Peristaltic transport of Conducting Bingham fluid in contact with a Newtonian fluid in a channel

Peristaltic transport of Conducting

Bingham fluid in contact with a Newtonian

fluid in a channel

M.Arun kumar*, S.Sreenadh*, A.N.S.Srinivas** and S.Venkata Ramana*

* Department of Mathematics, Sri Venkateswara University, Tirupati – 517 502 (A.P), India **School of Advanced Sciences, VIT University, Vellore –632 014 (T.N), India

Abstract:

Peristaltic pumping by a sinusoidal traveling wave in the walls of a two dimensional channel filled with two immiscible fluids with magnetic effect is investigated. The core region of the channel is occupied by a Bingham fluid where as the peripheral region is occupied by a Newtonian fluid. The flow is examined in a wave frame of reference moving with the velocity of the wave. The expressions for the stream function, the velocity and the pressure rise are obtained. The equation for the interface separating the two fluids is obtained. Numerical results are reported for several of the physical parameters of interest. We observed that the lower values of and give rise to a thicker peripheral layer in the constricted region of the pump and low viscosity ratios are associated with thinner peripheral layers in the constricted region of the pump.

Keywords: Peristaltic transport, Magnetic effect, Bingham fluid, Newtonian fluid, channel

.

1. Introduction:

Peristalsis is well known to physiologists to be one of the major mechanisms for fluid transport in many biological systems. In particular peristalsis may be a main mechanism for urine transport from kidney to bladder through the ureter, movement of chyme in the gastrointestinal tract, transport of lymph in the lymphatic vessels and the vasomotion of small blood vessels. In addition peristaltic pumps are designed by engineers for pumping corrosive fluids without contact with the walls of the pumping machinery.

Applying a wave frame of reference Jaffrin and Shapiro (1971) made a detailed analysis on the peristaltic pumping of a viscous fluid under long wave length and low Reynolds number assumptions. It is observed in some physiological systems such as the oesophagus, the ureter, that the wall of the structure doing the pumping is typically coated with a fluid with different properties from those of the fluid being pumped. In order to have an understanding about the effect of fluid coating on the transport, the single fluid analysis of peristaltic pumping is extended to two fluid analysis by including peripheral layer of different viscosity. Such an analysis was first done by Shukla et al., (1980) for channel and axisymmetric geometries. For non-uniform axisymmetric tubes Srivastava et al., (1983) made an important contribution in peristaltic pumping. All these authors have specified the interface shape.

Brasseur et al., (1987) made a significant contribution on the peristaltic transport of two immiscible fluids in a channel with flexible walls and have proved the invalidity of the analysis mentioned above in the limit of infinite peripheral layer viscosity. This problem is solved for axisymmetric case by Ramachandra Rao and Usha et al., (1995). Usha and Ramachandra Rao (1997) discussed the peristaltic pumping of two layered power-law fluids in an axisymmetric tube. The interface between the two layers is determined from a transcendental equation in the core radius. Comparani and Mannucci (1998) analysed the flow of a Bingham fluid in contact with a Newtonian fluid in a channel. Existence and uniqueness theorems are proved for the solution of the problem. Vajravelu et al., (2005 a,b) studied Peristaltic Transport of a Herschel-Bulkley fluid in a channel and an inclined tube. Radhakrishnamacharya et al., (2007) studied Influence of wall properties on peristaltic transport with heat transfer. Srinivas et al., (2009) studied the influence of slip conditions, wall properties and heat transfer on MHD peristaltic transport. Hayat et al., (2010) analyzed the Series solution for MHD channel flow of Jeffrey fluid. The slip and induced magnetic field effects on the Peristaltic transport with heat and mass transfer is analysed by Hayat et al.,(2012).

2. Mathematical formulation and solution:

Consider the peristaltic transport of a bio-fluid consisting of two immiscible and incompressible fluids of different viscosities and occupying the core by a Bingham fluid and peripheral layer by a Newtonian fluid in a channel. The half width of the channel is ‘a’. The wall deformation due to the propagation of an infinite train of peristaltic waves is given by

,

(1)

where



is the wavelength b is the amplitude and c is the wavespeed.

The subsequent deformation of the interface separating the core and the peripheral layers is denoted by

Y  H₁



X , t



(Figure.1) which is not known a priori

.

2.1Equations of motion:

Under the assumptions that the channel length is an integral multiple of the wavelength , the pressure difference across the ends of the channel is a constant and the periodicity of the interface is same as that of the peristaltic wave. The flow becomes steady in the waveframe , moving with the velocity c away from the fixed frame , called laboratory frame. The transformation between these two frames is given by

, , , , ,

, , , , , Ψ

(2)

where and Ψ are the stream functions in the wave and laboratory frames respectively. Using the non-dimensional quantities.

, , , , , , ,

, , , ,

, ,

,

,

(3)

Figure 1: Physical Model

Plug flow Bingham fluid Newtonian fluid

X Y

0

where and are the - and - components of velocities in the wave frame, in the equations governing the motion, under the lubrication approach, we get (dropping the bars),

ψ

(4)

(5)

ψ

(6)

The dimensionless boundary conditions are

Ψ

at

(7)

Ψ

at

(8)

Ψ

q

at

(9)

Ψ

q

at

(10)

Ψ

at

(11)

where is the Reynolds number and . and are the total and the core fluxes respectively across any cross-section in the wave frame. Further the velocity and the shear stress are continuous across the interface. The peripheral – layer flux is given by . It follows from the incompressibility of the fluids that , and are independent of . The average non-dimensional volume flow rate over one period of the peristaltic wave is defined as

(12)

The stream function is obtained by applying the boundary conditions (7) to (11) together with the boundary conditions at the ends of the channel given by specifying or the pressure difference ∆ _{across one} wavelength

.

2.2 Solution:

Solving the equations (4) to (6) with the corresponding boundary conditions (7) to (11), we obtain the stream function in the core and peripheral layer as . (13)

SinhMh (14)

Where , , Substituting equations (13) and (14) in momentum equations, one obtains the pressure gradient as (15)

The constants and are independent of . By prescribing in the above equation, we get

q q α S M h α SinhM Cosh M CoshMα S M h α CoshM Sinh Mα

SinhM

(17)

Where

,

Integrating equation (15) over one non-dimensional wavelength, we get

∆

(18)

3 Discussions of the Results:

The shape of the interface for different values of Hartmann number M with . , . ,

. , . is evaluated using the equation (17) and is shown in Figure 2. It is observed that the lower values of M give rise to a thicker peripheral layer in the constricted region of the pump.

The shape of the interface for different viscosity ratios with . , . , . , . is evaluated from equation (17) and is depicted in Figure 3. It is observed that low viscosity ratios are associated with thinner peripheral layers in the constricted region of the pump.

The shape of the interface for different values off yield stress with . , . , . , . is evaluated from equation (17) and is depicted in Figure 4. It is observed that low viscosity ratios are associated with thinner peripheral layers in the constricted region of the pump.

From the Figure (5), we observe that the variation of pressure rise ∆ , with time averaged flow rate with .9, . , . and for different values of Hartmann number M. We observe that for a given ∆ , the flux increases with increasing the Hartmann number M. For a given flux , the pressure difference decreases with increasing M.

The variation of pressure rise with time averaged flux is calculated from the Equation (18) for different values of yield stress to with .9, , . and is shown in figure 6, we observe that for a given ∆ , the flux depends on the yield stress and it decreases with increasing yield stress. For a given flux the pressure difference ∆ increases with decreasing yield stress .

From equation (18), we have also calculated the pressure difference as a function of for different values of amplitude ratios , and this is shown in Figure 7 for a fixed value of .9, . , . We observe that the larger the amplitude ratio, the greater the pressure rise against which the pump works. For a given p, the flux

Q

increases with increasing .

3.1 Trapping:

Another important phenomenon in peristaltic transport is trapping. The formation of an internally circulating bolus of fluid by closed streamlines is called trapping and is pushed a head along with the peristaltic wave. The physical phenomenon may be responsible for thrombus formation in blood and the movement of flood bolus in gastrointestinal tract. The streamlines in the wave frame for fixed values of .9, ,

Fig 2: The shape of the interface for . , . , . , . and different values of Hartmann number .

Fig 3: The shape of the interface for . , . , . , . and different values of Yield stress .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.2 0.4 0.6 0.8 1 1.2

h(x)

M = 0.1

M=0.5

M=1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.2 0.4 0.6 0.8 1 1.2

h(x)

τ0 = 0

τ0 = 0.1

Fig 4: The shape of the interface for . , . , . , . and different values of viscosity .

Fig 5: The variation of ∆ with for different values of M with .9, . , . .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.2 0.4 0.6 0.8 1 1.2

h(x)

µ = 0.1

µ = 0.3

Fig 6: The variation of ∆ with for different values of with .9, , . .

Fig 8: Streamlines for different values of yield stress (i) (ii) . (iii) . with

.9, , . , .

4.References:

[1] Brasseur, J.G., Corrsin, S. and Lu nan Q (1987)., The influence of a peripheral layer of different viscosity onperistaltic pumping with Newtonian fluids, J.Fluid Mech., 174, 495-519.

[2] Comparini, E and Mannucci, P (1998)., Flow of Bingham fluid in contact with a Newtonian fluid, J.Math. Anal. Appl. 227, 2, 359-381.

[3] Hayat T, Sajjad R and Asghar S (2010b)., Series solution for MHD channel flow of Jeffrey fluid. Communication in Nonlinear Science and Numerical Simulation 15, 2400-2406.

[4] Hayat .T. Noreen.S and A Alsaedi (2012)., The slip and induced magnetic field effects on the Peristaltic transport with heat and mass transfer, J. Mech. Med. Biol., Vol.12, Issue. 04, 1250068.

[5] Jaffrin, M.Y.and Shapiro, A.H,(1971),. Peristaltic Pumping, Ann. Rev. Fluid Mech., 313-36.

[6] Radhakrishnamacharya G and Srinivasulu Ch (2007). Influence of wall properties on Peristaltic transport with heat transfer. Compter Rendus Mecanique 335, pp. 369-373.

[7] Ramachandra rao, A., and Usha, S,(1995). Peristaltic transport of two immiscible viscous fluid in a Circular tube, J.Fluid Mech., 298, 271-285.

[8] Ramachandra rao, A., and Usha, S. (1995), Peristaltic pumping in a circular tube in the presence of an eccentric catheter, Trans. ASME J.Biomech. Engng., 117,448-454.

[9] Shukla, J.B., Parihar R.S., Rao, B.R.P. and Gupta,(1980),. S. P. Effects of peripheral – layer viscosity on peristaltic transport of a bio-fluid, J.Fluid Mech., 97, 225-237.

[10] Srinivas S, Gayathri R and Kothandapani M, (2009),. The influence of slip conditions, wall properties and heat transfer on MHD peristaltic transport. Computer Physics Communications 180, 2115-2122.

[11] Vajravelu K, Sreenadh S and Ramesh Babu V (2005a). Peristaltic Transport of a Herschel-Bulkley fluid in an inclined tube. International Journal of Nonlinear Mechanics. 40, 83 – 90.

[12] Vajravelu K, Sreenadh S and Ramesh Babu V (2005b),. Peristaltic pumping of Herschel-Bulkley fluid in a channel. Applied Mathematics and Computation 169, 726-735.

-0.2 0 0.2 0.4 0.6

0 0.25

0.5 0.75 1 1.25

1.5

-0.2 0 0.2 0.4 0.6

0 0.25

0.5 0.75 1 1.25

1.5

-0.2 0 0.2 0.4 0.6