Abstract

The existing analysis deals with heat transfer occurrence on peristaltic transport of a Carreau fluid in a rectangular duct. Flow is scrutinized in a wave frame of reference moving with velocity away from a fixed frame. A peristaltic wave propagating on the horizontal side walls of a rectangular duct is discussed under lubrication approximation. In order to carry out the analytical solution of velocity, temperature, and pressure gradient, the homotopy perturbation method is employed. Graphical results are displayed to see the impact of various emerging parameters of the Carreau fluid and power law index. Trapping effects of peristaltic transport is also discussed and observed that number of trapping bolus decreases with an increase in aspect ratio .

1. Introduction

The applications of peristaltic flows in medical and engineering sciences have attracted the attention of a number of researchers. Applications of peristalsis occur in swallowing food through the esophagus, urine transport from the kidney to the bladder through the ureter, transport of the spermatozoa in the efferent ducts of the male reproductive tract, movement of the ovum in the fallopian tube, movement of the chyme in the gastrointestinal tract, the transport of lymph in the lymphatic vessels, and vasomotion in the small blood vessels such as the arterioles, veins, and capillaries. The peristaltic phenomenon was first discussed by Latham [1]. Based on his experimental theory, numerous researchers have inspected the phenomenon of peristaltic transport under many conjectures [2–10].

Another fascinating area in connection with peristaltic motion is the heat transfer which has industrial applications like sanitary fluid transport, blood pumps in the heart-lungs machine and transport of corrosive fluids where the contact of fluid with the machinery parts are prohibited. Only a limited attention has been focused to the study of peristaltic flows with heat transfer [11–15].

An immense amount of literature is presented on two-dimensional peristaltic flow problems. The study of peristaltic phenomenon in a rectangular channel was first examined by [16]. Based on the theory of [16], several researchers have studied the phenomenon of peristaltic transport in a rectangular duct under various approximations [17–23]. In the papers cited above, the phenomena of heat transfer are not taken into account. Keeping in mind the present information, the heat transfer phenomena on the peristaltic flow of non-Newtonian fluid have not been discussed in a three-dimensional channel. So, the aim of the present problem is to discuss the effects of heat transfer on peristaltic flow of a non-Newtonian fluid in a rectangular duct with different wave forms. The governing equations for the three-dimensional rectangular channel are first modeled for Carreau fluid and then simplified under the long wavelength and low Reynolds number approximation. Homotopy perturbation technique is carried out to calculate the analytical solution of the highly nonlinear partial differential equations. The expressions for velocity, pressure rise, pressure gradient, and temperature have been computed and discussed through graphs. Three-dimensional graphical representations for the velocity field are also discussed through graphs. In the end, different wave forms are also used.

2. Mathematical Formulation of the Problem

Let us consider the peristaltic flow of an incompressible Carreau 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 the is taken along the axial direction, is taken along the lateral direction, and is along the vertical direction of a rectangular duct.

The peristaltic waves on the walls are represented as where and are the amplitudes of the waves, is the wave length, is the velocity of propagation, is the time, and is the direction of wave propagation. The walls parallel to the plane remain undisturbed and are not subjected to any peristaltic wave motion. We assume that the lateral velocity is zero as there is no change in the lateral direction of the duct cross section. Let be the velocity for a rectangular duct. The governing equations for the flow problem are in which is the density, is the pressure, is the time, and is the stress tensor for the Carreau fluid; is the specific heat; and is the temperature. The stress tensor for the Carreau fluid is defined by [17].

Let us define a wave frame moving with the velocity away from the fixed frame by the transformation

Defining the following nondimensional quantities,

Using the above nondimensional quantities in Equations (2), (3), (4), (5) and (7), the resulting equations after dropping the bars can be written as where

Under the assumption of long wavelength and low Reynolds number , Equations (10), (11), (12), (13), (14) and (21) take the form

The corresponding boundary conditions are where , for straight duct and corresponds to total occlusion.

3. Solution of the Problem

3.1. Homotopy Perturbation Method

The solution of the nonlinear partial differential Equations (22) and (23) has been calculated using homotopy perturbation technique. The homotopy perturbation method for Equations (22) and (23) can be defined as [24–30]. or

Here, is an embedding parameter which has the range , under the condition that for , we get the initial solution, and for , we seek the final solution. Here, £ is the linear operator which is taken here as . We define the initial guess as

Let us define substituting Equation (30) into Equations (26) and (27), and then comparing the like powers of , one obtains the following problems with the corresponding boundary conditions.

3.2. Zeroth Order System

3.3. First-Order System

3.4. Second-Order System

The resulting series solutions after three iterations are determined using Equation (30) as (when ) and are evaluated as The volumetric flow rate is given by