Peristaltic Transport Of A Conducting Bingham Fluid In A Channel

Peristaltic Transport Of A Conducting Bingham Fluid In A Channel

c: Wave speed

E: Mechanical efficiency

c: Wave speed

E: Mechanical efficiency

1. Dr P Ramesh Kumar 2. Dr G Krishnaiah 3. Dr S Srinadh

Narayana EngineeringCollege,Gudur SV University,Tirupathi SV University,Tirupathi

Abstract

Peristaltic transport of a conducting Bingham fluid in a channel under long wavelength and low Reynolds number assumptions is studied. This model is probably suitable for the blood flow in the sense that erythrocytes region and the plasma regions may be described as plug flow and non-plug flow regions. Further the blood is experimentally proved to be a conducting fluid in several physiological applications. Motivated by this fact, it is proposed to study the peristaltic transport of a Bingham fluid. It is observed that for a Bingham fluid the pressure difference and the mechanical efficiency of pumping depend on the yield stress and the magnetic field.

1. Introduction

Peristaltic pumping is a form of fluid transport generally from a region of lower to higher pressure, by means of a progressive wave of area contraction or expansion, which propagates along the length of a tube- like structure. Peristalsis occurs naturally as a means of pumping physiological fluids from one place in the body to another. Some electro-chemical reactions are held responsible for this phenomenon. This mechanism occurs in swallowing of food through oesophagus, in the ureter, the gastro intestinal tract, the bile duct and even in small blood vessels.

1. Nomenclature

   a: Half width of the wave B0: Applied magnetic field b: Amplitude of the wave

   F: Dimensionless friction force h: Height of the wave in the direction of y

   M: Hartmann number P: Pressure

   p: Dimensionless pressure P: Pressure rise/drop

   q: Volume flux

   Q: Time averaged flow rate Re: Reynolds number

   t: Time

   U,V: Axial and transverse velocity components in laboratory frame

   u,v: Axial and transverse velocity components in wave frame

   up: Velocity in plug flow region

   x,y: Axial and transverse co ordinates X,Y: Dimensionless axial and transverse co ordinates

   y0: Plug flow region : wave length

   e: Electrical conductivity : Stream function

   : Amplitude ratio (b/a) 0: Yield stress

   : Viscosity index

2. Mathematical formulation and solution

Consider the peristaltic pumping of a conducting Bingham fluid 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 figure 4.1. The region between Y = 0 and Y

= y0 is called plug flow region. In this region, |y,x| o. In the region between Y =

Y0 and Y = H, |y,x| > o. The wall deformation is given by

assume that the wavelength is infinite. So

the flow is of Poiseuille type at each local cross-section. Using the non-dimensional

H(x, t) = a + b sin

2 x ct

(1)

quantities,

u u ; x c

x ;

y y ; h a

h ; t ct a

F Fa

c

; a0 ;

0 c

; p ac

Pa2

c

y y0 ; q q ; b

0 a ac a

The non-dimensional form of equations governing the motion (dropping the bars) is

d

dy 0

yy M 2 y dp

dx

(3)

Fig. 1 Physical Model

1. Equations of Motion

   0 p

   y

   B 2a 2

   (4)

   Under the assumptions that the tube length is an integral multiple of the wave length and the pressure difference across

   where M =

   number

   o e , the Hartmann

   µ

   the ends of the tube 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 (x-ct, Y) – c

   v (x,y) = V (x-ct, Y) (2)

   In many physiological situations it is proved experimentally that the Reynolds

   The non-dimensional boundary conditions are

   = 0; yy = o at y = 0

   u= y = -1 at y = h (5) Where h = 1 + Sin2x (6)

2. Solution

   Solving equation (4) with the boundary conditions (5), we obtain the velocity as

   number of the flow is very small (for example in ureter, Re 1). So, we assume that the flow is inertia free. Further, we

   u = A1eMy + B1e-My +

   p (7)

   M2

   where A1 = a1p+a2, B1 = a3p + a4

3. The Pumping characteristics

   Integrating the equation (10) with

   1

   a = M 2

   1 Mh

   e

   e

   M 2

   respect to x over one wavelength, we get the pressure rise over one cycle of the wave as

   1 2Sinh M

   1 a

   Q 1

   P = 6 dx

   (13)

   0

   0

   1 e Mh

   0 a5

   a2 =

   2Sinh M

   1 eMh 1

   The pressure rise required to produce zero average flow rate is denoted

   M 2 M 2

   by P0 and is given by

   a3 =

   2Sinh M

   Mh

   Mh

   1 a 1

   0 e 1

   P0 = 6 dx

   (14)

   a4 =

   2 Sinh M

   0 a5

   p p

   x

   (8)

   The dimensionless friction force F at the wall across one wave length is given by

   Taking y = y0 in equation we get the

   velocity in plug flow region as

   1 dp

   p F = h

   dx

   0

   0

   0

   0

   up = A1eMy

   + B1e-My +

   (0 y y0)

   M2

   0 dz

   (9) the

   1 Q 1 a

   volume flux q through each cross-section in

   = h 6 dx

   (15)

   the wave frame is given by

   0 a5

   y0 y

   q = updy udy = a5P + a6 (10)

   0 y0

   where a5 =

   a eMy0 y 1 a e My0 y 1 a1 eMh a3 e Mh h

4. Mechanical Efficiency:

   The mechanical efficiency of pumping for one wave length is given by

   1 0 3

   M

   0

   M M M M 2

   E = Qp

   a6 =

   2 4 x

   2 4 x

   a eMy0 1 1 a

   eMy0 1 1 a2 eMh a4 eMh

   1

   p h 1dx

   M

   M

   M

   M

   M

   M

   M

   M

   0

   From the above equation

   p a6 q

   Q I Q 1I

   1 2

   1 2

   3

   3

   4

   4

   x a5

   (11)

   = Q I Q 1I

   (16)

   Averaging equation over one period yields the time mean flow (time-averaged

   Q

   1

   a

   a

   Where I1 = 6 dx

   volume flow rates)

   Q =

   as

   T

   T

   1 T

   Qdt

   0

   0 a5

   1 1

   I2 = a dx

   = q+1 (12) 0 5

   3 6

   3 6

   1 a

   I = Sin 2xdx

   0 a5

   of Q for fixed amplitude ratio, and is depicted in Fig. 5. It is observed that the frictional force F has the opposite

   1 1 behaviour compared to pressure rise (P).

   a

   a

   I4 =

   0 5

   Sin 2x dx

   As 0 increases the frictional force also increases with the flux Q . For a given Q the frictional force increases with decrease

   8 Discussion of the Results.

   From the equation 13 we have calculated the pressure difference as a function of Q for differen values of yield stress 0 for a fixed and shown in Fig. 2. It is observed that for a given P the flux Q

   depends on yield stress and it decreases with increase in yield stress. For a given flux Q the pressure rise P decreases with increase in yield stress 0. For free pumping, the flux Q increases with decreasing yield stress.

   From the variation of pressure rise with time averaged flow rate is calculated from equation (13) for different amplitude ratios and shown in Fig. 3. For given P the flux Q for Bingham fluid depends on and we find that the larger the amplitude ratio the greater the pressure rise against which the pump works. This observed for

   small values of P (P < 0001) and free pumping cases.

   From equation (13) we have calculated the variation of pressure rise with time averaged flow rate Q for different magnetic parameters M and is shown in Fig. 4 we observed that the larger the magnetic

   parameter the greater the pressure rise

   against which the pump works. For free pumping the flux Q depends on magnetic field and it increases with increasing magnetic parameter M.

   From equation (15), we have calculated the frictional force as a function

   in yield stress.

   From equation (16) we have calculated the mechanical efficiency as a function of Q and is depicted in Fig. 6 and Fig. 7. for different values of magnetic parameters M, yield stress and amplitude

   ratio . For the chosen parameter the

   mechanical efficiency is positive for Q less than or equal to 6 and is negative for Q greater than 6. Further we observed that the

   pumping efficiency is greater for higher

   values of magnetic parameters. It is also found that the mechanical efficiency decreases with the increasing yield stress.

   For lower value flux Q , the efficiency remains constant and is unaffected by variation in M and 0.

   From equation (14), we have calculated the pressure rise required to produce zero average flow rate P0 as a function of amplitude ratio for different values of yield stress 0 and is shown in Fig.

5. It is observed that for a conducting Bingham fluid, the value of P0 increases with decrease in yield stress 0. It is also observed that the increase in the value of

   P0 is more for the amplitude ratio beyond 0.7.

   From equation (14), we have calculated the pressure rise required to produce zero average flow rate P0 as a function of amplitude ratio for different values of magnetic parameter m and is shown in Fig.

6. It is observed that for a conducting

Bingham fluid, the value of P0 increases with increase in magnetic parameter.

Fig. 2 The variation of P with averaged flow rate Q for different 0 with = 0.6, m = 0.02, y0 = 0.2

Fig. 3 The variation of P with averaged flow rate Q for different , with 0 = 0.01, m = 0.01, y0 = 0.2

Fig. 4 The variation of P with averaged flow rate Q for different m with 0 = 0.01, = 0.6, y0 = 0.2

Fig. 5 The variation of F with averaged flow rate Q for different 0 with = 0.6, m = 0.02, y0 = 0.2

Fig. 6 The variation of Mechanical Efficiency E with averaged flow rate Q

for different m with 0 = 0.01, = 0.6, y0 = 0.1

Fig. 7 The variation of Mechanical Efficiency E with averaged flow rate Q

for different 0 with = 0.6, m = 0.02, y0 = 0.1

Fig. 8 The variation of P0 with Amplitude ratio for different 0 with m = 0.02, y0 = 0.2

Fig. 9 The variation of P0 with Amplitude ratio for different m with

0 = 0.01, y0 = 0.2

References

1. SUD, V. K., SEKHON, G. S. & MISHRA, R. K. 1977 Pumping action on blood by a magnetic field, Bull. Math. Biol. 39, no. 3, 385-390.

2. LATHAM, T. W. 1966 Fluid motions in a peristaltic pump. M. S. Thesis, M.I.T.

3. BURNS, J. C. & PARKES, T. 1967 Peristaltic motion. J. Fluid Mech. 29, 731-743.

4. SHAPIRO, A. H., JAFFRIN, M. Y. & WIENBERG, S. L.1969 Peristaltic pumping with long wavelengths at low Reynolds number. J. Fluid Mech. 37, 799-825.

5. BRASSEUR, J. G. 1987 A fluid mechanical perspective on esophageal bolus transport. Dysphagia. 2, 32-39.

6. USHA, S. & RAMACHANDRA RAO, A. 1997 Peristaltic transport of two-layered power-law fluids.

   ASME J. Biomech. Engg. 104, 182-186

7. USHA, S. & RAMACHANDRA RAO, A. 1995 Peristaltic transport of a bio-fluid in a pipe of elliptic cross section. J. Biomechanics.28, 45-52.