Long wavelength approximation to MHD peristaltic flow of a Bingham fluid through a porous medium in an inclined channel.

Reddy, M.V. Subba ; JayaramiReddy, B. ; Reddy, M. Sudhakar 等

Introduction

In past five decades, many mathematical and computational models were developed to describe fluid flow in a tube under going peristalsis with prescribed wall motions. In earlier analytical studies, simplifying assumptions were made, including zero Reynolds number, small--amplitude oscillations, infinite wave length, as well as symmetry of the channel (e.g. Burns and Parkes [1]; Jaffrin and Shaprio [16]). Subsequent studies have been less restrictive and have captured features such as finite wave length, non uniform channel geometry, and effects of finite--length channels (e.g., Eytan and Elad [3]; Fauci [4]; Li and Brasseur [8]; Pozrikidis [10]; Takabatake et al. [16]; Subba Reddy et al. [14]).

Most of the investigations on peristaltic flow deal with Newtonian fluids. The complex rheology of biological fluids has motivated investigations involving different non-Newtonian fluids. Raju and Devanathan [12] have studied the peristaltic flow of non-Newtonian fluid in a tube by considering the blood as a power-law fluid. Peristaltic motion of a power-law fluid in a channel under long wavelength approximation was studied by Radhakrishnamacharya [11]. Kapur [7] have made theoretical investigations regarding blood as a Casson and Herschel-Bulkley fluids. Sreenadh et al. [13] have investigated the effect of yield stress on peristaltic pumping of non-Newtonian fluids in a channel. Vajravelu et al. [17,18] made a detailed study on the effect of yield stress on peristaltic pumping of a Herschel--Bulkley fluid in an inclined tube and a channel. All these investigations are confined to hydrodynamic study of a physiological fluid obeying some yield stress model.

Moreover, flow through a porous medium has practical applications especially in geophysical fluid dynamics. Examples of natural porous medium are beach sand, sandstone, limestone, rye bread, wood, the human lung, bile duct, gall bladder with stones and in small blood vessels. In view of this, El Shehawey et al. [2] investigated the peristaltic flow of a Newtonian fluid through a porous medium. Mekheimer and Al-Arabi [9] have discussed the peristaltic flow of a Newtonian fluid through a porous medium in a channel under the effect of magnetic field. The peristaltic flow of electrically conducting fluid through a porous medium in a planar channel was investigated by Hayat et al. [5]. Sudhakar Reddy et al. [15] have studied peristaltic motion of a Carreau fluid through a porous medium in a channel under the effect of a magnetic field.

In view of these, the effect of magnetic field on the peristaltic pumping of a Bingham fluid through a porous medium in an inclined channel is studied under long wavelength and low Reynolds number assumptions. The expressions for the velocity field in the plug flow and non- plug flow regions, the pressure rise in the channel and the volume flow rate are obtained analytically. The effects of the magnetic field, Darcy number, yield stress and amplitude ratio on the axial pressure gradient, pumping characteristics and frictional force are discussed in detail with the help of graphs.

Mathematical formulation

We consider the peristaltic flow of a conducting Bingham fluid flow through a porous medium in a channel of half-width a. A longitudinal train of progressive sinusoidal waves takes place on the upper and lower walls of the channel. For simplicity, we restrict our discussion to the half-width of the channel as shown in the figure. The region between y = 0 and y = [y.sub.o] is called plug flow region. In the plug flow region, [absolute value of [[tau].sub.yx]] [less than or equal to] [[tau].sub.y]. In the region between y = [y.sub.0] and y = H, [absolute value of [[tau].sub.yx]] > [[tau].sub.y]. Fig. 1 shows the physical model of the problem. The wall deformation is given by

H(X,t)=a + b cos[2[pi]/[lambda](x -ct)] (2.1)

where b is the amplitude, [lambda] the wavelength and c is the wave speed.

Under the assumptions that the channel length is an integral multiple of the wavelength [lambda] and the pressure difference across the ends of the channel is a constant, the flow becomes steady in the wave frame (x, y)moving with velocity c away from the fixed (laboratory) frame(X, Y). The transformation between these two frames is given by

x = X - ct, y = Y, u(x, y) = U - c, and v(x, y) = V (2.2)

where U and V are velocity components in the laboratory frame and u and v are velocity components in the wave frame.

[FIGURE 1 OMITTED]

The equations governing the flow in wave frame are given by

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

where [[tau].sub.ij] denote the stresses. For the Bingham Plastic these are related to the strains through the constitutive model,

[[tau].sub.ij] = ([mu] + [[tau].sub.y]/[??])[[??].sub.ij] for [tau] [greater than or equal to] [[tau].sub.y] (2.6)

and

[[tau].sub.ij] = [[??].sub.ij] = 0for [tau ] < [[tau].sub.y] (2.7)

where [[??].sub.ij] is the rate of strain tensor,

[[??].sub.ij] = [partial derivative][u.sub.i]/[partial derivative][x.sub.j] + [partial derivative][u.sub.j]/[partial derivative][x.sub.i] (2.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

Introducing the non-dimensional quantities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

into Eqs. (2.3)-(2.5), we get (dropping the bars)

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 (2.10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)

where

[[tau].sub.ij] = ([mu] + [[tau].sub.y]/[??])[[??].sub.ij] for [tau] [greater than or equal to] [[tau].sub.y] (2.13)

[[tau].sub.ij] = [[??].sub.ij] = 0for [tau] < [[tau].sub.y] (2.14)

[[??].sub.xy] = [[??].sub.yx] = [partial derivative]u/[partial derivative]y + [[delta].sup.2] [partial derivative]v/[partial derivative]x (2.15)

[[??].sub.xx] = 2[delta] [partial derivative]u/[partial derivative]x [[??].sub.yy] = 2[delta] [partial derivative]v/[partial derivative]y (2.16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

[tau] = [square root of [[tau].sup.2.sub.xy] + [[delta].sup.2][[tau].sup.2.sub.xx]] (2.18)

Re = [[rho]ac/[mu]] is the Reynolds number, M = [square root of [sigma]/[mu]] [B.sub.0] a is the Hartman number and

Da = k/[a.sup.2] is the Darcy number .

Under the assumptions of long wavelength and low Reynolds number, the Eqs. (2.11) and (2.12) reduce to

0 = - [partial derivative]p/[partial derivative]x + [partial derivative][[tau].sub.yx]/[partial derivative]y - [N.sup.2] (u + 1) + Re/Frsin [alpha] (2.19)

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

where [N.sup.2] = [M.sup.2] + 1/Da,

[[tau].sub.xy] = [partial derivative]u/[partial derivative]y - [[tau].sub.y] for [tau] [greater than or equal to] [[tau].sub.y] (2.21)

[[tau].sub.xy] = 0 for [tau] < [[tau].sub.y] (2.22)

here [[tau].sub.y] is the yield stress.

Here Eq. (2.20) indicates that p is independent of y and depends only upon x. Therefore, Eq. (2.19), can be rewritten as

[[d.sup.2]u/d[y.sup.2]] - [N.sup.2]u = [dp/dx] + [N.sup.2] (2.23)

The non-dimensional boundary conditions are

[partial derivative]u/[partial derivative]y = [[tau].sub.y] at y = 0 (2.24)

u = -1 at y = h = 1 + [phi]cos2[pi]x (2.25)

The volume flux q through each cross section in the wave frame is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.26)

The instantaneous volume flow rate Q(X,t) in the laboratory frame between the centre line and the wall is

Q(X, t) = [[integral].sup.H.sub.0] UdY = [[integral].sup.h.sub.0] (u + 1)dy = q + h (2.27)

The average volume flow rate [bar.Q] over one wave period (T = [lambda]/c) of the peristaltic wave is defined as

[bar.Q] = 1/T [[integral].sup.T.sub.0] Q(X, t) dt = q + 1 (2.28)

Solution

Solving Eq. (2.23) using the boundary conditions (2.24) and (2.25) we obtain the velocity as

u = [1/[N.sup.2]](dp/dx - Re/Fr sin [alpha])[cosh Ny/cosh Nh - 1] - [[tau].sub.y] tanh Nh cosh Ny + [[[tau].sub.y]/N]] sinh Ny - 1 (3.1)

We find the upper limit of plug flow region using the boundary condition [partial derivative]u/[partial derivative]y = 0 at y = [y.sub.0]. So, we have

[[tau].sub.y] = -1/N(dp/dx - Re/Fr sin [alpha]) sinh N[y.sub.0]/cosh N(h - [y.sub.0]) (3.2)

Taking y = [y.sub.0] in equation (3.1), we get the velocity in plug flow region as

[u.sub.p] = 1/[N.sup.2](dp/dx - Re/Fr sin [alpha])[1 - cosh N(h - [y.sub.0]]/cosh N(h - [y.sub.0]) - 1 (3.3)

The volume flux q through each cross section in the wave frame is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

where [c.sub.1] = 1 - cosh N(- [y.sub.0])/cosh N(h - [y.sub.0]), [c.sub.2] = [y.sub.0] - sin N[y.sub.0]/N cosh Nh and [c.sub.3] = sinh Nh - sinh N[y.sub.0]/N cosh Nh

From equation (3.4), we have

dp/dx = [N.sup.2](q + h)/[C.sub.1][C.sub.2] + [C.sub.3] - h + [y.sub.0] + Re/Fr sin [alpha] (3.5)

The pressure rise over one wavelength of the peristaltic is defined as

[DELTA]p = [[integral].sup.1.sub.0] dp/dx dx (3.6)

Discussion of the results

In order to see the effects of Da,M,[y.sub.0] and [phi] on the axial pressure gradient and the pumping characteristics we have plotted Figs. 2-15.

Fig.2 shows the variation of axial pressure gradient dp/dx with Darcy number Da for [phi] = 0.6, [phi] = -1, M = 1, Re = 5, [alpha] = [pi]/6, Fr = 2 and [y.sub.0] = 0.2. It is found that, increasing the Darcy number Da decreases the axial pressure gradient.

The variation of axial pressure gradient dp/dx with Hartmann number M for [phi] = 0.6, [bar.Q] = -1, Da = 0.1, Re = 5, [alpha] = [pi]/6, Fr = 2 and [y.sub.0] = 0.2 is depicted in Fig. 3. It is observed that, the axial pressure gradient increases with increasing Hartmann number M.

Fig. 4 illustrates The variation of axial pressure gradient dp/dx with [y.sub.0] for [phi] = 0.6, [bar.Q] = -1, Da = 0.1, Re = 5, [alpha] = [pi]/6, Fr = 2 and M = 1. . It is found that, the axial pressure gradient increases with an increase in [y.sub.0].

Fig. 5 shows the variation of axial pressure gradient dp/dx with amplitude ratio [phi] for [phi] = 0.6,[bar.Q] = -1, Da = 0.1, Re = 5, [alpha] = [pi]/6, Fr = 2 and [y.sub.0] = 0.2 is shown in Fig. 5. It is observed that, the axial pressure gradient increases with increasing amplitude ratio [phi].

Fig. 6 shows the variation of axial pressure gradient dp/dx with inclination angle [alpha] for [phi] = 0.6, [bar.] = -1, Da = 0.1, Re = 5, M = 1, Fr = 2 and [y.sub.0] = 0.2. It is found that, the axial pressure gradient increases with an increase in [alpha].

The variation of axial pressure gradient dp/dx with Froude number Fr for [phi] = 0.6, [bar.Q] = -1, Da = 0.1, Re = 5, [alpha] = [pi]/6, Fr = 2 and [y.sub.0] = 0.2 is shown in Fig. 7. It is observed that, the axial pressure gradient decreases with increasing Fr.

Fig. 8 depicts the variation of axial pressure gradient dp/dx with Reynolds number Re for [phi] = 0.6, [bar.Q] = -1, Da = 0.1, M = 1, [alpha] = [pi]/6, Fr = 2 and [y.sub.0] = 0.2. it is noted that, the axial pressure gradient increases on increasing Re.

Fig. 9 depicts the variation of pressure rise [SELTA]p with [bar.Q] for different values of Darcy number Da with [phi] = 0.6, M = 1, Re = 5, [alpha] = [pi]/6, fr = 2 and [y.sub.0] = 0.2. It is found that, in the pumping region ([DELTA]p > 0) the time averaged flux [bar.Q] decreases with increasing Darcy number Da, while it increases with increasing Da in both the free pumping ([DELTA]p = 0) and co-pumping ([DELTA]p< 0) regions.

The variation of pressure rise [DELTA]p with [bar.Q] for different values of Hartmann number M with [phi] = 0.6, [bar.Q] = -1, Da = 0.1, Re = 5, [alpha] = [pi]/6, Fr = 2 and [y.sub.0] = 0.2 is presented in Fig. 10. It is noted that, in the pumping region the time averaged flux [bar.Q] increases with an increasing Hartmann number M, while it decreases with increasing M.

Fig. 11 shows the variation of pressure rise [DELTA]p with [bar.Q] for different values of [y.sub.0] with 0 = 0.6, [bar.Q] = -1, Da = 0.1, Re = 5, a = [pi]/6, Fr = 2 and M = 1. It is found that, the time averaged flux increases with increasing [y.sub.0] in both the pumping and free pumping regions, while in the co-pumping region it decreases with increasing [y.sub.0] for appropriately chosen [DELTA]p(<0).

The variation of pressure rise [DELTA]p with [bar.Q] for different values of amplitude ratio [phi] with M = 1, [bar.Q] = -1, Da = 0.1, Re = 5,[alpha] = [pi]/6, Fr = 2 and [y.sub.0] = 0.2 is depicted in

Fig. 12. It is observed that, the time averaged flux [bar.Q] increases with increasing 0 in both pumping and free pumping regions, while it decreases with increasing 0 in the co-pumping region for appropriately chosen [DELTA]p (< 0).

Fig. 13 shows the variation of pressure rise [DELTA]p with [bar.Q] for different values of inclination angle [alpha] with [phi] = 0.6, [bar.Q] = -1, Da = 0.1, Re = 5,M = 1, Fr = 2 and [y.sub.0] = 0.2. It is found that, the time averaged flux [bar.Q] increases with increasing [alpha] in all the tree regions.

The variation of pressure rise [DELTA]p with [bar.Q] for different values of Froude number Fr with [phi] = 0.6, [bar.Q] = -1, Da = 0.1, Re = 5, [alpha] = [pi]/6, M = 1 and [y.sub.0] = 0.2 is depicted in Fig. 14. It is noted that, the time averaged flux [bar.Q] decreases with an increase in Fr.

Fig. 15 illustrates the variation of pressure rise [DELTA]p with [bar.Q] for different values of Reynolds number Re with 0 = 0.6, [bar.Q] = -1, Da = 0.1,M = 1, [alpha] = [pi]/6, Fr = 2 and [y.sub.0] = 0.2.

It is observed that, the time averaged flux [bar.Q] increases on increasing Re.

Conclusions

In this chapter, we modelled the MHD peristaltic flow of a Bingham fluid through a porous medium in an inclined channel under assumptions of low Reynolds number and long wavelength. The expressions for the velocity and pressure gradient are obtained analytically. It is found that, the pressure gradient and the time averaged flux increases with increasing Hartmann number M, half width of the plug flow region [y.sub.0], amplitude ratio [phi], inclination angle [alpha] and Reynolds number Re while they decreases Darcy number Da and Fr.

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[FIGURE 15 OMITTED]

References

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[12] Raju, K.K. and Devanathan, R. Peristaltic motion of a non-Newtonian fluid part -I, Rheol. Acta., 11(1972), pp. 170-178.

[13] Sreenadh, S., Narahari, M. and Ramesh Babu, V., 2004, Effect of yield stress on peristaltic pumping of Non-Newtonian fluids in a channel, International Symposium on advances in fluid Mechanics, UGC-Centre foe Advanced Studies in Fluid mechanics, Bangalore University, pp. 234-247.

[14] Subba Reddy, M. V., Mishra, M., Sreenadh, S. and Ramachandra Rao, A., 2005, Influence of lateral walls on peristaltic flow in a rectangular duct, Transaction of the ASME Journal of Fluids Engineering, 127, pp. 824-827.

[15] Sudhakar Reddy, M., Subba Reddy, M. V. and Ramakrishna, S., 2009, Peristaltic motion of a carreau fluid through a porous medium in a channel under the effect of a magnetic field, Far East Journal of Applied Mathematics, 35, pp. 141-158.

[16] Takabatake, S. Ayukawa, K. and Mori, A., 1988, Peristaltic pumping in circular tubes: A numerical study of fluid transport and its efficiency, J. Fluid Mech., 193, pp. 267-283.

[17] Vajravelu, K., Sreenadh, S. and Ramesh Babu, V., 2005, Peristaltic transport of a Hershel-Bulkley fluid in an inclined tube, Int. J. Non linear Mech. 40, pp. 83-90.

[18] Vajravelu, K. Sreenadh, S. and Ramesh Babu, V. 2005, Peristaltic pumping of a Hershel-Bulkley fluid in a channel, Appl. Math. and Computation, 169, pp. 726-735.

M.V. Subba Reddy (a) *, B. JayaramiReddy (b), M. Sudhakar Reddy (c) and N. Nagendra (c)

(a) Professor, Department of Information Technology, Sri Venkatesa Perumal College of Engineering & Technology, Puttur-517583, Chittoor, A.P., India

(b) Principal & Professor of Civil Engineering, YSR Engineering College of Yogi Vemana University, Proddatur-516360, A.P., India.

(c) Department of Mathematics, Sri Sai Institute of Technology and Science, Rayachoty-516 2 70, A.P., India.

* Corresponding Author E-mail: drmvsubbreddy@yahoo.co.in ; bjreddyyvuce@gmail.com