Influence of Lateral Walls on Peristaltic Flow of a Third Grade Fluid in a Rectangular Duct

In the present study we have discussed the influence of lateral walls on peristaltic flow of a third grade fluid in a rectangular duct. The mathematical equations of the third grade fluid for the rectangular duct are first modeled and then simplified under the assumptions of long wave length and low Reynolds number approximation. The reduced equations are solved analytically using Homotopy perturbation method and the Eigen function expansion method. The graphical results of the present problem are also discussed to see the effects of various emerging parameters. It is observed that with an increase in third grade parameter the pressure rise, pressure gradient and number of the trapping bolus decreases. analytically with the help of Homotopy perturbation method and eigen function expansion method. The expressions of pressure rise, pressure gradient and stream functions are plotted and discussed for various physical parameters of interest. Mathematical Modelling Let us consider the peristaltic flow of an incompressible third grade fluid in a duct of rectangular cross section having the channel width 2d and height 2a. We are considering the Cartesian coordinates system in such a way that X-axis is taken along the axial direction, Y-axis is taken along the lateral direction and Z-axis is along the vertical direction of a rectangular duct as shown in Figure 1. The peristaltic waves on the walls are represented as 2 ( , ) cos ( ) π λ   = = ± ± −     Z H X t a b X ct (1) where a and b are the amplitudes of the waves, λ is the wave length, t is the velocity of propagation, X is the time and is the direction of wave propagation. The walls parallel to XZ plane remain undisturbed and are not subject to any peristaltic wave motion. We assume that the lateral velocity is zero as there is no change in lateral direction of the duct cross section. Let (U, 0, W) be the velocity for a rectangular duct. The governing equations for the flow problem are 0 ∂ ∂ + = ∂ ∂ U W X Z (2) , ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂   + + = − + + +   ∂ ∂ ∂ ∂ ∂ ∂ ∂   XX XY XZ U U U P U W S S S t X Z X X Y Z (3) Journal of Applied Mechanical Engineering J o u r n al o f A pp lied ical Eninee r i n g ISSN: 2168-9873 Citation: Akram S, Nadeem S, Hussain A (2014) Influence of Lateral Walls on Peristaltic Flow of a Third Grade Fluid in a Rectangular Duct. J Appl Mech Eng 3: 140. doi:10.4172/2168-9873.1000140


Introduction
Experimental and theoretical research in peristaltic flows has received increased attention during the past several decades. This is due to importance of this area in biomedical engineering and physiology. Several pertinent area of interest (where the peristaltic flows are functional) are urine transport from kidney to bladder, the movement of chyme in the gastrointestinal tract, fluids in the lymphatic vessels, biles from the gall bladder into the duodenum, the movement of spermatozoa in the ductus efferent of the male reproductive tract, the movement of the ovum in the fallopian tube, circulation of blood in small blood vessels, roller and finger pumps, heart lung machine, blood pump machine and dialysis machine etc. After the pioneering work done by Latham [1], various researchers have discussed the peristaltic flows with different geometries [2][3][4][5][6][7][8][9][10]. Recently, Subba Reddy et al. [11] have examined the flow of a viscous fluid due to symmetric peristaltic waves propagating on the horizontal sidewalls of a rectangular duct. They pointed out that the peristaltic flow of traditional asymmetric two dimensional channels may not better approximate the motion of intrauterine fluid in a sagittal cross section of the uterus. Therefore, they consider the rectangular duct instead of two dimensional channels. Later on Mandiwalla and Archer [12] extended the idea of Subba Reddy et al. [11] and discussed the influence of slip boundary condition on peristaltic pumping in a rectangular channel. Another area of focus in fluid mechanics is the study of non-Newtonian fluids. Some important examples of non-Newtonian fluids are blood, mustard, mayonnaise, tooth paste, asphalt, lava and ice, mud slides, snow avalanches, flow of plasma, nuclear fuel slurries, flow of nuclear fuel slurries flow of liquid metals and alloys, flow of mercury amalgams and lubrications with heavy oils and greases [13][14][15][16][17]. The flow characteristics of non-Newtonian fluids are quite different from those of Newtonian fluids. There are many models of non-Newtonian fluids which exhibits different flow properties. However, Second and third order fluids are those which exhibits both shear thinning and shear thickening effects [18][19][20][21][22].
The main goal here is to present the peristaltic flow of a third grade fluid in a rectangular duct. The governing equations of third grade fluid are simplified under the assumptions of long wave length and low Reynolds number and then the resulting nonlinear equations with the corresponding boundary conditions of rectangular duct are solved analytically with the help of Homotopy perturbation method and eigen function expansion method. The expressions of pressure rise, pressure gradient and stream functions are plotted and discussed for various physical parameters of interest.

Mathematical Modelling
Let us consider the peristaltic flow of an incompressible third grade fluid in a duct of rectangular cross section having the channel width 2d and height 2a. We are considering the Cartesian coordinates system in such a way that X-axis is taken along the axial direction, Y-axis is taken along the lateral direction and Z-axis is along the vertical direction of a rectangular duct as shown in Figure 1.
The peristaltic waves on the walls are represented as where a and b are the amplitudes of the waves, λ is the wave length, t is the velocity of propagation, X is the time and is the direction of wave propagation. The walls parallel to XZ plane remain undisturbed and are not subject to any peristaltic wave motion. We assume that the lateral velocity is zero as there is no change in lateral direction of the duct cross section. Let (U, 0, W) be the velocity for a rectangular duct. The governing equations for the flow problem are in which ρ is the density, P is the pressure, t is the time and S s is the stress tensor for third grade fluid. The stress tensor for third grade fluid is defined as [23] ( ) Let us define a wave frame (x,y) moving with the velocity c away from the fixed frame (X, Y) by the transformation , , Defining the following non-dimensional quantities Using the above non-dimensional quantities in Equations (2) to (6), the resulting equations after dropping the bars can be written as Under the assumption of long eave length 1 δ ≤ and Low Reynolds number. Equations. (8) to (11) take the form The corresponding boundary conditions are Where 0 1, φ ≤ ≤ 0 φ = for straight duct and 1 φ = corresponds to total occlusion.

Solution of the problem
Homotopy perturbation method: The Homotopy Perturbation Method for Eq. (12) can be defined as For our convenience we have taken as the liner operator. We define the initial guess as Let us define (17) Substituting Equation (17) into Equation (15) and then comparing the like powers of q one obtains the following problems with the corresponding boundary conditions.
First order system ( ) ( ) Solution of Zeroth order system With the help of Equation (17) the closed form solution of Equation (19) satisfying the boundary conditions is written as ( ) The solution of the above non-homogeneous partial differential equation can be expressed as  Where constants appearing in Equations (28) are defined Equation (26) The volumetric flow rate is given by The instantaneous flux is defined as The average volume flow rate over one period ( λ = T c ) of the peristaltic wave is defines as The pressure gradient is obtained from Equation (29) and (30) as ( ) n n n n n n n n n n n n n n n n n n dp Where dp dx is defined in Equation (31) It is noticed here that the limit 0 β → (keeping a fixed and → ∞ d ), the rectangular duct reduces to a two dimensional channel. It is also noticed that when 1 β = the rectangular duct becomes a square duct.

Numerical Result and Discussion
This section deals with the graphical and numerical results of the present problem under discussion. The expression for pressure rise and pressure gradient is calculated numerically using a mathematics software Mathematica. Figures 2-4 show the variation of pressure rise with volume flow rate Q for different values of aspect ratio β , amplitude ratio φ , and third grade parameter Γ . It is observed from Figure 2 that in the retrograde pumping ( 0, 0) ∆ > < p Q region the pressure rise increases with an increase in aspect ratio β , while in the peristaltic ) and copumping ( 0, 0) ∆ < < p Q regions the behavior is quite opposite. Here pressure rise decreases with an increase in aspect ratio β . Figure 3 shows the variation of pressure rise with volume flow rate Q for different values of amplitude ratio φ . It is observed that the pressure rise increases with an increase in the retrograde pumping and free regions, while in the copumping region the pressure rise decreases with an increase in amplitude ratio φ . It is observed from Figure 4 that the pressure rise decreases with an increase in third grade parameter Γ in all the regions. Figures 5-8  , pressure gradient increases with an increase in  in the retrograde pumping region, while in the peristaltic pumping, (free pumping 0 ∆ = p ) and copumping regions the behavior is quite opposite.
• The pressure rise increases with an increase in amplitude ratio in the retrograde pumping and free pumping regions, while in the copumping region the pressure rise decreases with an increase in amplitude ratio.
• The pressure rise decreases with an increase in third grade parameter Γ in all the regions.
• The pressure gradient increases with an increase in aspect ratio β and amplitude ratio φ and decreases with an increase in volume flow rate Q and third grade parameter Γ .
• The size of the trapped bolus decreases with an increase in φ , β and Q, while in the middle of the channel the size of the stream lines increases with an increase in φ , β and Q.
•The number of the trapping bolus decreases with an increase in Γ . aspect ratio φ and amplitude ratio φ and decreases with an increase in volume flow rate Q and third grade parameter Γ . Figures 9-11 show the streamlines for different values of amplitude ratio φ , aspect ratio β , Volume flow rate Q and third grade parameter Γ. It is observed from Figures 9 and 10 that the size of the trapped bolus decreases with an increase in φ , β and Q, while in the middle of the channel the size of the stream lines increases with an increase in φ , β and Q.
It is observed from Figure 11 that the number of the trapping bolus decreases with an increase in Γ , while in the middle of the channel the stream lines increases with an increase in Γ .

Concluding Remarks
In this paper effect of lateral walls on peristaltic flow of a third grade fluid in a rectangular duct is discussed. Assumptions of long wave length and low Reynolds number approximation is used to develop the simplified mathematical equations of third grade fluid for the rectangular duct. The reduced equations are solved analytically using Homotopy perturbation method and the eigen function expansion method. The results are discussed through graphs. The main findingcan be summarized as follows: • The pressure rise increases with an increase in aspect ratio β