Abstract
We investigate the unsteady peristaltic transport of a viscoelastic fluid with fractional Maxwell model through two coaxial vertical tubes. This analysis has been carried under low Reynolds number and long wavelength approximations. Analytical solution of the problem is obtained by using a fractional calculus approach. The effects of Grashof number, heat parameter, relaxation time, fractional time derivative parameter, amplitude ratio, and the radius ratio on the pressure gradient, pressure rise, and the friction forces on the inner and outer tubes are graphically presented and discussed.
1. Introduction
Peristaltic transport is a mechanism of pumping fluids in tubes when progressive wave of area contraction or expansion propagates along the length on the boundary of a distensible tube containing fluid. In the physiological process, peristalsis or the blood flow is the major applications of this transport phenomenon. The contractions and relaxations of the walls propel the fluid. The first investigation to study the peristaltic pumping was by Latham [1]. The peristaltic transport of a Newtonian fluid at long wavelengths at low Reynolds number has been studied with particles suspended by Kaimal [2] or without particles by Shapiro et al. [3] and by Mishra and Rao [4]. The non-Newtonian effects of fluids on the peristaltic transport have been studied for a third-grade fluid by Ali et al. [5], for a Jeffrey fluid by Abd-Alla et al. [6], for a Maxwell model by Rachid and Ouazzani [7], for a Herschel-Bulkley fluid by Medhavi [8], or for a power-law fluid by Shukla and Gupta [9]. The effect of an endoscope on peristaltic flow for a Newtonian fluid through a nonuniform annulus has been analyzed by Mekheimer [10]. This effect has been also studied for a Johnson Segalman fluid by Akbar and Nadeem [11] or for a Maxwell fluid by Husseny et al. [12]. Recently, the peristaltic transport of non-Newtonian fluids has gained considerable interest because of its applications in industry and biology. Mathematically, the non-Newtonian fluids have a nonlinear relationship between the stress and the rate of strain. It is difficult to propose a single model which can exhibit all the properties of non-Newtonian fluids. To describe the viscoelastic properties of such fluids recently, constitutive equations with ordinary and fractional time derivatives have been introduced. Fractional calculus has proved to be very successful in the description of constitutive relations of viscoelastic fluids. The starting point of the fractional derivative model of viscoelastic fluids is usually a classical differential equation which is modified by replacing the time derivative of an integral order with fractional order and may be formulated both in the Riemann-Liouville and in the Caputo sense [13]. This generalization allows one to define precisely noninteger order integrals or derivatives and this plays an important role in studying the valuable tool of viscoelastic properties. In recent years and considering the relevance of fractional models of viscoelastic fluids, a number of articles have addressed unsteady flows of viscoelastic fluids with the fractional Maxwell model [14, 15], fractional second-grade fluid [16, 17], fractional Oldroyd-B model [18, 19], and fractional Burgers model [20]. For example, Tripathi [21] studied a mathematical model based on viscoelastic fractional Oldroyd-B model for the peristaltic flow of chyme in small intestine, which is assumed to be in the form of an inclined cylindrical tube. The peristaltic transport for a fractional second-grade model through a cylindrical tube has been investigated by Tripathi [22]. Rathod and Channakote [23] analyzed the interaction of heat transfer on peristaltic pumping of a fractional second-grade fluid through a vertical cylindrical tube by using Caputo’s definition. For the fractional Maxwell model, Tripathi et al. [24] studied the peristaltic flow through a channel where the homotopy perturbation method (HPM) and Adomian decomposition method (ADM) are used. For the fractional Burger’s model Tripathi et al. [25] studied the influence of slip condition on peristaltic transport by using a homotopy analysis method (HAM). This generalized model of fluid has been investigated for an infinite oscillating plate by Khan et al. [26], for an orthogonal rheometer by Siddiqui et al. [27] or between concentric cylinders by Shah [28]. Heat transfer analysis is one of the principle axes of research in chemical engineering, but nowadays due to its wide applications in biofluid mechanics it has great interaction with peristaltic motion. The effect of heat transfer on peristaltic transport of a Newtonian fluid through a porous medium in a vertical tube under the effect of a magnetic field has been studied by Vasudev et al. [29]. This effect on peristaltic flow with slip condition and variable viscosity in an asymmetric channel has been analyzed by Hayat and Abbasi [30]. Hayat et al. [31] investigated the effects of heat and mass transfer and wall properties on the peristalsis in a power-law fluid.
The aim of this paper is to analytically study the effect of heat transfer on peristaltic flow of a viscoelastic fluid with fractional Maxwell model in the gap between two coaxial vertical tubes. The influence of different physical parameters on the pressure gradient, pressure rise, and the fictional forces has been graphically shown and discussed.
2. Formulation and Analysis
Consider the peristaltic flow of a viscoelastic fluid with fractional Maxwell model through the gap between two coaxial vertical tubes. The inner tube is rigid (endoscope) maintained at a temperature , and the outer tube has a sinusoidal wave traveling down its wall and it is exposed to temperature as seen in Figure 1.
In a cylindrical coordinate system , the equations for the tube walls in their dimensional form are given bywhere is the radius of the endoscope, is the average radius, is the amplitude of the wave, is the wavelength, and is the wave speed of the outer tube.
The equation of the fractional Maxwell fluid is given bywith , , , and being the time, shear stress, rate of shear strain, and the viscosity, respectively. is the relaxation time and is the fractional time derivative parameter such that . is the upper convected fractional derivative defined byin whichand is the fractional differentiation operator of order with respect to and may be defined as [32]Here denotes the Gamma function.
We choose a cylindrical coordinate system where -axis is the longitudinal direction and the -axis is transverse to it.
The equations of motion of the flow for an incompressible fluid are given bywhere is the fluid density, is the temperature, is the coefficient of linear thermal expansion of the fluid, denotes thermal conductivity, denotes specific heat at constant pressure, is the constant heat addition/absorption, and are the velocity components in the wave frame, and is the pressure.
We assume that the extra stress depends on and only. After using the initial condition , (2)–(4) yield andwhere is the tangential stress.
For carrying out further analysis, we introduce the following dimensionless parameters:where is the radius ratio, is the wave number, is the Reynolds number, is the Prandtl number, is the Grashof number, is the nondimensional heat source/sink parameter, and is the amplitude ratio with .
Using the above nondimensional quantities and under the assumptions of long wavelength approximation (i.e., or ) and the low Reynolds number (i.e., ), (6) yield
The boundary conditions arewhere is the dimensionless equation of the outer tube radius in the wave frame.
Integrating (12) with the boundary conditions (13), we find the dimensionless temperature as follows:
Use the transformations between the laboratory and the wave frames, in the dimensionless form, given bywhere and are the velocity components in the fixed frame.
Substituting (14) in (10) and using the same boundary conditions, we obtain the velocity profile of the fluid aswhere .
Using the definition of the fractional differential operator (5) we find the expression of as follows:
The volume rate of flow in the fixed coordinate system (, ) is given asUsing the transformation (15) we findwhere is the volume flow rate in the moving coordinate system given by
The time-averaged flow rate is defined by
Eliminating the flow rate between (19) and (21) we find
From (18) and (22) we obtain the pressure gradient as follows:
3. The Pumping Characteristics
The pressure rise and the frictional force at the walls of the inner and outer tubes, in the nondimensional form, are given by
4. Results and Discussions
We note that the fractional Maxwell model reduces to a Newtonian fluid when .
The analytical expressions for the pressure gradient , the pressure rise , and the frictional forces on the inner and outer tubes, respectively, and , are derived in the previous sections. In order to compute these physical quantities with respect to interest parameters of problem we observe that the integrals in (24) are not integrable in the closed form. They are evaluated numerically using a mathematics software.
Pressure gradient against the axial distance on one wavelength for different given values of the Grashof number , the heat parameter , the relaxation time , the fractional time derivative parameter , the amplitude ratio , and the radius ratio is plotted in Figures 2–4. These figures show that, in the wider part of the tube , is relatively small, where the flow can easily pass without giving any large pressure gradient. However, in the narrow part a much larger pressure gradient is required to maintain the same flux to pass it, especially for the narrowest position near . Moreover the maximum of the pressure gradient increases with increasing , , , and while it decreases with the increase in and .
The effects of various parameters on the pressure rise versus the time-averaged flow rate are shown in Figures 5–7. It is known that we have the following peristaltic regions, pumping region (), free-pumping region (), and copumping region (). These figures show a linear relationship between and in all the regions; that is, the pressure rise decreases with increasing the time-averaged flow rate. From Figure 5 we observe that the time-averaged flow rate increases with increasing while it decreases with the increase in in all the regions. In addition, and the flow rate and the pumping increase with increasing while they decrease with the increase in . Figures 6-7 show that for different values of , , , and the curves are intersecting in the pumping region. In addition, for a given value of the pressure rise we observe that in both the free-pumping and copumping regions the time-averaged flow rate increases with increasing while it decreases with the increase in . The same figures show that decreases in the free-pumping and copumping regions with increasing and .
In Figures 8–13 we plotted the frictional forces and on the inner and outer tubes, respectively. These figures show that and have an opposite behavior compared to the pressure rise versus the physical parameters. In addition the frictional force on the outer tube is greater than the frictional force on the outer tube .
5. Conclusions
We have analytically studied the effect of heat transfer on peristaltic flow of fractional Maxwell model in the gap between two vertical coaxial tubes. The problem is simplified under the assumptions of long wavelength approximation and low Reynolds number. Analytical solution of problem is obtained by using a fractional calculus approach. The pressure gradient, pressure rise, and frictional forces on the endoscope and on the outer tube are discussed with the physical parameters, Grashof number , the heat parameter , the relaxation time , the fractional time derivative parameter , the amplitude ratio , and the radius ratio . The graphical solutions have shown the following:(1)The maximum of the pressure gradient increases with increasing , , , and while it decreases with the increase in and .(2)The time-averaged flow rate increases with increasing while it decreases with the increase in in all the regions of pumping for a given value of the pressure rise .(3) increases with the increase in in both the free-pumping and co-pumping regions while we observe an opposite behavior versus in these two regions.(4) decreases with increasing and in the free-pumping and copumping regions.(5)The frictional forces and have opposite behavior compared to the pressure rise versus the physical parameters.
Appendix
Consider
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author is grateful to the referees and editors for their constructive suggestions.