Peristaltic flow of Burgers’ fluid with compliant walls and heat transfer
Introduction
The peristaltic waves generated along the flexible walls of tube are expected to provide a mathematical model in living organisms and industrial applications. Such flows are specifically observed in the functioning of ureter, food mixing and chyme movement in the intestine, movement of egg in the fallopian tube, the transport of spermatozoa in cervical canal, transport of bile in the bile duct, transport of cilia, circulation of blood in small blood vessels, roller and finger pumps, sanitary fluid transport, transport of corrosive fluids and many others. The theory of peristaltic transport is first addressed by Latham [1]. Later, Shapiro [2] showed good agreement between theoretical analysis and experiment. Based on the experimental work, Burns and Parkes [3] studied peristaltic motion of viscous fluid through a pipe and channel when there is sinusoidal variation of the waves at the walls. The peristaltic flow of viscous fluid in the channel/tube subject to long wavelength and low Reynolds number assumptions has been investigated by Shapiro et al. [4]. Now the relevant studies on this topic are extensive. Few representative studies in this direction are mentioned in the Refs. [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Very recently, Ellahi et al. [16], [17] discussed the peristaltic flow in a rectangular duct. The pipe flow of nanofluid in presence of temperature dependent viscosity is analyzed by Ellahi [18]. Nadeem et al. [19] studied the three-dimensional stretch flow of Casson fluid.
On the other hand, peristaltic flow with heat transfer is of considerable interest in hemodialysis and oxygenation. Peristaltic transport with heat transfer for the motion of viscous incompressible fluid in a two-dimensional non-uniform channel has been studied by Radhakrishnamacharya and Radhakrishna Murthy [20]. Vajravelu et al. [21] discussed the peristaltic activity in a vertical porous annulus with heat transfer. Mekheimer and Abd Elmaboud [22] studied heat transfer effects on peristaltic motion of MHD Newtonian fluid in a vertical annulus. The influence of heat transfer on the peristaltic flow in an asymmetric channel has been studied by Srinivas and Kothandapani [23]. Srinivas and Gayathri discussed peristaltic transport of a Newtonian fluid in a vertical asymmetric channel with heat transfer and porous medium [24]. Hayat et al. studied the effect of heat transfer on peristaltic flow of an electrically conducting fluid flowing in a porous space [25]. Sobh studied the heat transfer in a slip flow of peristaltic transport of a magneto-Newtonian fluid through a porous medium [26]. The space porosity and heat transfer effects on the peristaltic flow in an asymmetric channel have been analyzed by Mekheimer et al. [27]. The effects of heat transfer and slip on the peristaltic transport of an electrically conducting viscous fluid have been discussed by Nadeem and Akram [28]. Sobh et al. discussed peristaltic flow of viscoelastic fluid in an asymmetric channel with heat transfer [29]. Mixed convective heat and mass transfer in an asymmetric channel with peristalsis has been investigated by Srinivas et al. [30]. Tripathi and Beg [31] reported the influence of heat transfer on unsteady physiological magneto-fluid flow. Tripathi [32] further studied the effect of heat transfer on peristaltic flow pattern through a finite length channel.
In the above mentioned attempts, the compliant wall effects on the peristalsis have not been taken into account. Abd Elnaby and Haroun [33] discussed peristaltic flow of viscous fluid in a two-dimensional channel with compliant walls. The mathematical analysis has been carried out under the assumption of small amplitude ratio. Hayat et al.[34] analyzed the peristaltic flow of Maxwell fluid in a channel with compliant walls. Hayat et al. [35] have also investigated the effects of compliant walls and porous space on the MHD peristaltic flow of a Jeffrey fluid. Radhakrishnamacharya and Srinivasulu [36] have examined the influence of wall properties on peristaltic transport with heat transfer. Kothandapani and Srinivas [37] have studied the influence of wall properties in the MHD peristaltic transport with heat transfer and porous medium. Hayat et al. [38] investigated the effect of wall properties on the peristaltic flow of a third grade fluid in a curved channel with heat and mass transfer. The interaction between peristaltic transport of a dusty fluid (Saffman’s model) in a two-dimensional uniform channel and the elasticity of the channel’s flexible walls was studied by Rathod and Kulkarni [39]. To the best of our knowledge the influence of heat transfer analysis on peristaltic flow of Burgers’ fluid in a channel with compliant walls has not been reported yet. Thus, the aim of this work is to analyze such problem. Such investigation of peristaltic flow in Burgers’ fluid is important from both mathematical and practical view points. The paper is organized as follows. In section two, we provide the mathematical modelling and problem statement. The nonlinear differential equations subject to relevant boundary conditions have been solved in section three. Section four contains the graphical results. The key points of this work have been summarized in section five.
Section snippets
Problem statement
Consider an incompressible homogenous Burgers’ fluid in a channel with uniform width as shown in the figure.
Fig. Geometry of the problem.
The temperatures of the lower and upper walls in a channel are maintained at and respectively. Assume an infinite wave train travelling with velocity c along the walls. The walls of the channel are represented by [36]in which d is the half channel width, the wavelength, c the wave speed, a the wave amplitude and x and y the
Solution procedure
Since our interest lies in seeking the perturbation solution, so we expand the flow quantities in terms of the small wave number as follows
One can observe that under the long wavelength and low Reynolds number approximations, the problem is reduced to the case of viscous fluid.
Substituting expressions (27), (28), (29), (30), (31), (32) into (18), (19), (20), (21), (22), (23)
Discussion
The aim of this section is to see the effects of various important parameters. In particular the behaviors of the longitudinal velocity, temperature and heat transfer coefficient are displayed and discussed. Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6 can serve such purpose. Fig. 1 is plotted to examine the effect of various parameters on the longitudinal velocity, u. It is observed from Fig. 1(a) that the longitudinal velocity, u, decreases when the relaxation time, , increases. On the
Closing remarks
The main theme of this work is to see the effect of wall compliance on the peristaltic flow of a Burgers’ fluid. The main results of the present study are as follows.
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The behaviors of relaxation and retardation time on the longitudinal velocity are different.
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The longitudinal velocity is greater for the Newtonian fluid when compared with the other fluids.
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Temperature increases when relaxation time, , is increased.
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The temperature is minimum for the Newtonian fluid when compared with the other
Acknowledgment
The first author gratefully acknowledges the financial support of Higher Education Commission (HEC) of Pakistan (IPFP/HRD/HEC/2014/1658). We are grateful to the reviewer for the useful suggestions.
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