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Peristaltic pumping of a particulate suspension under long wavelength approximation.

Introduction

Fluid transport through flexible tubes by means of progressive wave of area contraction or expansion, which propagates along the length of the distensible tube, has been the subject of scientific and engineering research since the first investigation of Latham (1966). Physiologists term the phenomenon of such transport as peristalsis. It is well known that the pumping of fluids through muscular tubes by means of peristaltic waves is an important biological mechanism. Most of the early studies on peristalsis were concerned with ureteral physiology, although the ureteral transport has not been the only motivation for the study of peristaltic flow. Besides its role in physiology, some aquatic animals use peristalsis as a means of locomotion. Finger and roller pumps also use this mechanism which is used to transport blood, slurries, corrosive fluids and foods, etc. whenever it is required to prevent the transported fluid from coming in contact with the mechanical parts of the pump. Jaffrin and Shapiro (1971) explained the basic principles of peristaltic pumping and brought out clearly the significance of the various parameters governing the flow. The literature on the topic is quite extensive by now and an account of most of the experimental and theoretical investigations reported up to the 1983, arranged in various details was presented by Srivastava and Srivastava (1984). The important contributions to the subject between the years 1984 and 1994 are well referenced in Srivastava and Saxena (1995). The recent years investigations include the studies of Srivastava and Srivastava (1997), Mekheimer et al. (1998), Muthu et al. (2001), Srivastava (2002), Misra and Pandey (2002), Hayat et al. (2002,2003), Mekheimer (2002,2003), Hayat et al. (2004), Misra and Rao (2004), Hayat et al. (2005), Hayat and Ali (2006a,b), Srivastava (2007), Medhavi and cowrkers (2008a,b,2009), Hayat and coworkers (2008a,b), Ali and Hayat (2008), etc.

The study of the theory of particle-fluid mixture is very useful in understanding a number of diverse physical problems (Srivastava and Srivastava, 1989). Recently, interest is developing in applying the two-phase theory to physiological flows as it provides an improved understanding of subjects such as the diffusion of protein, the rheology of blood, the swimming of micro-organism, and the particle deposition on respiratory tract, etc. A good number of studies on two-phase flows with and without peristalsis have been well referred by Srivastava and Srivastava (1989).

Hung and Brown (1976), in order to investigate various dynamic and geometric effects on the particle transport in fluid, was the first to study the problem of peristaltic transport through the details of an experimental work in a two-dimensional vertical channel with flexible walls on which a peristaltic wave of finite amplitude to the wavelength ratio was imposed. Since then several researchers have attempted the peristaltic transport problem involving two-phase fluid which include the investigations of Brown and Hung (1977), Kaimal (1978), Takabatake and Ayakawa (1982), Srivastava and Srivastava (1989, 1997), Mekheimer et al. (1998), Srivastava (2002), Medhavi and Singh (2008b,2009), and a few others.

Barring a few, most of the studies conducted in the literature on two-phase peristaltic transport have been carried out in free pumping case (Fung and Yih, 1968). The free pumping case may occur in certain engineering applications but in physiological flows the initial flow is either Poiseuille or pulsatile. The purpose of this paper is to study the peristaltic pumping of a particle-fluid suspension in a two-dimensional channel. A viscous incompressible fluid, in which small spherical rigid particles are suspended, is assumed to be confined in a channel with flexible walls upon which symmetric traveling transverse waves are imposed. The motion of two phases is studied with and without externally imposed pressure gradient under the assumption that the wavelength is long compared to the mean half width of the channel and that Reynolds number is of order unity which corresponds to the assumption that the frequency of the peristaltic wave is small compared to the reciprocal of a characteristic time for the vorticity diffusion. The study is finally aimed at possible application to urine flow from Kidney to bladder (infected) in human ureter.

Formulation of the Problem

Consider a two-dimensional infinite channel of mean width 2d (Fig.1), filled with a mixture of small spherical particles in an incompressible Newtonian Viscous fluid. The walls of the channel are flexible on which are imposed traveling sinusoidal waves of small amplitude. Using a continuum approach, the equations governing conservation.

[FIGURE 1 OMITTED]

of mass and linear momentum in terms of stream functions for both the fluid and particulate phase, after eliminating the pressure term are expressed (Drew, 1979; Srivastava and Srivastava, 1989) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

= q [F.sub.o] [[DELTA].sup.2] ([PSI] - [[PSI].sup.p]), (2)

where [[DELTA].sup.2] denotes the Laplacian operator ([[partial derivative].sup.2] / [partial derivative] [x.sup.2] + [[partial derivative].sup.2] / [partial derivative] [y.sup.2]) and the subscripts x and y stand for partial derivatives, x and y are the Cartesian coordinates with x measured in the direction of wave propagation and y measured in the direction normal to the mean position of the channel walls, ([[rho].sub.f], [[rho].sub.p]) be the actual densities of the materials constituting (fluid, particle) phase respectively, (1-q) [[rho].sub.f] is the fluid phase density and q [[rho].sub.p] is the particle phase density, q denotes the volume fraction density which is assumed to be constant (Drew, 1979, Srivastava and Srivastava, 1989) [rho] is the pressure already eliminated, [[mu].sub.s] (q) is the mixture viscosity, [F.sub.o] is the drag coefficient of interaction for the force exerted by one phase on the other, T is the time, [PSI] and [[PSI].sup.p] are stream functions for the fluid and particulate phase respectively, and are expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

with (U,V) and ([U.sub.p], [V.sub.p]) are the velocity components for the fluid and particulate phase respectively.

The expression for the drag coefficient of interaction, [F.sub.o] and the empirical relation for the viscosity of suspension, [[mu].sub.s] (q) for the present problem are selected (Srivastava and Srivastava, 1997) as

[F.sub.o] = 4.5 ([[mu].sub.o] / [a.sup.2.sub.o]) {4 + 3 [(8q - 3 [q.sup.2]).sup.1/2] + 3q / [(2 - 3q).sup.2],} (4)

[[mu].sub.s] [congruent to] [[mu].sub.s] (q) = [[mu].sub.o]/1 - rq,

r = 0.07 exp [2.49q + 1107 / T' exp (-1.6 9 q)], (5)

with T' measured in absolute temperature ([sup.o]K).

The boundary conditions that must be satisfied by the fluid on the walls of the channel are the no-slip and impermeability conditions. The walls of the channel are assumed to be extensible with a traveling sinusoidal wave, and the displacement in the channel walls is in the transverse direction only. The appropriate boundary conditions are therefore stated as

U = [[PSI].sub.X] = O; on Y = [eta] (X,T),

V = - [[PSI].sub.X] = [+ or -] [partial derivative] [eta] / [partial derivative] T

[V.sup.p] = [[PSI].sup.p.sub.X] = [+ or -] [partial derivative] [eta]/[partial derivative] T. (6)

The transverse displacement, [eta] (X, T) of the walls is given as

[eta] (X,T) = [+ or -] {d + cos 2[pi] / [lambda] (X - CT),} (7)

with a is the amplitude, [lambda] is the wavelength and C is the wave propagation speed of the peristaltic wave.

Introducing the following dimensionless variables

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where [v.sub.s] = [[mu].sub.s] / [[rho].sub.f] and R is the suspension Reynold's number, into equations (1)-(3), (6) and (7), yields the following set of equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

= q kp ([[delta].sup.2] [[partial derivative].sup.2] / [partial derivative] [x.sup.2] + [[partial derivative].sup.2] / [partial derivative] [y.sup.2]) ([psi] - [[psi].sup.p]), (10)

n = [+ or -] [1 + [[epsilon] cos 2 [pi] (x - t)], (11)

and

[[psi].sub.y] = O,

[[psi].sub.x] = [[psi].sup.p.sub.x] = {- or +] 2 [pi] [member of] sin 2[pi] (x - t), (12)

where [k.sub.f] = [F.sub.o] d/C [[rho].sub.f] and [k.sub.p] = [F.sub.o] d/C [[rho].sub.p]. The equations for the axial pressure gradient for fluid and particulate phase are obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Analysis

A long wave approximation is now applied to obtain the solution for stream functions as a power series in [delta], by expanding [psi], [[psi].sup.p] and [bar.P] under the limit [delta] <<1, in the form

[psi] (x, y, t; [delta], [epsilon], R) ~ [[psi].sub.o] (x, y, t; [epsilon], R) + [delta] [[psi].sub.1] (x, y, t; [epsilon], R) + [[delta].sup.2] [[psi].sub.2] (x, y, t; [epsilon], R) +...., (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

[bar.P] ~ [[bar.P].sub.o] + [delta] [[bar.P].sub.1] + [[delta].sup.2] [[bar.P].sub.2] + ...., [bar.P] = [partial deriviate] p/[partial derivative]x / [[rho].sub.f] [C.sup.2]/d). (17)

The first term on the right hand side of equation (17) corresponds to the imposed pressure gradient associated with the primary flow and the other terms correspond to the peristaltic motion or higher imposed pressure gradient. Substitution of equations (15)-(17) into equations (9)-(14), yields three sets of coupled differential equations with their corresponding boundary conditions in [[psi].sub.o], [[psi].sub.1], [[psi].sub.2]; [[psi]sup.p.sub.o], [[psi].sup.p.sub.1], [[psi].sup.p.sub.2] and [bar.P.sub.o], [bar.P.sub.1], [bar.P.sub.2] for the first three powers of [delta].

For Fluid Phase

1/R [[psi].sub.oyyyy] = q [k.sub.f] [([[psi].sup.p.sub.o] - [[psi].sub.o]).sub.yy], 1/R [[psi].sub.lyyyy] = (1 - q) ([[psi].sub.otyy] + [[psi].sub.oy] [[psi].sub.oxyy] - [[psi].sub.ox] [[[psi].sub.oyyy]), (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

For Particle Phase

q [k.sub.p] [([[psi].sub.o] - [[psi].sup.p.sub.o]).sub.yy] = O, (21)

q kp [([[psi].sub.1] - [[psi].sup.p.sub.1]).sub.yy] = q ([[psi].sup.p.sub.otyy] + [[psi].sup.p.sub.oy] [[psi].sup.p.sub.oxyy] - [[psi].sup.p.sub.oyyy]), (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

Axial Pressure Gradient for the Fluid Phase

(1-q) [[bar.P].sub.o] = 1/R [[psi].sub.oyyy] + q [k.sub.f] [([[psi].sup.p.sub.o] - [psi]).sub.y], (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

Axial Pressure Gradient for the Particulate Phase

q [[bar.P].sub.o] = q [k.sub.f] [([[psi].sub.o] - [[psi].sup.p.sub.o].sub.y], (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

The corresponding boundary conditions for the above equations are obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

Zeroth Order Solution

The zeroth order differential equations in [[psi].sub.o], [[psi].sup.p.sub.o] and [[bar.P].sub.o], subject to the steady parallel flow and transverse symmetry assumption for a time dependent pressure gradient in the x-direction, yields the following

[[psi].sub.o] = [[bar.A].sub.1] y + [[bar.A].sub.3] [y.sup.3], (31)

[[psi].sup.p.sub.o] = [[psi].sub.o] - [C.sub.p] (t)y, (32)

[[bar.P].sub.o] = 6/R [[bar.A].sub.3], (33)

where [[bar.A].sub.1] and [[bar.A].sub.3] are functions of x and t, and are given as

[[bar.A].sub.1] = - 3 C (t) - [epsilon]/2 cos [beta] / [eta], [C.sub.p] (t) = [[bar.P].sub.o] / [k.sub.f], [[bar.A].sub.3] = C(t) - [epsilon]/2 cox [beta] / [[eta].sup.3], (34)

where [beta] = 2 [pi] (x-t) and [eta] = 1+ [epsilon] cos [beta]. C (t) is the constant of integration and is proportional to the time dependent applied pressure gradient.

The expressions for the zeroth order velocity profiles for the fluid and the particulate phase are therefore given as

[u.sub.o] = [[psi].sub.oy] = [[bar.A].sub.1] + 3 [[bar.A.sub.3] [y.sup.2], (35)

[u.sup.p.sub.o] = [[psi].sup.p.sub.oy] = [u.sub.o] - [C.sub.p] (t). (36)

The zeroth order flow rate, [Q.sub.o] is now calculated as

[Q.sub.o] = (1 - q) [[psi].sub.o] = [absolute value of y = [eta] + q [[psi].sup.p.sub.o]] y = [eta] = [[bar.A].sub.1] [eta] + [[bar.A].sub.3] [[eta].sup.3] - q [C.sub.p] (t) [eta], (37)

where [[bar.A].sub.1] and [[bar.A].sub.3] assume the same values as in (34).

Time- averaged Flow Values for Zeroth Order Solutions

The time-averaged flow variables [[bar.u].sub.o], [[bar.u].sup.p.sub.o] and [[??].sub.o] are obtained by averaging the corresponding values over one period of the wave as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (38)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (39)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (40)

where [bar.Q].sub.o] is the time averaged mean flow over one period of the wave and [S.sub.1] = 1/R kf.

The dimensionless time averaged mean pressure is therefore obtained as

[[??].sub.o] / ([[??].sub.o]) [[bar.Q].sub.o] = o = 1 - 2/3 (1/2 + 1/[[epsilon].sup.2]) [[bar.Q].sub.o]. (41)

First Order Solutions

Now solving the first order equations form, [[psi].sub.1], [[psi].sup.p.sub.1] and [[bar.P].sub.1], subject to the boundary conditions given in (30), and one obtains

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (42)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (43)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (44)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (45)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (46)

where [K.sub.p] (t) = [[bar.P].sub.o] / [k.sub.p].

The expressions for the first order velocity profiles are obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (47)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

The volume flow, Q1 due to the pressure gradient of O ([delta]) is now obtained as

[Q.sub.1] = (1-q) [[psi].sub.1] [absolute value of y = [eta] + q [[psi].sup.p.sub.1] y = [eta] = R K (t) + q Kp (t). (49)

Zero Mean Values for the First Order Solutions

We now attempt to investigate the first order solutions for zero mean flow. In this case C (t) = [C.sub.p] (t) = K (t) = [K.sub.p] (t) = O, and upon substituting the zero mean quantities simplified under the limit [member of] [right arrow] O (i.e., ignoring the terms of order [[epsilon].sup.3] and above) in the equations (47), (48) and (44), one determines

[u.sub.1] = R [1 - q (1 - [[rho].sub.p] / [[rho].sub.f])] ((- 1/40 [y.sup.6] + 3/4 [y.sup.4] - 159/280 [y.sup.2] + 3/70)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (50)

- (3 [y.sup.2] + [[eta].sup.2]) [epsilon] [pi] sin [beta], (51)

[bar.P].sub.1] = 12/5 [1 - q (1 - [[rho].sub.p]/[[rho].sub.f])] [12/7 [[epsilon].sup.2] [pi] sin 2 [beta] - [epsilon] [pi] sin [beta]. (52)

Integration of equations (50)-(52), now yields

[[bar.u].sub.1] = [[bar.u].sup.p.sub.1] = [[??].sub.1] = O. (53)

This concludes that the peristaltic motion contributes nothing to the mean axial velocity and mean pressure gradient is of O ([delta]).

Second Order Solution

Next, solving equations (20), (23) and (26) under the boundary conditions (30), yields the following solutions for the second order approximation as

1/R [[psi].sub.2] = [a.sub.1] [y.sup.11] + [a.sub.2] [y.sup.9] + [a.sub.3] [y.sup.7] + [a.sub.4] [y.sup.5] + [z.sub.3] [y.sup.3] + [z.sub.1] y, (54)

[[psi].sup.p.sub.2] = [[psi].sub.2] - 1 / [k.sub.p] ([b.sub.1] [y.sup.9] + [b.sub.2] [y.sup.4] + [b.sub.3] [y.sup.5] + [b.sub.4] [y.sup.3]) + wy, (55)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (56)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

Zero Mean Flow values For Second Order solutions

Similar to the case of first order solutions, the second order values are now investigated for the case for zero mean flow. Thus, C (t) = K (t) = [C.sub.p] (t) = [K.sub.p] (t) = O. Again to simplify the algebra, the quantities are investigated under the limit [epsilon] [right arrow] O by neglecting all terms of orders O ([[epsilon].sup.3]. Upon integrating the expressions for [u.sub.2], [u.sup.p.sub.2] and [bar.P].sub.2] thus obtained over one period of the wave, yields the following results for zero mean flow as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (58)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (59)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (60)

Results, Discussion and Concluding Remarks

It has been observed that bacteria or other materials some time passes from the bladder to the kidney or from one Kidney to the other in opposite direction to the urine flow. The phenomenon of such backward flow is called reflux and physiologists term this as 'uretral reflux". Two different definitions of reflux exist in the literature; Shapiro et al. (1969) call a flow reflux when there is a negative net displacement of a particle trajectory, while Yin and Fung (1971) define a flow reflux whenever there is a negative mean velocity in the flow field. In the present analysis the later definition of reflux is adopted.

Since it has already been shown above that the case of zero mean flux corresponds to an adverse mean pressure gradient, it appears likely that the reflux might occur in this case. From equation (38), it is noticed that in the absence of [bar.Q].sub.o], the mean axial velocity, [u.sub.o] along the axis (i.e., at y = O) is always negative for all [member of] (0 < [member of] < 1). Thus, the conditions that will make [bar.u]sub.o] negative are essentially the reflux conditions. At y = O, equation (35) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (61)

where, for brevity [omega] = q / [S.sub.1] (1 + [[epsilon.sup.2] / 2) [(1 - [[epsilon].sup.2]).sup.-5/2] is used.

An inspection of equation (61), establishes the following criteria for reflux as

(i) O [less than or equal to] [bar.Q].sub.o] < 1/2 (2 + [omega]) [1 - [(1 - [[epsilon].sup.2]).sup.-1/2]] - 3/4 [[epsilon].sup.2] / 1 + [[epsilon].sup.2] / 2. (62)

This range of [bar.Q].sub.o] makes the values of [u.sub.o] negative, a condition for backward flow at y = O (Fig. 2 (a)).

[FIGURE 2 (a) OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (63)

is the critical value below which reflux always occurs (Fig.2 (b).

[FIGURE 2(b) OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Corresponding to this [bar.Q].sub.oc], the critical mean pressure gradient [[??].sub.oc], is obtained by replacing [[bar.Q].sub.o] by the value of [bar.Q].sub.oc] from equation (63) in equation (37) as

[[??].sub.oc] 3/R [(1 - [[epsilon].sup.3]).sup.-5/2] [1 + [[epsilon].sup.2]/2 - [(1 - [[epsilon].sup.2]).sup.-1/2], (64)

which is a positive quantity.

(iii) 3 [[epsilon].sup.2] (2 + [omega])) > [bar.Q].sub.o] > 1/2 (2 + [omega]) [1 - [(1 - [[epsilon].sup.2]).sup.-1/2] - 3/4 [[epsilon].sup.2] / (1 + [[epsilon].sup.2] / 2) [omega]. (65)

This range of [[bar.Q].sub.o] will never result in to reflux (Fig. 2 (c).

[FIGURE 2 (c) OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (66)

This value of [[bar.Q].sub.o] will yield a uniform velocity profile and a zero mean resultant pressure gradient (Fig. 2 (d)).

[FIGURE 2(d) OMITTED]

[[bar.Q].sub.o] = 3 [[epsilon].sup.2] (2 + [omega]) / (2 + [[epsilon].sup.2]) = [[bar.Q].sub.op]. (v) [[bar.Q].sub.o] > 3 [[epsilon].sup.2] (2 + [omega]) / 2 + [[epsilon].sup.2], (67)

is the stage where the velocity profile shows an opposite curvature (Fig. 2 (e)).

[FIGURE 2(e) OMITTED]

[[bar.Q].sub.o] = 3 [[epsilon].sup.2] (2 + [omega]) / (2 + [[epsilon].sup.2]) (vi ) [[bar.Q].sub.o] >> 3 [[epsilon].sup.2] (2 + [omega]) / 2 + [[epsilon].sup.2]. (68)

[[bar.Q].sub.o] gets exceedingly large, reflux occurs near the walls (Fig. 2 (f)).

[FIGURE 2(f) OMITTED]

[[bar.Q].sub.o] >> 3 [[epsilon].sup.2] (2 + [omega])/(2 + [[epsilon])

Next, it should be noted that these criteria apply to the case of finite amplitude ratio, i.e., [epsilon] > O. In the absence of peristalsis, i.e., for [epsilon] = O, any finite positive [[bar.Q].sub.o] will result in an exact Poiseuillian profile, hence reflux will never occur.

Various results obtained above for the fluid phase of the suspension are in good agreement with those of Zien and Ostrach (1970), with slight modifications. This arises due to the interacting term between the two phases. For the particle free flow (i.e., q = o), the results obtained above for fluid phase reduce to the same results as Zien and Ostrach (1970).

It is also observed, that for the zeroth order solutions of the fluid phase, the peristaltic waves for the case of mean zero volume flux correspond to an 'adverse' mean pressure gradient and reflux occurs near the plane of symmetry. Hence for reflux not to occur there must be an additional pressure gradient. This additional pressure gradient measured in term of the man volume flow [[bar.Q].sub.o] is given in the various criteria for reflux conditions.

The higher order solutions for the fluid phase show the effects of long wavelength peristaltic waves on the man flow quantities. For the first order solution, the peristaltic motion of the walls does not produce any axial velocity for the zero man volume flux, whereas, for the second order solution, it gives rise to a mean axial velocity and a mean axial pressure gradient.

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Amit Medhavi and U. K. Singh

Department of Mechanical Engineering, Kamla Nehru Institute of Technology, Sultanpur-228118, India
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Author:Medhavi, Amit; Singh, U.K.
Publication:International Journal of Dynamics of Fluids
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Date:Dec 1, 2009
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