Homogeneous-heterogeneous reactions in peristaltic flow with convective conditions.

This article addresses the effects of homogeneous-heterogeneous reactions in peristaltic transport of Carreau fluid in a channel with wall properties. Mathematical modelling and analysis have been carried out in the presence of Hall current. The channel walls satisfy the more realistic convective conditions. The governing partial differential equations along with long wavelength and low Reynolds number considerations are solved. The results of temperature and heat transfer coefficient are analyzed for various parameters of interest.

Introduction

In the last few decades, the peristaltic motion of non-Newtonian fluids is a topic of major contemporary interest both in engineering and biological applications. To be more specific, such motion occurs in powder technology, fluidization, chyme movement in the gastrointestinal tract, vasomotion of small blood vessels, locomotion of worms, gliding motility of bacteria, passage of urine from kidney to bladder, reproductive tracts, corrosive and sanitary fluids transport, roller, finger and hose pumps and blood pump through heart lung machine. There is no doubt that viscoelasticity has key role mostly in all the aforementioned applications. Viscoelastic materials are non-Newtonian and possess both the viscous and elastic properties. Most of the biological liquids such as blood at low shear rate, chyme, food bolus etc. are viscoelastic in nature. Another aspect which has yet not been properly addressed is the interaction of rheological characteristics of fluids in peristalsis with convective effects. The significance of convective heat exchange with peristalsis cannot be under estimated for instance in translocation of water in tall trees, dynamic of lakes, solar ponds, lubrication and drying technologies, diffusion of nutrients out of blood, oxygenation, hemodialysis and nuclear reactors. The heat and mass transfer effects in such processes have prominent role. The magnetohydrodynamic (MHD) character of fluid especially in physiological and industrial processes seems too much important. Such consideration is useful for blood pumping and magnetic resonance imaging (MRI), cancer therapy, hyperthermia etc. The controlled application of low intensity and frequency pulsating fields modify the cell and tissue behavior. Magnetically susceptible of chyme is satisfied from the heat generated by magnetic field or the ions contained in the chyme. Also the magnetotherapy is an application of magnets to human body which is used for the treatment of diseases. The magnets could heal inflammations, ulceration, several diseases of bowel (intestine) and uterus. With all such motivations in mind, the recent researchers are engaged in the development of model of peristalsis of non-Newtonian liquids through different aspects including heat/mass transfer, MHD etc. Few representative attempts in this direction can be mentioned through the recent researchers [1]–[13] and several studies therein.

To our knowledge, no study has been undertaken yet to discuss the effects of homogeneous-heterogeneous reactions in peristaltic flows of non-Newtonian fluids. Even such study is yet not presented for the viscous fluid. However such consideration is quite important because many chemically reacting systems involve both homogeneous and heterogeneous reactions, with examples occurring in combustion, biochemical systems, catalysis, crops damaging through freezing, cooling towers, fog dispersion, hydrometallurgical processes etc. Hence the main objective of present investigation is to model and analyze the peristalsis of Carreau fluid in a compliant wall channel with convective conditions and homogeneous-heterogeneous reactions. Effects of Hall current and viscous dissipation are also considered. The resulting mathematical systems are solved and examined in the case of long wavelength and small Reynolds number. This article is structured as follows. Section two consists of mathematical modelling and solution expressions up to first order. The behaviors of sundry variables on the temperature, heat transfer coefficient and concentration are discussed graphically in section three. Main results of present study are also included in this section.

Mathematical Formulation

Consider the peristaltic transport of an incompressible Carreau fluid in two-dimensional compliant wall channel. The channel walls satisfy the convective conditions. The Cartesian coordinates and are considered along and transverse to the direction of fluid flow respectively. The flow is generated by the peristaltic wave of speed travelling along the channel walls. The Hall effects are also considered in the flow analysis. Further we consider the flow in the presence of a simple homogeneous and heterogeneous reaction model. There are two chemical species and with concentrations and respectively. The physical model of the wall surface can be analyzed by the expression:where represents the wave amplitude, the wavelength, the half width of symmetric channel, the time, displacement of upper wall and displacement of lower wall.

Consider the uniform magnetic field in the form

Application of generalized Ohm's law leads to the following expressionwhere represents the current density, the electrical conductivity, the velocity field, the electric charge and the number density of electrons. Also the effects of electric field are considered absent i.e.

If is the velocity with components in the and directions respectively then from Eqs. (2) and (3) we havewhere serves as the Hall parameter. The reaction model is considered in the form [14]–[16]:

while on the catalyst surface we have the single, isothermal, first order chemical reaction.

in which and are the rate constants. Both reaction processes are assumed isothermal.

The corresponding flow equations are as follows: where are the components of extra stress tensor for the Carreau fluid and extra stress tensor (see refs. [17] and [25]) here is given by

Here is the infinite shear-rate viscosity, the zero shear-rate viscosity, the time constant and the dimensionless form of power law index . Also is defined as follows:where denotes the second invariant strain tensor defined by and Here we consider the case for which and Therefore the extra stress tensor takes the form

It is worth mentioning that the above model reduces to viscous model for or The component forms of extra stress tensor are

In above equations is the material time derivative, the density of fluid, the fluid viscosity, the kinematic viscosity, the fluid temperature, the concentration of fluid, and the temperatures at the lower and upper walls respectively, the measure of the strength of homogeneous reaction, and the diffusion coefficients for homogeneous and heterogeneous reactions, the specific heat at constant pressure, the thermal conductivity, and the concentrations of homogeneous and heterogeneous reactions with serves as uniform concentration of reactant and the rate constant.

The exchange of heat at the walls is given by

Here and indicate the heat transfer coefficients at the upper and lower walls respectively.

The no-slip condition at the boundary wall is represented by the following expressions

The compliant wall properties are described through the expressionin which is the elastic tension in the membrane, the mass per unit area and the coefficient of viscous damping. The mass conditions under the homogeneous and heterogeneous reactions are given through the following expressions:

where indicates the rate constant.

Performing (7) we get

Introducing the stream function and defining the following dimensionless variables:

Eqs. (8), (9) and (22) yield with the dimensionless conditions

Also Eqs. (14–16) become

In above equations asterisks have been omitted for simplicity. Here is the dimensionless wave number, the Reynolds number , the Prandtl number , the amplitude ratio , the chemical reaction parameter , the Hartman number the non-dimensional elasticity parameters , , the Schmidt number , the Eckert number the Brinkman number , the heat transfer Biot numbers , the Weissenberg number , the ratio of diffusion coefficient the strength measuring parameters and (for homogeneous and heterogeneous reaction respectively) and (non-dimensional form of ) are given through the following variables:

We now employ the approximations of long wavelength and low Reynolds number [17]–[23] and equality of diffusion coefficients and i.e. The assumption leads to the following relation:and we obtain the following set of equations

2.1 Method of solution

It is seen from Eqs. (39) and (41) that these Eqs. are non-linear and involve Weissenberg number and homogeneous reaction parameter respectively. Therefore the problem at hand cannot be solved exactly, but can be linearized about "small" parameter to the mathematical description of the exactly solvable problem. The technique is referred as perturbation. Perturbation method represent a very powerful tool in modern mathematical physics and, in particular, in fluid dynamics and leads to a series solution of resulting system of equations having small paramter. Therefore we have applied this method to form the series solutions for stream function , temperature and concentration corresponding to the involved non-linear quantities ( and ). For this we write the flow quantities in the forms:

2.2 Zeroth order system and its solution

The zeroth order system is given by

The solutions of Eqs. (46–48) subject to the boundary conditions (49–52) are

2.3 First order system and its solution

At this order we have

Solving Eqs. (57–59) and then applying the corresponding boundary conditions we get the solutions in the forms given below: and the heat transfer coefficient is

in which

Results and Discussion

This section is prepared to explore the effects of influential parameters on the temperature, heat transfer coefficient and concentration.

3.1 Temperature profile

Figs. (1–8) are formulated to examine the impact of various involved parameters on temperature distribution Fig. 1 indicates the increasing behavior of temperature profile with wall parameters and while corresponds to reduction in temperature profile. It is in view of the fact that elastic properties of the wall depicted by and cause less resistance to flow of fluid velocity as well as energy. On the other hand the damping characteristic of the wall identified by reduces the velocity and temperature of the fluid (see Fig. 1). The temperature profile is an increasing function of Brinkman number (see Fig. 2). This is because of the increase in internal resistance of fluid particles which increases the fluid temperature. The Biot numbers and on the lower and upper walls have similar effect on the temperature profile increase in and decreases the temperature profile near upper and lower channel walls respectively (see Figs. 3 and 4). It is seen that increasing and reduces the thermal conductivity which causes reduction of temperature profile. The Hall parameter increases the temperature. This is due to the fact that electrical conductivity increases with increasing values of (see Fig. 5). It is observed from Fig. 6 that Hartman number lessens the temperature distribution. Also the results drawn in Figs. 7 and 8 show opposite effects of Weissenberg number and the power law index increasing values of reduces the temperature whereas an increase in enhances the temperature of fluid. The obtained results are in good agreement with the articles presented in [17]–[19].

Figure 1

Plot of temperature for wall parameters with , , and

Figure 8

Plot of temperature for power law index with , , and

Figure 2

Plot of temperature for Brinkman number with , , and

Figure 3

Plot of temperature for Biot number with , , and

Figure 4

Plot of temperature for Biot number with , , and

Figure 5

Plot of temperature for Hall parameter with , , and

Figure 6

Plot of temperature for Hartman number with , , and

Figure 7

Plot of temperature for Weissenberg number with , , and

3.2 Heat transfer coefficient

Figs. 9–16 demonstrate the influence of embedded parameters on the heat transfer coefficient The graphs signify the oscillatory behavior of because of the propagation of peristaltic waves. Fig. 9 reveals that magnitude of heat transfer coefficient increases for compliant wall parameters and . Since and describes the elastic nature of wall that offer less resistance to heat transfer. Increasing values of Brinkman number show similar behavior on heat transfer as of wall parameters. However the results obtained are much more distinguished in case of (see Fig. 10). The Biot number causes reduction in magnitude of heat transfer coefficient on the upper wall. Here thermal conductivity decreases with an increase in which lessens the impact of heat transfer coefficient near positive side as depicted in Fig. 11. Reverse effect of Biot number has been observed in the region from Fig. 12 as heat transfer being directly related to Biot number dominates with an increase in which in turn increases the heat transfer distribution. Fig. 13 shows decrease in heat transfer coefficient with Hall parameter . Also in absence of Hall parameter the results are much more distinguished. The Hartman number is an increasing function of heat transfer coefficient as fluid viscosity decreases with an increase in . The less viscous fluid particles will move through gain of higher kinetic energy that causes rise in transfer of heat (see Fig. 14). The effects of Weissenberg number are displayed in Fig. 15. The obtained results show increase in transfer of heat when increases as speed of wave increases with an increase in that supports the transfer of heat. The increasing values of power law index show decline in heat transfer distribution (see Fig. 16).

Figure 9

Plot of heat transfer coefficient for wall parameters with , , and

Figure 16

Plot of heat transfer coefficient for power law index with , and

Figure 10

Plot of heat transfer coefficient for Brinkman number with , and

Figure 11

Plot of heat transfer coefficient for Biot number with , and

Figure 12

Plot of heat transfer coefficient for Biot number with , and

Figure 13

Plot of heat transfer coefficient for Hall parameter with , and

Figure 14

Plot of heat transfer coefficient for Hartman number with , and

Figure 15

Plot of heat transfer coefficient for Weissenberg number with , and

3.3 Homogeneous-Heterogeneous reactions effects

Effects of homogeneous and heterogeneous reaction parameters and and Schmidt number are displayed in the Figs. 17–19. The results drawn in Fig. 17 illustrates the dual behavior of homogeneous reaction parameter on the concentration profile. It is observed that concentration increases in the region as in this region increase in enhances the fluid density hence concentration rises while in the region the concentration decreases because viscosity reduces. On the other hand the heterogeneous reaction parameter shows the opposite behavior when compared with it increases along positive side of the coordinate axes (since diffusion reduces with an increase in and less diffused particles will rise the concentration) and decreases along negative side of coordinate axes (as increase in rate of reaction dominates the decrease in diffusion in this region). It is evident from Figs. 17 and 18 that the concentration distribution of reactants increases from to in both cases and after a certain value of it starts decaying. This critical value of depends on the strength of homogeneous reaction and it is prominent for increasing . The effects of Schmidt number are depicted in Fig. 19. The exhibited results are quite similar to Fig. 17. The drawn results follow by the fact that viscosity of fluid increases with an increase in Schmidt number that provides resistance to flow of fluid. The slow moving fluid particles have small molecular vibrations which lessen the concentration of fluid. As Schmidt number defines the ratio of viscous diffusion rate to molecular diffusion rate. Hence increasing values of enhances the viscous diffusion rate for fixed molecular diffusion rate which in turn helps to increase the concentration of fluid (see Fig. 19). The similar findings are reported by Shaw et al. [24].

Figure 17

Plot of concentration for homogeneous reaction parameter with , and .

Figure 19

Plot of concentration for Schmidt number with , and .

Figure 18

Plot of concentration for heterogeneous reaction parameter with , and .

3.4 Concluding remarks

The present analysis explores the effects of homogeneous and heterogeneous reactions in the peristalsis of Carreau fluid. Such analysis even for viscous fluid is yet not available. The major results of this study are listed below.

Similar behavior is observed for compliant wall parameters on temperature profile and heat transfer coefficient.

Temperature is increasing function of Brinkman number and Hall parameter.

The Biot numbers and Hartman number decrease the temperature of fluid.

Opposite effects of Weissenberg number and power law index are observed on the temperature profile and heat transfer coefficient.

Concentration of the reactants is more signified in case of homogeneous reaction parameter than heterogeneous reaction parameter .