JOURNAL OF APPLIED PHYSICS 103, 124905 共2008兲
Waves in a cylindrical borehole filled with micropolar fluid Dilbag Singha兲 and S. K. Tomarb兲 Department of Mathematics, Panjab University, Chandigarh 160 014, India
共Received 27 December 2007; accepted 17 April 2008; published online 23 June 2008兲 Dispersion equation is derived for the propagation of surface waves in a cylindrical borehole filled with a micropolar viscous fluid and hosted in an infinite micropolar elastic solid medium. These waves are found to be dispersive and attenuated. The effects of fluid viscosity, micropolarity of the fluid, and radius of the borehole on the dispersion curve are noticed and depicted graphically. For a particular model, the dispersion curve is found to be significantly affected by the fluid viscosity and radius of the borehole, but not much by the micropolarity of the fluid. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2946448兴 I. INTRODUCTION
Eringen’s micropolar theory of elasticity1 is now well known and does not need much introduction. A historical development of the theory of micropolar elasticity is given in a monograph by Eringen.2 In this theory, the load across a surface element is transmitted not only by a force stress vector but also by a couple stress vector and the motion is characterized by six degrees of freedom 共three of translation and three of microrotation兲. Micropolar elastic solids can be thought of as being composed of dumb-bell type molecules and these molecules in a volume element can undergo rotation about their center of mass along with the linear displacement. The theory of microfluids was also introduced by Eringen,3 which deals with a class of fluids exhibiting certain microscopic effects arising from the local structure and the micromotions of the fluid elements. A subclass of these fluids is the micropolar fluid, which has the microrotational effects and microrotational inertia.4Likewise, as in the case of micropolar elastic solids, a micropolar fluid can also support couple stress, the asymmetric stress tensor and also possesses a rotational field, which is independent of the velocity of the fluid. A large class of fluids such as anisotropic fluids, liquid crystals 共with dumb-bell type molecules兲, magnetic fluids, muddy fluids, biological fluids, animal blood, and dirty fluids 共dusty air, snow兲 over airfoil can be nicely modeled more realistically as micropolar fluids. Biot5 studied the propagation of elastic waves in a cylindrical bore filled with and without fluid and embedded in a uniform elastic solid of infinite extent. He studied two dimensional problems and obtained dispersion relations for the waves propagating along the boundary of such a cylindrical bore. Stilke6 obtained solutions for the propagation of elastic waves at the surface of a tunnel-like hole with a circular border embedded in a three dimensional uniform elastic medium and found that the phase and group velocities depend on the ratio between the wavelength and the circumference of the cylindrical hole. Since then several problems concerning the cylindrical bore have been attempted by several authors. Some of them are by Banerji and Sengupta,7 Sengupta a兲
Electronic mail:
[email protected]. b兲 Electronic mail:
[email protected]. 0021-8979/2008/103共12兲/124905/8/$23.00
and Chakrabarti,8 Shrama and Gogna,9 Tomar and Kumar,10 Deswal et al.,11 Kumar and Deswal,12 Bhujanga Rao and Rama Murthy,13 Vashishth and Khurana,14 and Arora and Tomar.15 Recently, Cheng and Blanch17 reviewed the methods of simulating elastic wave propagation in a borehole by considering two different approaches, a quasianalytic approach known as the discrete wavenumber summation method and a finite difference method. In this paper, we have investigated a problem of propagation of surface waves in a cylindrical borehole situated in an infinite micropolar elastic solid and filled with a micropolar viscous fluid. The cylindrical bore is assumed to be of infinite length and the frequency equation relevant to the propagation of surface waves is derived and then solved numerically for a particular model. The effect of borehole radius, micropolarity, and viscosity of the contained fluid column is noticed on the dispersion curves. The present model may be viewed in a situation arising in the field of oil-well exploration. The oil inside the oil-well is generally found in crude form containing several impurities and therefore can be best modeled with muddylike/dusty viscous fluid of micropolar nature. Thus the present problem may be of great help to oil companies. II. FIELD EQUATIONS AND RELATIONS
Following Eringen,1,4 the equations of motion and constitutive relations for micropolar viscous fluid and micropolar solid media in the absence of body and surface loads are given as follows. For micropolar viscous fluid medium, ˙f 共c21f + c23f 兲 ⵜ ⵜ · u˙ f − 共c22f + c23f 兲 ⵜ ⫻ ⵜ ⫻ u˙ f + c23f ⵜ ⫻ ⌽ 共1兲
= u¨ f , ˙ f兲 ˙ f − c2 ⵜ ⫻ ⵜ ⫻ ⌽ ˙ f + c2 共ⵜ ⫻ u˙ f − 2⌽ 共c24f + c25f 兲 ⵜ ⵜ · ⌽ 6f 5f ¨ f, =⌽
共2兲
and for micropolar solid medium, 2 2 2 2 2 + c3s 兲 ⵜ ⵜ · us − 共c2s + c3s 兲 ⵜ ⫻ ⵜ ⫻ us + c3s ⵜ ⫻ ⌽s 共c1s
= u¨ s , 103, 124905-1
共3兲 © 2008 American Institute of Physics
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124905-2
J. Appl. Phys. 103, 124905 共2008兲
D. Singh and S. K. Tomar
2 2 2 2 共c4s + c5s 兲 ⵜ ⵜ · ⌽s − c5s ⵜ ⫻ ⵜ ⫻ ⌽s + c6s 共ⵜ ⫻ us − 2⌽s兲
¨ s, =⌽
共4兲
冉
共 f + K f 兲 ⵜ2 −
冊
˙f 2urf ˙ 2f 1 f f f e f f ˙ = − K u + 共 + 兲 , r z t2 r2 r 共9兲
where 2 c1R = 共R + 2R兲/R,
2 c2R = R/ R,
2 c4R = 共␣R + R兲/R jR,
2 c3R = K R/ R ,
2 c5R = ␥ R/ R j R,
共 f + K f 兲ⵜ2u˙zf + 共 f + f 兲
2 2 c6R = c3R /jR ,
R is the density of the medium, jR is the microinertia, and uR and ⌽R are, respectively, the displacement and microrotation vectors. Here, the quantity having superscript R corresponds to the micropolar viscous fluid or solid medium as per the following definition: R=
再
f for micropolar viscous fluid medium s for micropolar elastic solid medium.
冎
f f f f ˙ f 兲, klf = f u˙r,r ␦kl + f 共u˙k,l + u˙l,k 兲 + K f 共u˙l,k − klp p
skl
=
s sur,r ␦kl
+
s
s 共uk,l
+
s ul,k 兲
+K
s
s 共ul,k
−
klpsp兲,
˙ f ␦kl +  f ˙ f + ␥f ˙f , mklf = ␣ f r,r k,l l,k mskl
=␣
s
s r,r ␦kl
+
s
s k,l
+␥
s
s l,k ,
共5兲
⌽f = 共0, 2f ,0兲.
冊
共11兲 and Eqs. 共3兲 and 共4兲 become
冉
冊
s 2urs s2 1 s s s e = s 2 , − Ks 2 ur + 共 + 兲 z t r r
共12兲 共s + Ks兲ⵜ2uzs + 共s + s兲
2uzs es Ks 共rs2兲 = s 2 , 共13兲 + t z r r
冋冉 冊 册 ␥s ⵜ2 −
冉
冊
2s2 urs uzs 1 s s s s s − − 2K + K = j , 2 t2 r2 z r 共14兲
where 1 共rurR兲 uzR + , r r z
ⵜ2 =
2 1 2 + + . r2 r r z2
共7兲
Introducing the potentials ⬘R, R, and ⌫R as follows
共8兲
urR =
Consider a circular cylindrical bore of radius a through a micropolar elastic medium of infinite extent. Taking the cylindrical polar coordinates 共r , , z兲 such that the z axis is pointing vertically upward along the axis of the cylinder. Our aim is to investigate the frequency equation relevant to the propagation of axial symmetric waves which are harmonic along the axial direction. To discuss the surface waves at micropolar fluid/micropolar solid interface, we consider the following forms of the displacement and microrotation vectors as
u f = 共urf,0,uzf兲,
冉
22f u˙rf u˙zf 1 f ˙f f f f − − 2K + K = j , 2 t2 r2 z r
共6兲
III. PROBLEM AND FREQUENCY EQUATION
⌽s = 共0, s2,0兲,
␥ f ⵜ2 −
eR =
where Rkl is the force stress tensor, mRkl is the couple stress tensor, the “comma” in the subscript denotes the spatial derivative, and ␦kl and klp are Kronecker delta and the alternating tensors, respectively. Other symbols have their usual meanings.
us = 共urs,0,uzs兲,
冋冉 冊 册 共 s + K s兲 ⵜ 2 −
The quantities f , f , and K f are the fluid viscosity coefficients and the quantities ␣ f ,  f , and ␥ f are the fluid viscosity coefficients responsible for gyrational dissipation of the micropolar fluid, while the quantities s and s are the Lame’s constant and the quantities Ks, ␣s, s, and ␥s are the micropolar elastic constants for the micropolar elastic solid medium. The superposed dot represents the temporal derivative of that quantity. The constitutive relations are given by
2uzf e˙ f K f 共r˙ 2f 兲 = f 2 , 共10兲 + t z r r
Since we are considering axially symmetric waves, therefore, the quantities would remain independent of . With these considerations, the above Eqs. 共1兲 and 共2兲 become
⬘R 2 R + , r r z
uzR =
冉
冊
⬘R 2 − ⵜ2 − 2 R , z z 共15兲
⌫R , R2 = − r into Eqs. 共9兲–共14兲, we obtain 2 2 + c3R 兲ⵜ2 − 씲R兴⬘R = 0, 关共c1R
共16兲
2 2 2 + c3R 兲ⵜ2 − 씲R兴R + c3R ⌫R = 0, 关共c2R
共17兲
2 2 2 ⵜ2 − 2c6R − 씲R兴⌫R − c6R ⵜ2R = 0, 关c5R
共18兲
where 씲R =
再
2t for R = s t for R = f .
冎
We note that Eq. 共16兲 is uncoupled in the potential ⬘R, while Eqs. 共17兲 and 共18兲 are coupled in the potentials R and ⌫R. Now, we shall find the solutions of these equations for time harmonic waves propagating along the z direction. In order to solve Eq. 共16兲, we take
⬘R = R1 共r兲exp兵ı共kz − t兲其, where the symbols , k, and c共= / k兲 represent the angular frequency, the wavenumber, and the phase velocity, respectively. Inserting it into Eq. 共16兲, we obtain
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124905-3
J. Appl. Phys. 103, 124905 共2008兲
D. Singh and S. K. Tomar
2R1 1 R1 − 共R1 兲2R1 = 0, + r2 r r
共19兲
2 where 共R1 兲2 = k2 − 共2 / VR1 兲. The expressions of quantities Vs1 2 2 2 and V f1 are given by Vs1 = c1s + c3s , V2f1 = −ı共c21f + c23f 兲. From Eqs. 共17兲 and 共18兲, one can obtain
兵Aⵜ4 + Bⵜ2 + C其共R,⌫R兲 = 0,
共20兲
2 2 2 2 2 2 2 2 2 共c2R + c3R 兲, B = c3R c6R − 씲Rc5R − 共c2R + c3R 兲共2c6R where A = c5R 2 + 씲R兲 and C = 씲R共2c6R + 씲R兲. Equation 共20兲 can be further written as
兵共ⵜ2 − ␦1兲共ⵜ2 − ␦2兲其共R,⌫R兲 = 0,
1 关− B − 冑B2 − 4AC兴. 2A
Let us find the solution of Eq. 共21兲 corresponding to R, by taking
=
− t兲其,
Inserting it into Eq. 共21兲, we obtain
2R2 2 r
+
1 R2 r r
− 共Ri 兲2R2 = 0共i = 2,3兲,
1 关− b⬘ ⫾ 共b⬘2 − 4a⬘c⬘兲1/2兴, 2a⬘
a = 1 − 220/2, 2 + c3s 兲20/2,
a⬘ = + 2ıc26f , + c23f c26f 兴,
2 2 2 2 b = c2s + c3s + c5s − 共2c2s 2 2 2 c = c5s 共c2s + c3s 兲,
关A2f I0共2f r兲
+ A2⬘ f I0共3f r兲兴其exp兵ı共kz − t兲其,
共24兲 关A3f I0共2f r兲
2 20 = c6s ,
b⬘ = 关ıc25f + ı共c22f + c23f 兲共 + 2ıc26f 兲 c⬘ = − 3c25f 共c22f + c23f 兲.
We note that the Eqs. 共19兲 and 共22兲 are the modified Bessel differential equations of order zero. Their solutions are the modified Bessel functions of first and second kind, i.e., I0共Rj r兲 and K0共Rj r兲共j = 1 , 2 , 3兲. Note that the function I0 is bounded as r → 0, the function K0 → 0 as 兩r兩 → ⬁ and they represent incoming and outgoing waves in cylindrical coordinates, respectively. Now, we intend to apply the boundary conditions at the fluid-solid interface. For the type of waves considered in a fluid-filled cylindrical borehole, there are three boundary conditions at the surface of the cylindrical borehole: 共i兲 the motion 共i.e., displacement and microrotation兲 must remain finite at the center of the borehole, 共ii兲 there are no incoming waves from infinity, and 共iii兲 the displacement, microrota-
共25兲
where the quantities As1 , A1f , As2 , A2⬘s , A2f , A2⬘ f , As3 , A3⬘s , A3f , and A3⬘ f are arbitrary constants. Note that the solution of coupled equations corresponding to ⌫R can be obtained by plugging the solution of R from Eq. 共24兲 into Eq. 共17兲, where the unknown coefficients are given by A3⬘s = bs3A2⬘s,
As3 = bs2As2,
共22兲
1 = 关b ⫾ 共b2 − 4ac兲1/2兴, 2a
V2f2,f3 =
兵s, f 其 = 兵关As2K0共s2r兲 + A2⬘sK0共s3r兲兴,
s b2,3 =
2 兲. The expressions of the quantities where 共Ri 兲2 = k2 − 共2 / VRi Vsi and V fi are given by 2 Vs2,s3
共23兲
+ A3⬘ f I0共3f r兲兴其exp兵ı共kz − t兲其,
and
R2 共r兲exp兵ı共kz
A1f I0共1f r兲其exp兵ı共kz − t兲其,
兵⌫s,⌫ f 其 = 兵关As3K0共s2r兲 + A3⬘sK0共s3r兲兴,
1 ␦1 = 关− B + 冑B2 − 4AC兴 2A
R
兵⬘s, ⬘ f 其 = 兵As1K0共s1r兲,
共21兲
where
␦2 =
tion, and stresses at the fluid-solid interface should be continuous. Thus, condition 共i兲 implies that we only have I0 in the inner fluid column, and condition 共ii兲 implies that we only have K0 in the outer formation. Hence, the general solutions of Eqs. 共16兲–共18兲 satisfying the boundary conditions 共i兲 and 共ii兲 can be written as
f = b2,3
2 2 c2s + c3s 2 c3s
c22f + c23f c23f
冋 冋
A3⬘ f = b3f A2⬘ f ,
A3f = b2f A2f ,
册 册
k2 −
2 s 2 2 2 − 共2,3兲 , c2s + c3s
k2 −
ı f 2 − 共2,3 兲 . c22f + c23f
In our present problem, the fluid column inside the micropolar solid formation is micropolar as well as viscous in nature. Since the micropolar viscous fluid can support couple stresses and shear stresses, therefore, both the shear and couple stresses must be taken into account while formulating the boundary conditions at the surface of cylindrical borehole. Thus, the boundary condition 共iii兲 implies that the radial displacement, microrotation, radial force stress, shear force stress, and couple stress across the fluid-solid interface must be continuous. Mathematically, these boundary conditions can be expressed as: At the fluid-solid interface, r = a s f rr = rr ,
s f rz = rz ,
mrs = mrf ,
urs = urf,
uzs = uzf, s2
= 2f .
共26兲
Using Eqs. 共5兲–共8兲, 共15兲, and 共23兲–共25兲 into the above boundary conditions given in Eq. 共26兲, we obtain a system of six homogeneous equations in six unknowns, namely, As1 , As2 , A2⬘s , A1f , A2f , and A2⬘ f , given by
再
关共s + 2s + Ks兲共s1兲2 − sk2兴K0共s1a兲 + 共2s + K s兲
冎
s1 K1共s1a兲 As1 + ık共2s + Ks兲共s2兲2K0⬙共s2a兲As2 a
再
+ ık共2s + Ks兲共s3兲2K0⬙共s3a兲A2⬘s + ı 关共 f + 2 f
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124905-4
J. Appl. Phys. 103, 124905 共2008兲
D. Singh and S. K. Tomar
+ K f 兲共1f 兲2 − f k2兴I0共1f a兲 − 共2 f + Kf兲
冎
1f I1共1f a兲 A1f − k共2 f + K f 兲共2f 兲2I0⬙共2f a兲A2f a
− k共2 f + K f 兲共3f 兲2I0⬙共3f a兲A2⬘ f = 0, − 共2 + K s
s
兲ıks1K1共s1a兲As1
+
s2关共s
+K
verge to the dispersion equation of Stoneley-type surface waves at micropolar solid/micropolar fluid interface. For this to be achieved, making ka → ⬁ and using the asymptotic expansions of the modified Bessel functions given by
共27兲 s
兲共s2兲2
K0共u兲 = K1共u兲 =
+ k
s 2
+ Ksbs2兴K1共s2a兲As2 + s3关共s + Ks兲共s3兲2 + sk2
=
+ Ksbs3兴K1共s3a兲A2⬘s − k共2 f + K f 兲1f I1共1f a兲A1f − ı2f 关共 f + K f 兲共2f 兲2 + f k2 + K f b2f 兴I1共2f a兲A2f − ı3f 关共 f + K f 兲共3f 兲2 + f k2 + K f b3f 兴I1共3f a兲A2⬘ f = 0,
bs2s2
冋
册
− s K1共s2a兲 + ␥ss2K1⬘共s2a兲 As2 a
+ bs3s3
冋
+ ıb3f 3f
冋 冋
册 册 册
+
f I1共2f a兲 − ␥ f 2f I1⬘共2f a兲 A2f a f I1共3f a兲 − ␥ f 3f I1⬘共3f a兲 A2⬘ f = 0, a ıks2K1共s2a兲As2
+
共29兲
as
u → ⬁,
Here, we investigated the dispersion relation given by Eq. 共33兲 numerically, for a specific model. Since this equation is an implicit functional relation of wavenumber and phase velocity of Stoneley waves, therefore one can proceed to find the variation of phase velocity with wavenumber. Once the phase velocity is computed at different wavenumbers, the corresponding group velocity Vg can be determined from the formula given by Vg = c + k
共30兲 ıkK0共s1a兲As1 − 共s2兲2K0共s2a兲As2 − 共s3兲2K0共s3a兲A2⬘s
= 0,
共31兲
bs2s2K1共s2a兲As2 + bs3s3K1共s3a兲A2⬘s + b2f 2f I1共2f a兲A2f 共32兲
For a nontrivial solution of these equations, the determinant of the coefficient matrix must vanish. This will provide us the frequency equation for the propagation of surface waves at the micropolar solid/micropolar fluid interface, given by 共33兲
where D is the determinant of the coefficient matrix 关amn兴6⫻6 of the homogeneous linear system of Eqs. 共27兲–共32兲. Here, the parameter F involves the geometrical and material constants. The entries of the matrix 关amn兴 in nondimensional form are given in the Appendix A. It can be noticed from Eq. 共33兲 that for a fixed value of parameter F, it is an implicit functional relationship between the wavenumber and the phase velocity. Moreover, some of the coefficients are involving complex quantities, therefore, it is expected that the relevant surface waves are dispersive and attenuated. For the waves of very short wavelengths, i.e., for sufficiently large values of ka, the dispersion 关Eq. 共33兲兴 will con-
dc . dk
For numerical computations, we take the following values of the relevant parameters for micropolar solid 共aluminum epoxy兲 and micropolar fluid as Symbol
− ıkI0共1f a兲A1f + 共2f 兲2I0共2f a兲A2f + 共3f 兲2I0共3f a兲A2⬘ f
D共k,c,F兲 = 0,
冑2u exp共u兲,
I0共u兲 = I1共u兲
it can be verified that the dispersion relation in Eq. 共33兲 reduces to Eq. 共24兲 of Singh and Tomar16 for the propagation of Stoneley waves at micropolar solid/micropolar fluid interface.
ıks3K1共s3a兲A2⬘s
+ 1f I1共1f a兲A1f + ık2f I1共2f a兲A2f + ık3f I1共3f a兲A2⬘ f = 0,
+ b3f 3f I1共3f a兲A2⬘ f = 0.
exp共− u兲, 2u
IV. NUMERICAL RESULTS AND DISCUSSION
− K1共s3a兲 + ␥ss3K1⬘共s3a兲 A2⬘s a s
+ ıb2f 2f
s1K1共s1a兲As1
共28兲
1
冑
s s Ks s ␥s js s f f Kf f ␥f jf f
Value 7.59⫻ 1010 dyn/ cm2 1.89⫻ 1010 dyn/ cm2 0.0149⫻ 1010 dyn/ cm2 0.0226⫻ 1010 dyn 0.0263⫻ 1010 dyn 0.00196 cm2 2.192 gm/ cm3 1.0⫻ 1010 dyn s / cm2 0.5⫻ 1010 dyn s / cm2 0.0110⫻ 1010 dyn s / cm2 0.0122⫻ 1010 dyn s 0.0126⫻ 1010 dyn s 0.00140 cm2 1.0 gm/ cm3
The radius a of the cylindrical borehole is taken as a = 10 cm, whenever not mentioned.
Since some of the entries of the determinant in Eq. 共33兲 are complex, therefore it is not analytically possible to find the roots of this equation for a given value of wavenumber. So, for a given value of wavenumber, Eq. 共33兲 is solved numerically by taking the above data of the physical parameters. Suppose the roots of Eq. 共33兲 lie along a smooth curve C in the phase velocity-wavenumber domain, then for a particular value of wavenumber k = k0, ∃ some c = c0 僆 C such that D共k0 , c0 , F兲 = 0. The dispersion relation c共k , F兲 for this mode is obtained by tracing the locus of the root in the c
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124905-5
D. Singh and S. K. Tomar
J. Appl. Phys. 103, 124905 共2008兲
FIG. 1. Comparison of real parts of phase and group velocities at different radii of the borehole.
FIG. 2. Comparison of imaginary parts of phase velocities at different radii of the borehole.
domain as k takes values greater than k0. We require that the dispersion curve c共k , F兲 僆 C should also be a smooth function of k in order to avoid a mix-up with other modes at possible points of degeneracy where different dispersion curves intersect. This notion of dispersion leads directly to a numerical method for computing modal dispersion curves practically. Starting from c0, one or two 共depending on whether c0 is an end point of C or not兲 sequences of sufficiently close phase velocity on C are computed. Using the initial guess, c共k0 , F兲 is determined by finding the zero of D共k , c , F兲 with the help of MATHEMATICA. Subsequently, stepping along k away from k0, all the samples of c共k , F兲 are computed for each k, using the value of c found at the previous wavenumber as initial guess. Thus the dispersion curve is obtained by this mode tracking procedure. In the present computation, we have computed nondimensional phase velocity 共c / Vs1兲 from Eq. 共33兲 at different real values of nondimensional wavenumber ka using MATHEMATICA. It is found that the phase velocity is complex in nature, which means that the concerned waves are not only dispersive but possess attenuation too. Figure 1 depicts the variations of the real parts of the nondimensional phase and group velocities with the real nondimensional wavenumber corresponding to two different values of the radii of the borehole. The solid curves correspond to the case when the radius of the borehole is 10 cm, while the dotted curves correspond to the case when the radius of the borehole is 20 cm. We observe that for small values of the nondimensional wavenumber, there is significant effect of the radius of the borehole. The increase in radius of the borehole results in decrease in the phase velocity of the surface waves. As the value of the nondimensional wavenumber increases and takes higher and higher values, the nondimensional phase velocity of the surface waves, for both the radii, tends to the same value and remains constant, which is equal to 0.588676. We also notice that the group
velocity is less than the phase velocity for small values of wavenumber for both the radii. However, for higher values of wavenumber, the values of group and phase velocities also tend to the same value. Figure 2 represents the variation of the imaginary parts of the nondimensional phase velocity versus nondimensional wavenumber at two different radii of the borehole. We see that at a given value of ka, the value of the imaginary part of the nondimensional phase velocity corresponding to a = 10 cm is greater than that of corresponding to a = 20 cm. However, corresponding to both the radii considered, the values of the imaginary parts of the phase velocity increase with increase of wavenumber. Since the imaginary part of the phase velocity is connected with attenuation of the corresponding waves, therefore, we may conclude that the concerned surface waves are more attenuated when the borehole radius is relatively small. Figures 3 and 4 show the effect of fluid viscosity on the dispersion curves corresponding to the real and imaginary parts of the nondimensional phase velocity. The solid curves in Fig. 3 correspond to low viscosity fluid and the dotted curves correspond to high viscosity fluid. For limitedly high and low viscous fluids, we have taken the numerical values of the relevant coefficient as f = 2.0⫻ 1010 dyn s / cm2 and f = 0.000 01⫻ 1010 dyn s / cm2, respectively. However, we have kept the density of the fluid to be fixed for both types of fluids, which may not be the same in general. These numerical values of the coefficient f are taken for computational purposes only. We see that the real part of the phase velocity for highly viscous fluid is greater than that for the low viscous fluid up to certain values of the nondimensional wavenumber ka. In Fig. 4, we have depicted the variation of the imaginary part of the phase velocity with the wavenumber at two different values of f , the viscosity of the fluid. The solid curve corresponds to highly viscous fluid and the dashed curve corresponds to low viscosity. We see that the
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J. Appl. Phys. 103, 124905 共2008兲
D. Singh and S. K. Tomar
FIG. 3. Comparison of real parts of phase velocities and group velocities at low and high fluid viscosity.
imaginary part of nondimensional phase velocity which corresponds to highly viscous fluid is greater than that which corresponds to low viscous fluid. Hence, we may conclude that the attenuation of the surface waves decrease with the decrease of the viscosity of the fluid. Figures 5 and 6 depict the variation of the real and imaginary parts of the phase velocity versus viscosity f , when a = 10 cm and ka = 5. We observe that the real and imaginary parts of the nondimensional phase velocity increase with the increase of f . Figures 7 and 8 depict the variation of the real and
FIG. 4. Comparison of the imaginary parts of phase velocities at low and high fluid viscosity.
FIG. 5. Variation of real part of phase velocity vs fluid viscosity 共 f 兲.
imaginary parts of the nondimensional phase velocity versus micropolarity K f , the micropolarity of the fluid. It is clear from these figures that both the parts of the phase velocity increase very slowly with K f . Hence, the effect of the micropolar parameter, K f on the dispersion curve is not appreciable as compared to the effect of viscous parameter, f on the dispersion curve. V. CONCLUSIONS
A problem of propagation of Stoneley-type waves at the surface of a cylindrical bore hole situated in an infinite micropolar elastic medium is investigated. The cylindrical bore
FIG. 6. Variation of imaginary part of phase velocity vs fluid viscosity 共 f 兲.
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J. Appl. Phys. 103, 124905 共2008兲
D. Singh and S. K. Tomar
共3兲 On the real part of the phase velocity of the surface waves, the effect of viscosity is found to be more dominant as compared to micropolarity of the fluid column. It is found that the higher the viscosity of the fluid, the slower the phase velocity of the surface waves. 共4兲 The phase and the group velocities are found to be affected only at small values of the wavenumber, while at higher values of the wavenumber, both the phase and group velocities are found to be same and constant. This constant phase velocity corresponds to the wave speed of Stoneley wave at micropolar solid/fluid interface. 共5兲 For a given value of the wavenumber, the imaginary part of the phase velocity at a = 10 cm is found to be greater than that of at a = 20 cm. ACKNOWLEDGMENTS
FIG. 7. Variation of real part of phase velocity vs micropolarity of fluid 共K f 兲.
hole is assumed to be vertical and filled with micropolar viscous fluid. Using appropriate boundary conditions, the frequency equation corresponding to the surface wave propagation is derived and solved numerically for a particular model. From the present analysis, it can be concluded that: 共1兲 The frequency equation corresponding to surface wave propagation is found to be dispersive and attenuated in nature. 共2兲 The increase in radius of the borehole results in decrease in the phase velocity of the surface waves.
One of the authors 共D.S.兲 is thankful to the Council of Scientific and Industrial Research, New Delhi for providing financial assistance in the form of Senior Research Fellowship to complete this work. The authors are thankful to the unknown reviewer for his valuable suggestions on the manuscript. APPENDIX A
The nondimensional entries of the matrix amn in the Eq. 共33兲 are given by
兵关共 +2 +K 兲共s1a兲2 − 共ka兲2兴K0共s1a兲 + 共 2+K 兲 ⫻共s1a兲K1共s1a兲其 , s
a11 = 共ka兲
s
s
s
s
s
s
兵关共 +2 +K 兲共1f a兲2 − 共ka兲2兴I0共1f a兲 − 共 2 +K 兲共1f a兲I1共1f a兲其 , f
a14 = ı共ka兲
f
f
f
s
f
s
f
s
a1i = ı共ka兲
共 2+K 兲共si a兲2K0⬙共si a兲, s
s
s
a1j = − 共ka兲
f f a兲2I0⬙共 j−3 a兲, 共 2 +K 兲共 j−3 f
f
s
a21 = − ı共ka兲2共s1a兲
关
a2i = 共si a兲 共si a兲2
关 2+K 兴K1共s1a兲, s
s
s
共 +K 兲 + 共ka兲2 + 共bsi a2兲 K 兴K1共si a兲, s
s
s
a24 = − 共1f a兲共ka兲2共1f a兲
f
f
s
关
共 +K 兲 + 共ka兲2
兴
f
f f + 共b j−3 a2兲 Ks I1共 j−3 a兲,
a3i = 共bsi a2兲共si a兲
s
s
共 2 +K 兲I1共1f a兲,
f f a2j = − ı共 j−3 a兲2 a兲 共 j−3 f
s
s
f
f
s
s
a31 = 0,
关 −a K1共si a兲 + ␥a 共si a兲K1⬘共si a兲兴 , s
s
s 2
s 2
a34 = 0, f a兲 关  a I1共 j−3 f f − ␥ a 共 j−3 a兲I1⬘共 j−3 a兲兴 ,
f f a2兲共 j−3 a兲 a3j = ı共b j−3
FIG. 8. Variation of imaginary part of phase velocity vs micropolarity of fluid 共K f 兲.
f
s 2
f
s 2
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J. Appl. Phys. 103, 124905 共2008兲
D. Singh and S. K. Tomar
a41 = 共ka兲共s1a兲K1共s1a兲,
a4i = ı共ka兲共si a兲K1共si a兲,
a44 = 共ka兲共1f a兲I1共1f a兲, f a兲I1共2f a兲, a4j = ı共ka兲共 j−3
a51 = ı共ka兲2K0共s1a兲, a5i = − 共si a兲2K0共si a兲, a54 = − ı共ka兲2I0共1f a兲,
f f a兲2I0共 j−3 a兲, a5j = 共 j−3
a61 = 0,
a6i = 共bsi a2兲共si a兲K1共si a兲,
a64 = 0,
f f f a6j = 共b j−3 a2兲共 j−3 a兲I1共 j−3 a兲
for i = 2,3, and j = 5,6.
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