This means that when the micropressure is dropped from our analysis, our solution reduces to the micropolar solution of Turk et al. [14]. Furthermore, our solution (3.27) for microstretch has no counterpart in the micropolar case. Again, it can be readily verified that when we set jð1Þ ¼ jð0Þ in our solutions for the microrotation and velocity fields, (3.48) and (3.57), we find after some lengthy but straightforward algebra, that the latter reduce to the micropolar solutions of Turk et al. [14]. For purposes of obtaining a graphical illustration of the solutions, a complete numerical computation is necessary. The coefficients cn ðrÞ present in the Legendre series expansion in (3.18), (3.25), (3.27) and (3.29) can be readily computed as shown in (3.20)–(3.23). Moreover, the Legendre polynomials are extensively tabulated in [18]. Thus, the solutions mð0Þ ðr; tÞ; ð1Þ mð1Þ ðr; tÞ; jð1Þ ðr; tÞ and p0 ðr; tÞ can all be numerically evaluated. Next, the positive roots gi of the equation J0 ðagi Þ ¼ 0 are furnished by the tables ([18], pp. 359 and 409). It is clear from [18] that the graphs of the functions J0 ðrgi Þ and J1 ðrgi Þ,regarded as functions of gi for each given value of r in 0 6 r 6 a, possess the character similar to that of damped oscillations, rapidly converging and thus facilitating the evaluation of the sums of the series involved in the above solutions. Furthermore, the integrals involved in the expressions (3.56) and (3.57) can be readily evaluated by the use of SimpsonÕs rule. Thus, the numerical computations of the solutions can be performed in a straightforward manner leading to the graphical illustration of the solutions. Acknowledgement I sincerely thank Professor A. Cemal Eringen for helpful discussions and suggestions. References [1] A.C. Eringen, Int. J. Engng. Sci. 7 (1) (1969) 115–127. [2] A.C. Eringen, Int. J. Engng. Sci. 2 (1964) 205–217. [3] A.C. Eringen, Int. J. Engng. Sci. 28 (1990) 133–143.
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