Abstract
A theoretical analysis is presented for the peristaltic motion of a magneto-hydrodynamic (MHD) non-Newtonian fluid in channel with complaint walls. The fluid obeys viscoelastic non-Newtonian model with Burger’s constitutive equation. The relevant equations are first developed and then solved using perturbation technique. Expressions of stream function and velocity components are constructed under the assumption that (characteristic ratio of transversal and axial scales of peristaltic motion) is small. The results indicate the strong effects of Burger’s fluid parameter, Hartman number, Reynolds number, and complaint wall parameters on the velocity field and stream function. The obtained solutions are shown graphically for the different values of involved parameters.
1. Introduction
Peristaltic pumping is a mean of fluid transport caused by waves of contraction along the boundary of a channel/tube. To analyze the properties of peristalsis theoretically, the flow caused by sinusoidal traveling waves of contraction along the boundary has been considered in an infinite channel/tube. In general, flow in a peristaltic channel is driven by two independent mechanisms: the movement of the walls and due to the mean pressure gradient along the channel. After the pioneering work of Shapiro et al. [1] the fluid flow due to prescribed wall motion has been described theoretically by Ebaid [2]. The approach of Fung and Yih [3] was used to analyze the peristaltic flow by Hayat et al. [4] and Haroun [5]. In the above mentioned studies, peristaltic motion was investigated under various assumptions. One of the important aspects in such studies is the nature of the fluid. Since, it is a fact that most of the biological and industrial fluids are non-Newtonian in nature, therefore, peristaltic flow of these fluids has also been studied by Hayat et al. [6] considering non-Newtonian Maxwell fluid in a channel. Srinivas et al. [7] studied the MHD peristaltic motion with slip and heat characteristics. The effects of wall flexibility on the peristaltic motion have also attracted the attention of various researchers in the field. Mittra and Prasad [8] initiated the study of peristaltic transport of viscous fluid with compliant walls. Their work has been generalized by Muthu et al. [9] for viscous incompressible fluid in circular tube and Abd Elnaby and Haroun [10] for two-dimensional channel. The work in [10] has been extended for Maxwell fluid by Ali et al. [11]. Peristaltic motion of Jeffrey fluid in a channel having compliant walls was also studied by Hayat et al. [12]. Recently the heat and mass transfer effects on the peristaltic flow of an Oldroyd-B fluid in a channel with complaint walls were investigated by Hayat et al. [13]. Motivated by the above mentioned investigations, the aim of the present paper is to extend the work of [13] by considering the fluid which obeys the constitutive equation of Burger’s fluid model. The Burger’s fluid is a rate type viscoelastic fluid model [14] which has been found quite successful in predicting the behavior of viscoelastic material, food products such as cheese and soil. The viscoelastic behavior of fine-grained polycrystalline olivine can also be predicted by Burger’s fluid model [15]. Moreover, this model has also been used for calculation of transient creep proportion of the earth’s mantle [16] and in the interpretation of postglacial uplift [17].
The paper is arranged in the following manner. Flow geometry is explained in Section 2 while governing equations are developed in Section 3. Series solution is provided in Section 4. Results and discussion presented in Section 5. Some conclusions are made at the end of article in Section 6.
2. Flow Geometry
Let us consider the flow of magneto hydrodynamic (MHD) Burger’s fluid due to sinusoidal motion of the compliant walls in a channel. A constant magnetic field of strength is applied in the direction normal to the flow. The deformation of the walls of the channel is represented by the expression where is the amplitude of the peristaltic wave, is the wavelength, is the time, is the speed of the wave, and is the half-width of the channel.
3. Governing Equations
Let and denote the velocity components in the axial and transverse direction, respectively, then we can write The law of conservation of mass and momentum for the flow of an incompressible fluid are where is the velocity, is the density, is the pressure, and is the extra stress tensor, which satisfies the following relation for a Burger’s fluid: In the above equation is the dynamic viscosity, is the first Rivlin-Ericksen tensor, and are the relaxation and retardation times, respectively, is Burger’s fluid parameter, is the gradient of velocity, and is the upper convective derivative and defined as Using the definition of velocity and in (3) we get the following equations: where the components of extra stress appearing in the above equations satisfy three equations that can be obtained directly from (4a).
The relevant boundary conditions are [8] where is the tension in the wall, is the mass per unit area, is the coefficient of viscous damping, and is the electrical conductivity of fluid. Defining the stream function such that , , and the following dimensionless variables: we find that the continuity equation (5) is identically satisfied and (8)-(9), after dropping the primes, reduce to In the above equations , is the dimensionless amplitude, denotes the characteristic ratio of transversal and axial scales of peristaltic motion, is the Reynolds number, denotes the Hartman number, , , are the nondimensional elasticity parameters, and is the dimensionless Burger’s fluid parameter.
4. Solution of the Problem
Due to nonlinearity of (11) it seems impossible to get a closed form solution. Therefore, some plausible assumptions must be made to put these equations in tractable form. We would like to mention here that without such assumptions one cannot obtain solution of these equations even for Newtonian fluid. Following [1], we assume that the is small and proceed to write Substituting expressions (13) into (11) and (12) and collecting the terms of equal powers of , we get the following systems.
4.1. Zero-Order System
4.2. First-Order System
4.3. Second-Order System
The solution of these systems yields where and value of is given in the appendix.
The value of constants involved in (18) is given in the appendix.
5. Results and Discussion
The purpose of this section is to provide the graphical illustration of the solution obtained in previous section. Emphasis has been given to examine the influence of emerging parameters such as , and the rheological parameter of the Burger’s fluid on the axial velocity and the stream function.
In fact the novelty of the present analysis is based on the effects of Burger’s fluid parameter on various flow characteristics. This is because of the fact that the results in the absence of are available already in the literature [13] for comparison. Particularly the velocity is plotted for different values of the involved parameter including the Burger’s fluid parameter . Figure 1 interprets the effects of Hartman number and on . It is noted from this figure that is an increasing function of , while it decreases with . The decrease of with is a result of the fact that magnetic force acts in the transverse direction and hence resists the flow due to peristalsis.
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The effects of and on can be seen through Figure 2. This figure shows that increases by increasing and . The Reynolds number represents the relative magnitude of inertial to the viscous force and with the increase of Reynolds number, the viscous forces decrease resulting in increases in velocity. Similarly with an increase in the wavelength of wave decreases which causes the increase in the velocity. It is worth mentioning that these results are in accordance with those obtained in [11] for a Maxwell fluid. A maximum in the velocity at center of the channel and an increase of velocity with and are also observed from Figure 2.
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Figure 3 is prepared to see the effects of complaint wall parameters , and on longitudinal distribution of velocity . This figure reveals that the velocity increases when both wall parameters and increase because of the increase in flexibility and elasticity of the wall. However, velocity decreases with due to the increase in rigidity of the wall. Note that we recover the results reported in [11] for .
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Figure 4 illustrates the effects of and on . It is observed that when we increase the relaxation parameter the velocity gets increased due to the increment in . The effects of retardation parameter on the velocity are quite opposite to that of .
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The purpose of Figures 5, 6, 7, and 8 is to capture the variation in size and circulation of the trapped bolus, in particular, the effects of , and . It is observed from these figures that the size of bolus decreases by increasing and while it increases with and .
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6. Conclusions
Peristaltic motion of a (MHD) non-Newtonian fluid with Burger’s constitutive equation in channel with complaint walls is taken into account. The results for velocity and stream function are obtained and compared with the literature. It is observed that Reynolds number and cause an increase in the velocity while a decrease in velocity is noted by increasing Hartman number. In previous mathematical model [13], Burger’s parameter is absent and therefore such models cannot accurately describe the peristaltic motion of food stuff like cheese and so forth. The physical impact of this parameter is quite interesting. Our mathematical model predicts an increase in velocity by increasing Burger’s fluid parameter. The model further reveals that the size and circulation of the trapped bolus increase with Burger’s fluid parameter.
Appendix
Here we provide the value of along with various constants appearing in it and in the expressions of and , as follows:
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the reviewers for their valuable suggestions which improved the quality of the paper.