Non-linear peristaltic motion of a Carreau fluid under the effect of a magnetic field in an inclined planar channel.

Reddy, M.V. Subba ; Gangadhar, K.

Introduction

The mechanics of peristaltic has been examined by a number of investigators. Latham [12] discussed for the first time about peristalsis in his thesis. Later, Fung and Yih [9] and Shapiro et al. [15] worked on very similar lines. Lew et al. [13] suggested chyme in the small intestine as a non-Newtonian fluid. Bohme and Friedrich [4] discussed the peristaltic flow of a viscoelastic liquid assuming that the relevant Reynolds number to be small enough to neglect inertia forces and ratio of the wave length and channel hight to be large which implies that the pressure is constant over the cross-section. Pozrikids [14] considered peristaltic flow under the assumption of creeping motion and used boundary integral method for Stokes flow. Srivastava and Srivastava [17,18] showed the effects of power-law fluid in uniform and non-uniform tubes and in a channel under zero Reynolds number and long wavelength approximations. Siddiqui and Schwarz [16] illustrated the peristaltic flow of a second order fluid in tubes and used a perturbation method to second order in dimensionless wave number. El Misery et al. [5] studied peristaltic transport of Carreau fluid through a uniform channel, under zero Reynolds number and long wave length approximations. El Shahawey et al. [7] investigated peristaltic transport of Carreau fluid through non-uniform channel, under zero Reynolds number and long wave length approximations. El Shehaway et al. [8] analyzed peristaltic pumping of Carreau fluid through a porous medium in a channel. Abd El Hakeem et al. [1] investigated separation in the flow through peristaltic motion of a Carreau fluid in an axisymmetric tube. Ali and Hayat [3] investigated peristaltic motion of a Carreau fluid in an asymmetric channel.

Agrawal and Anwaruddin [2] studied the effect of moving magnetic field on blood flow. They studied a simple mathematical model for blood through an equally branched channel with flexible outer walls executing peristaltic waves. The result revealed that the velocity of the fluid increases with an increase in the magnetic field. Hayat et al. [10] discussed the peristaltic flow of a MHD third order fluid in a planar channel. The peristaltic flow of a MHD fourth grade fluid in a planer channel has studied by Hayat et al. [11]. Peristaltic transport of a Johnson-Segalman fluid under the effect of a magnetic field was developed by Elshahed and Haroun [6].

However, the interaction of the magnetic field with peristaltic flow of a Carreau fluid in an inclined symmetric channel has received little attention. Hence, an attempt is made to model the peristaltic flow of a Carreau fluid in an inclined symmetric channel under the long wavelength and low Reynolds number assumptions, in the presence of a transverse magnetic field. The flow is examined in a wave frame of reference moving with velocity of the wave. A regular perturbation technique is employed to solve the present problem and solutions are expanded in a power of small Weissenberg number. Expressions for the velocity, axial pressure gradient, pressure rise and frictional force over a one wavelength are obtained. The effects of various emerging parameters on pumping characteristics and frictional forces are discussed in detail.

Mathematical Formulation

A two-dimensional flow of an incompressible electrically conducting Carreau fluid in an inclined symmetric channel with inclination a induced by sinusoidal wave trains propagating with constant speed along the channel walls is considered. A uniform magnetic field [B.sub.0] applied in the transverse direction to the flow. The fluid is assumed to be of small electrically conductivity so that the magnetic Reynolds number is very small and hence the induced magnetic field is negligible in comparison with the applied magnetic field. The external electric field is zero and the electric field due to polarization of charges is also negligible. Heat due to Joule dissipation is neglected. Fig. 1 represents the physical model of the flow field.

The equation of the channel walls is given by

Y = [+ or -]H (X, t) = [+ or -]a [+ or -]b sin 2[pi]/[lambda](X-ct), 2.1)

where b, [lambda], c and a are amplitude, wave length, phase speed of the wave, mean-half width of the channel respectively, t is the time and (X,Y) are the Cartesian coordinates in a fixed frame.

[FIGURE 1 OMITTED]

We introduce a wave frame of reference (x, y) moving with the velocity c in which the motion becomes independent of time when the channel length is an integral multiple of the wave length and the pressure difference at the ends of the channel is a constant (Shapiro et al [15]).

The transformation from the fixed frame of reference (X,Y)to the wave frame of reference (x, y) is given by

x = X - ct, y = Y, u = U - c, v = V, p'(x) = P'(X,t). (2.2)

where (u, v) and (U,V) are the velocity components, p and P are the pressures in the wave and fixed frames of reference respectively.

The constitute equation for a Carreau fluid is

[tau] = - [[[eta].sub.[infinity]] + ([[eta].sub.0]+[[eta].sub.[infinity]])[(1+[[GAMMA][??]).sup.2]).sup.n-1/2][??] (2.3)

where [tau] is the extra stress tensor, [[eta].sub.[infinity]] is the infinite shear rate viscosity, [[eta].sub.0] is the zero shear rate viscosity, [GAMMA] is the time constant, n is the dimensionless power-law index and [??] is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

where [pi] is the second invariant of strain-rate tensor. We consider in the constitutive Equation (2.3) the case for which [[eta].sub.[infinity]] = 0 and so we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

The above model reduces to a Newtonian model for n = 1 (or) [GAMMA] = 0. The equations governing the flow field, in the wave frame of reference are

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 (2.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

where [rho] is the density, [sigma] - the electrical conductivity, [[mu].sub.e]--the magnetic permeability and [B.sub.0]--constant transverse magnetic field.

Due to symmetry, the problem is studied only for upper half of the channel. The boundary conditions for the velocity are

[partial derivative]u/[partial derivative]y = 0 at y = 0 (2.9)

u = -c at y = H (2.10)

In order to write the governing equations and the boundary conditions in dimensionless form, the following non-dimensional quantities are introduced.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

where Re is the Reynolds number, We - Weissenberg number and [delta])- the wave number.

In view of Equation (2.11), the Equations (2.6) - (2.8), after dropping bars, reduce to

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 (2.12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.14)

and Da = k/[a.sup.2] and M = a[[mu].sub.e][B.sub.0][square root of ([sigma]/[[eta].sub.0])] are the Darcy number and Hartman number respectively.

Under lubrication approach, neglecting the terms of order [sigma] and Re, from Equations (2.13) and (2.14), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.15)

[partial derivative]p'/[partial derivative]y = -[[eta].sub.1]cos[alpha] (2.16)

The corresponding dimensionless boundary conditions in wave frame of reference are given by

[partial derivative]u/[partial derivative]y = 0 at y = 0 (2.17)

u = -1 at y = h = 1 + [phi]cos2[pi]x, (2.18)

Let p = p'(x)-[[eta].sub.1](cos[alpha])y [partial derivative]p/[partial derivative]x = [partial derivative][p.sup.1] [partial derivative]x. Then Equations (2.15) and (2.16) become

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.19)

[partial derivative]p/[partial derivative]y = 0 (2.20)

Equation (2.20) implies that p [not equal] p(y) . Therefore Equation (2.19) can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.21)

The volume flow rate q in a wave frame of reference is given by

q = [[integral].sup.h(x).sub.0] udy (2.22)

The instantaneous flux Q(X,t) in a fixed frame is

Q(X,t) = [[integral].sup.h.sub.0] Udy = [[integral].sup.h.sub.0] (u=1)dy = q + h. (2.23)

The time average flux [bar.Q] over one period T(=[lambda]/c) of the peristaltic wave is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.24)

Solution of the Problem

The Equation (2.21) is non-linear and its closed form solution is not possible. Hence, we linearize this equation in terms of [We.sup.2] , since We is small for the type of flow under consideration. So we expand u, p and q as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

Substituting (3.1) in the Equation (2.21) and in the boundary conditions (2.17) and (2.18) and equating the coefficients of like powers of [We.sup.2] to zero and neglecting the terms of [We.sup.4] and higher order, we get the following equations:

Equation of order [We.sup.0]

[dp.sub.0]/dx - [eta]sin[alpha] = [[partial derivative].sup.2][u.sub.0]/[partial derivative][y.sup.2] - [M.sup.2] ([u.sub.0]+1) (3.2)

and the respective boundary conditions are

[partial derivative][u.sub.0]/[partial derivative]y = 0 at y = 0 (3.3)

[u.sub.0] = -1 at y = h (3.4)

Equation of order [We.sup.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

and the respective boundary conditions are

[partial derivative][u.sub.1]/[partial derivative]y = 0 at y = 0(3.6)

[u.sub.1] = 0 at y = h (3.7)

Solving the Equation (3.2) by using the boundary conditions (3.3) and (3.4), we get

[u.sub.0] = 1/[M.sup.2] [[dp.sub.0]/dx - [eta]sin[alpha]](cosh My/cosh Mh-1)-1 (3.8)

and the volume flow rate [q.sub.0] is given by

[q.sub.0] = [[integral].sup.h.sub.0] [u.sub.0]dy = 1/[M.sup.3]([dp.sub.0]/dx - sin[alpha])[sinh Mh-Mh cos Mh / cosh Mh]-h (3.9)

From Equation (3.9), we get

[dp.sub.0]/dx = [M.sup.3] [([q.sub.0]+h)cosh Mh/sinh Mh-Mh cosh Mh] + [eta] sin [alpha]. (3.10)

Solving the Equation (3.5) by using the Equation (3.8) and the boundary conditions (3.6) and (3.7), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the volume flow rate [q.sub.1] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Equations (3.12) and (3.10), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting Equations (3.8) and (3.11) into the Equation (3.1) and using the relation [dp.sub.0]/dx = dp/dx - [We.sup.2] [dp.sup.1]/dx and neglecting terms greater than O ([We.sup.2]), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.14)

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.15)

The dimensionless pressure rise and frictional force per one wavelength in the wave frame are defined, respectively as

[DELTA]p = [[integral.sup.1.sub.0] dp/dx dx (3.16)

and

F = [[lambda].sup.1.sub.0] h (-dp/dx)dx. (3.17)

Results and Discussion

In order to get a the physical insight of the problem, pressure rise and friction force per one wavelength are computed numerically for different values of the emerging parameters, viz., Weissenberg number We, power-law index n, amplitude ratio [phi], inclination angle [alpha] and gravity parameter [eta] and are presented in figures 2-13.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

The variation of pressure rise [DELTA]p with time averaged flux [bar.Q] for different values of Weissenberg number We with [phi] = 0.6, M = 1, [eta] = 1, [alpha] = [pi]/6 and n = 0.398 is shown in Fig. 2. It is observed that, both the pumping ([DELTA]p > 0) and free pumping decrease with an increase in We, whereas the co-pumping ([DELTA]p < 0) increases with an increase in We .

Fig.3 shows the variation of [DELTA]p with [bar.Q] for different values of n with [phi] = 0.6, M = 1, [eta] = 1, [alpha] = [pi]/6 and We = 0.2. It is observed that, the [bar.Q] increases with an increase in n in the pumping region, whereas in the co-pumping region the [bar.Q] decreases with an increase in n. Further, the pumping is more for Newtonian fluid (n = 1) than that of a Carreau fluid (0 < n < 1).

The variation of [DELTA]p as a function of [bar.Q] for different values of Hartmann number M with [phi] = 0.6, [eta] = 1, We = 0.2, [alpha] = [pi]/6 and n = 0.398 is depicted in Fig. 4. It is found that, any of two pumping curves intersect at a point in the first quadrant and to the left of this point [bar.Q] increases and to the right of this point [bar.Q] decreases with an increase in M.

Fig.5 shows the variation of [DELTA]p with [bar.Q] for different values of [phi] with We = 0.2, [eta] = 1, M = 1, [alpha] = [pi]/6 and n = 0.398. It is found that, the [bar.Q] increases with an increase in [phi] in both pumping and free pumping regions. But, in the co-pumping region, the [bar.Q] decreases with an increase in [phi] for appropriately chosen [DELTA]p (<0).

Fig. 6 depicts the variation of pressure rise [DELTA]p with [bar.Q] for different values of [alpha] with [phi] = 0.6, We = 0.2, M = 1, [eta] = 1 and n = 0.398 . It is found that, as [alpha] increases the [bar.Q] increases in pumping, free pumping and co-pumping regions.

The variation of pressure rise [DELTA]p with [bar.Q] for different values of [eta] with [phi] = 0.6, We = 0.2, M = 1, [alpha] = [pi]/6 and n = 0.398 is shown in Fig.7. it is observed that, as [eta] increases, the [bar.Q] increases in pumping, free pumping and co-pumping regions.

Figures 8-13 depict the effects of We, n, M , [alpha], [eta] and [phi] on the frictional force. From Fig.8, it is found that the friction force first increases and then decreases with an increase in We. From Fig.9, it is seen that the friction force initially decreases and then increases with an increase in n. From Fig. 10, it is observed that the friction force increases with an increase in M. Fig.11 indicates that friction force first decreases and then increases with an increase in [phi]. From Fig.12, it is observed that the friction force decreases with increasing [alpha]. From Fig.13, it is found that the friction force decreases with an increase in [eta]. In general, figures 2-13 show that the friction force has opposite character in comparison to the pressure rise.

Conclusions

The peristaltic flow of a MHD Carreau fluid in an inclined channel under lubrication approach is investigated. It is found that the pumping is more for Newtonian fluid (n = 1) than that of Carreau fluid (0 < n < 1). The magnitudes of pressure rise and friction force increase with an increase in M or [phi] or [alpha] or [eta], whereas, the magnitudes of pressure rise and friction force decrease with an increase in We.

References

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[2] Ali, N. and Hayat, T., 2007, Peristaltic motion of a Carreau fluid in an asymmetric channel, Appl. Math. Comput., 193, pp. 535-552.

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[4] Bohme, G., and Friedrich, R., 1983, Peristaltic flow of viscoelastic liquids, J.Fluid Mech., 128, pp. 109-122.

[5] El Misery, A. M., El Shehawey, E. F. and Hakeem, A., 1996, Peristaltic motion of an incompressible generalized Newtonian fluid in a planar channel, J. Phys. Soc. Jpn., 65, pp. 3524-3529.

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[7] El Shehawey, E.F., El Misery, A. M., and Hakeem, A. 1998, Peristaltic motion of generalized Newtonian fluid in a non-uniform channel, J. Phys. Soc. Jpn., 67, pp. 434-440.

[8] El Shehawey, E.F., Sobh, A.M.F. and Elbarbary, E.M.E., 2002, Peristaltic motion of a generalized Newtonian fluid through a porous medium, Phys. Soc. Jpn., 69, pp. 401- 407.

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[10] Hayat, T., Afsar, A., Khan, M. and Asghar, S., 2007, Peristaltic transport of a third order fluid under the effect of a magnetic field, Computers and Mathematics with Applications, 53, pp. 1074-1087.

[11] Hayat, T., Khan, M., Siddiqui, A. M. and Asghar, S., 2007, Non-linear peristaltic flow of a non-Newtonian fluid under the effect of a magnetic field in a planar channel, Communications in Nonlinear Science and Numerical Simulation, 12, pp. 910-919.

[12] Latham, T.W., 1966, Fluid motions in peristaltic pump, M.S. Thesis, MIT, Cambridge, Massachussetts.

[13] Lew, H.S., Fung, Y.C. and Lowenstein, C.B., 1971, Peristaltic carrying and mixing of Chyme in the small intestine ( An analysis of a mathematical model of peristaltic of the small intestine), J. Biomech., 4, pp. 297-315.

[14] Pozrikidis, C., 1987, A study of peristaltic flow, J. Fluid Mech., 180, pp. 515527.

[15] Shapiro, A.H., Jaffrin, M.Y and Weinberg, S.L., 1969, Peristaltic pumping with long wavelengths at low Reynolds number, J. Fluid Mech. 37, pp. 799-825.

[16] Siddiqui, A.M. and Schwarz, W.H., 1994, Peristaltic flow of a second order fluid in tubes, J. Non-Newtonian Fluid Mech., 53, pp. 257-284.

[17] Srivastava, L.M. and Srivastava, V. P., 1985, Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube, J. Biomech., 18, pp. 247-253.

[18] Srivastava, L.M. and Srivastava, V.P., 1988, Peristaltic transport of power law fluid: application to the ductus of efferentes of the reproductive tract, Rheol. Acta., 27, pp. 428-433.

M.V. Subba Reddy (a) and K. Gangadhar (b)

(a) Department of Information Technology, BITS, Adoni, A.P., India

(b) Department of Mathematics, ANUPG Center, Ongole, A.P., India

Corresponding Author E-mail: [email protected]