Flow of an Eyring-Powell Model Fluid between Coaxial Cylinders with Variable Viscosity

Abstract

We consider the flow of Eyring-Powell model fluid in the annulus between two cylinders whose viscosity depends upon the temperature. We consider the steady flow in the annulus due to the motion of inner cylinder and constant pressure gradient. In the problem considered the flow is found to be remarkedly different from that for the incompressible Navier-Stokes fluid with constant viscosity. An analytical solution of the nonlinear problem is obtained using homotopy analysis method. The behavior of pertinent parameters is analyzed and depicted through graphs.

1. Introduction

The analysis of the behaviour of the fluid motion of the non-Newtonian fluids becomes much complicated and subtle as compared to Newtonian fluids due to the fact that non-Newtonian fluids do not exhibit the linear relationship between stress and strain. Rivlin and Ericksen [1] and Truesdell and Noll [2] classified viscoelastic fluids with the help of constitutive relations for the stress tensor as a function of the symmetric part of the velocity gradient and its higher (total) derivatives. In recent years, there have been several studies [3–12] on flows of non-Newtonian fluids. It is a well-known fact that it is not possible to obtain a single constitutive equation exhibiting all properties of all non-Newtonian fluids from the available literature. That is why several models of non-Newtonian fluids have been proposed in the literature. Eyring-Powell model fluid is one of these models. Eyring-Powell model was first introduced by Powell and Eyring in 1944. However, the literature survey indicates that very low energy has been devoted to the flows of Eyring-Powell model fluid with variable viscosity. Massoudi and Christie [13] have considered the effects of variable viscosity and viscous dissipation on the flow of a third grade fluid in a uniform pipe. Massoudi and Christie [13] found the numerical solutions with the help of straight forward finite difference method. They also discussed that the flow of a fluid-solid mixture is very complicated and may depend on many variables such as physical properties of each phase and size and shape of solid particles. Later on, the influence of constant and space dependent viscosity on the flow of a third grade fluid in a pipe has been discussed analytically by Hayat et al. [14]. The approximate and analytical solution of non-Newtonian fluid with variable viscosity has been analyzed by Yürüsoy and Pakdermirli [15] and Pakdemirli and Yilbas [16]. The pipe flow of non-Newtonian fluid with variable viscosity keeping no slip and partial slip has been discussed analytically by Nadeem and Ali [17] and Nadeem et al. [18]. More recently, Nadeem and Akbar [19] studied the effects of temperature dependent viscosity on peristaltic flow of a Jeffrey-six constant fluid in a uniform vertical tube. The main aim of the present study is to venture further in the regime of Eyring-Powell model fluid with variable viscosity. To the best of the authors knowledge no attempt has been made to investigate Eyring-Powell model fluid in the annulus between two cylinders whose viscosity depends upon the temperature. The governing equations for Eyring-Powell model fluid are formulated considering cylindrical coordinates system. The equations are simplified using the assumptions of long wave length and low Reynolds number approximation. The obtained non-linear problem is solved using homotopy analysis method [20–28]. The effects of the emerging parameters are analyzed and depicted through graphs.

2. Mathematical Model

The constitutive equation for a Cauchy stress in an Eyring-Powell model fluid is given by 𝐒=𝜇∇1𝑉+𝛽sinh⁻¹1𝑐∇𝑉,sinh⁻¹1𝑐∇𝑉1∼𝑐∇1𝑉−61𝑐∇𝑉³,1𝑐∇𝑉≪1, (1) where 𝑉 is the velocity, 𝐒 is the Cauchy stress tensor, 𝜇 is the coefficient of shear viscosity, and 𝛽 and 𝑐 are the material constants. We take the velocity and stress as 𝐕(𝑟)=00𝑣(𝑟)=𝑆,𝐒^𝑟𝑟𝑆_𝑟𝜃𝑆_𝑟𝑧𝑆_𝜃𝑟𝑆_𝜃𝜃𝑆_𝜃𝑧𝑆_𝑧𝑟𝑆_𝑧𝜃𝑆_𝑧𝑧. (2)

3. Physical Model

Consider the steady flow of an Eyring-Powell model fluid with variable temperature dependent viscosity between coaxial cylinders. The motion is caused due to a constant pressure gradient and by the motion of the inner cylinder parallel to its length, whereas the outer cylinder is kept stationary. The heat transfer analysis is also taken into account. The dimensionless problem which can describe the flow is 𝜇𝑟𝑑𝑣+𝑀𝑑𝑟𝑟𝑑𝑣𝑑𝑑𝑟+𝜇²𝑣𝑑𝑟²𝑑+𝑀²𝑣𝑑𝑟²−3𝐾𝑑𝑣𝑑𝑟²𝑑²𝑣𝑑𝑟²−𝐾𝑟𝑑𝑣𝑑𝑟³−𝐵=0,𝜇Γ𝑑𝑣𝑑𝑟²+𝑀Γ𝑑𝑣𝑑𝑟²−Γ𝐾𝑑𝑣𝑑𝑟⁴−1𝑟𝑑𝜃−𝑑𝑑𝑟²𝜃𝑑𝑟²𝑣(𝑟)(𝑟)𝑣(𝑟)(𝑟)=0,=1,𝜃=1,𝑟=1,=0,𝜃=0,𝑟=𝑏, (3)(4) whence 𝑟=𝑟𝑅𝜇,Γ=^∗𝑉²₀𝑘𝜃_𝑚−𝜃_𝑤,𝜃=𝜃−𝜃_𝑤𝜃_𝑚−𝜃_𝑤,𝑉𝑧=²⁰𝑅²𝑐²,𝐶₁=𝜕𝑝1𝜕𝑧,𝑀=𝛽𝑐𝜇_∗,𝑉𝐾=²⁰𝑅²𝑐²𝐶,𝐵=¹𝑅²𝜇_∗𝑉₀,𝑣=𝑣𝑣₀,𝜇=𝜇𝜇_∗, (5) where 𝜇_∗, 𝜃_𝑚, 𝑉₀ and Γ are, respectively, the reference viscosity, a reference temperature (the bulk mean fluid temperature), and reference velocity Γ is related to the Prandtl number and Eckert number.

4. Series Solutions for Reynolds’ Model

Here the viscosity is expressed in the form 𝜇=𝑒^−𝑃𝜃 (6) which by Maclaurin’s series can be written as 𝜇=1−𝑃𝜃+𝑂𝜃². (7) Note that 𝑀=0 corresponds to the case of constant viscosity. Invoking the above equation into (3) one has 𝑀𝑟𝑑𝑣+1𝑑𝑟𝑟𝑑𝑣−𝑑𝑟𝑃𝜃𝑟𝑑𝑣+𝑑𝑑𝑟²𝑣𝑑𝑟²𝑑−𝑃𝜃²𝑣𝑑𝑟²𝑑+𝑀²𝑣𝑑𝑟²−3𝐾𝑑𝑣𝑑𝑟²𝑑²𝑣𝑑𝑟²−𝐾𝑟𝑑𝑣𝑑𝑟³Γ−𝐵=0,𝑑𝑣𝑑𝑟²−Γ𝑃𝜃𝑑𝑣𝑑𝑟²+𝑀Γ𝑑𝑣𝑑𝑟²−Γ𝐾𝑑𝑣𝑑𝑟⁴−1𝑟𝑑𝜃−𝑑𝑑𝑟²𝜃𝑑𝑟²=0. (8) For HAM solution, we choose the following initial guesses: 𝑣₀(𝑟)=(𝑟−𝑏)(1−𝑏),𝜃₀(𝑟)=(𝑟−𝑏)().1−𝑏 (9) The auxiliary linear operators are in the form £_𝑣𝑟(𝑣)=𝑣^′′,£_𝜃𝑟(𝜃)=𝜃^′′ (10)(11) which satisfy ℒ_𝑣𝑟𝐴₁+𝐵₁𝑟=0,ℒ_𝜃𝑟𝐴₂+𝐵₂𝑟=0, (12) where 𝐴₁, 𝐴₂, 𝐵₁, and 𝐵₂ are the constants.

If 𝑝∈[0,1] is an embedding parameter and ℎ_𝑣 and ℎ_𝜃 are auxiliary parameters, then the problems at the zero and 𝑚th order are, respectively, given by 1−𝑝ℒ_𝑣𝑣𝑟,𝑝−𝑣₀(𝑟)=𝑝ℏ_𝑣𝑁_𝑣𝑣𝑟,𝑝,𝜃,ℒ𝑟,𝑝1−𝑝_𝜃𝜃𝑟,𝑝−𝜃₀(𝑟)=𝑝ℏ_𝜃𝑁_𝜃𝑣𝑟,𝑝,𝜃,ℒ𝑟,𝑝_𝑣𝑣_𝑚(𝑟)−𝜒_𝑚𝑣_𝑚−1(𝑟)=ℏ_𝑣𝑅_𝑣(𝑟),ℒ_𝜃𝜃_𝑚(𝑟)−𝜒_𝑚𝜃_𝑚−1(𝑟)=ℏ_𝜃𝑅_𝜃(𝑟),𝑣𝑟,𝑝=𝜃𝑟,𝑝=1,𝑟=1,𝑣𝑟,𝑝=𝜃𝑟,𝑝=0,𝑟=𝑏. (13)(14)(15)(16)(17)(18) The boundary conditions at the 𝑚th order are 𝑣_𝑚𝑟,𝑝=𝜃_𝑚𝑟,𝑝=0,𝑟=1,𝑣_𝑚𝑟,𝑝=𝜃_𝑚𝑟,𝑝=0,𝑟=𝑏. (19) In (11)–(13) 𝑁_𝑣𝑣𝑟,𝑝,𝜃𝑟,𝑝=−𝑃𝜃𝑟𝑑𝑣+1𝑑𝑟𝑟𝑑𝑣+𝑀𝑑𝑟𝑟𝑑𝑣+𝑑𝑑𝑟²𝑣𝑑𝑟²𝑑−𝑃𝜃²𝑣𝑑𝑟²𝑑+𝑀²𝑣𝑑𝑟²−3𝐾𝑑𝑣𝑑𝑟²𝑑²𝑣𝑑𝑟²−𝐾𝑟𝑑𝑣𝑑𝑟³𝑁−𝐵,_𝜃𝑣𝑟,𝑝,𝜃𝑟,𝑝=Γ𝑑𝑣𝑑𝑟²−Γ𝑃𝜃𝑑𝑣𝑑𝑟²+𝑀Γ𝑑𝑣𝑑𝑟²−Γ𝐾𝑑𝑣𝑑𝑟⁴−1𝑟𝑑𝜃−𝑑𝑑𝑟²𝜃𝑑𝑟²,𝑅_𝑣𝑃=−𝑟^𝑚−1_𝑘=0𝑣^′_{𝑚−1−𝑘}𝜃_𝑘+1𝑟𝑣^′_𝑚−1+𝑀𝑟𝑣^′_𝑚−1+𝑣^′′_𝑚−1−𝑃^𝑚−1_𝑘=0𝑣^′′_{𝑚−1−𝑘}𝜃_𝑘+𝑀𝑣^′′_𝑚−1−3𝐾^𝑚−1𝑘_𝑘=0𝑙=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−𝑙𝑣^′′_𝑙−𝐾𝑟^𝑚−1𝑘_𝑘=0𝑙=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−𝑙𝑣^′_𝑙𝑅−𝐵,_𝜃=Γ^𝑚−1_𝑘=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−Γ𝑃^𝑚−1𝑘_𝑘=0𝑙=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−𝑙𝜃^′_𝑙+𝑀Γ^𝑚−1_𝑘=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−Γ𝐾^𝑚−1𝑘_𝑘=0^𝑙_𝑙=0𝑠=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−𝑙𝑣^′_𝑙−𝑠𝑣^′_𝑠−1𝑟𝜃^′_𝑚−1−𝜃^′′_𝑚−1. (20)(21) By Mathematica the solutions of (21) can be written as 𝑣_𝑚(𝑟)=^3𝑚_𝑛=0𝑎_𝑚,𝑛𝑟^𝑛𝜃,𝑚≥0,_𝑚(𝑟)=^3𝑚+1_𝑛=0𝑑_𝑚,𝑛𝑟^𝑛,𝑚≥0, (22) where 𝑎_𝑚,𝑛 and 𝑑_𝑚,𝑛 are constants which can be determined on substituting (22) into (15) and (16).

5. Series Solutions for Vogel’s Model

Here 𝜇=𝜇_∗𝐴exp(𝑡+𝜃)−𝜃_𝑤 (23) which by Maclaurin’s series reduces to 𝐵𝜇=𝑠1−𝐴𝜃𝑡². (24) Invoking the above expressions, (1) become −𝐴𝐵𝜃𝑟𝑠𝑡²𝑑𝑣+1𝑑𝑟𝑟𝑑𝑣+𝑀𝑑𝑟𝑟𝑑𝑣+𝐵𝑑𝑟𝑠𝑑²𝑣𝑑𝑟²−𝐴𝐵𝑠𝑡²𝜃𝑑²𝑣𝑑𝑟²𝑑+𝑀²𝑣𝑑𝑟²−3𝐾𝑑𝑣𝑑𝑟²𝑑²𝑣𝑑𝑟²−𝐾𝑟𝑑𝑣𝑑𝑟³Γ−𝐵=0,𝑑𝑣𝑑𝑟²−Γ𝐴𝐵𝑠𝑡²𝜃𝑑𝑣𝑑𝑟²+𝑀Γ𝑑𝑣𝑑𝑟²−Γ𝐾𝑑𝑣𝑑𝑟⁴−1𝑟𝑑𝜃−𝑑𝑑𝑟²𝜃𝑑𝑟²=0. (25) With the following initial guesses and auxiliary linear operators 𝑣_0𝑣(𝑟)=(𝑟−𝑏)(1−𝑏),𝜃_0𝑣(𝑟)=(𝑟−𝑏)(),£1−𝑏_𝑣𝑣(𝑣)=𝑣^′′,£_𝜃𝑣(𝜃)=𝜃^′′, (26) the 𝑚th-order deformation problems are £_𝑣𝑟𝑣_𝑚(𝑟)−𝜒_𝑚𝑣_𝑚−1(𝑟)=ℏ_𝑣𝑅_𝑣𝑟(𝑟),£_𝜃𝑟𝜃_𝑚(𝑟)−𝜒_𝑚𝜃_𝑚−1(𝑟)=ℏ_𝜃𝑅_𝜃𝑟(𝑟),𝑁_𝑣𝑣𝑟,𝑝,𝜃𝑟,𝑝=−𝐴𝐵𝜃𝑟𝑠𝑡²𝑑𝑣+1𝑑𝑟𝑟𝑑𝑣+𝑀𝑑𝑟𝑟𝑑𝑣+𝐵𝑑𝑟𝑠𝑑²𝑣𝑑𝑟²𝑑+𝑀²𝑣𝑑𝑟²−𝐴𝐵𝑠𝑡²𝜃𝑑²𝑣𝑑𝑟²−3𝐾𝑑𝑣𝑑𝑟²𝑑²𝑣𝑑𝑟²−𝐾𝑟𝑑𝑣𝑑𝑟³𝑁−𝐵,_𝜃𝑣𝑟,𝑝,𝜃𝑟,𝑝=Γ𝑑𝑣𝑑𝑟²−Γ𝐴𝐵𝑠𝑡²𝜃𝑑𝑣𝑑𝑟²+𝑀Γ𝑑𝑣𝑑𝑟²−Γ𝐾𝑑𝑣𝑑𝑟⁴−1𝑟𝑑𝜃−𝑑𝑑𝑟²𝜃𝑑𝑟²,𝑁_𝑣𝑣𝑟,𝑝,𝜃𝑟,𝑝=−𝐴𝐵𝑟𝑠𝑡^2𝑚−1_𝑘=0𝑣^′_{𝑚−1−𝑘}𝜃_𝑘+1𝑟𝑣^′_𝑚−1+𝑀𝑟𝑣^′_𝑚−1+𝐵𝑠𝑣^′′_𝑚−1−𝐴𝐵𝑠𝑡^2𝑚−1_𝑘=0𝑣^′′_{𝑚−1−𝑘}𝜃_𝑘+𝑀𝑣^′′_𝑚−1−3𝐾^𝑚−1𝑘_𝑘=0𝑙=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−𝑙𝑣^′′_𝑙−𝐾𝑟^𝑚−1𝑘_𝑘=0𝑙=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−𝑙𝑣^′_𝑙𝑁−𝐵,_𝜃𝑣𝑟,𝑝,𝜃𝑟,𝑝=Γ^𝑚−1_𝑘=0𝑣^′′_{𝑚−1−𝑘}𝑣^′_𝑘−Γ𝐴𝐵𝑠𝑡^{2𝑚−1𝑘}_𝑘=0𝑙=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−𝑙𝜃^′_𝑙+𝑀Γ^𝑚−1_𝑘=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−Γ𝐾^𝑚−1𝑘_𝑘=0^𝑙_𝑙=0𝑠=0𝑣^′_{𝑚−1−𝑘}𝑣^′_𝑘−𝑙𝑣^′_𝑙−𝑠𝑣^′_𝑠−1𝑟𝜃^′_𝑚−1−𝜃^′′_𝑚−1. (27) The expressions of 𝑣_𝑚 and 𝜃_𝑚 are finally given by 𝑣_𝑚(𝑟)=^3𝑚_𝑛=0𝑎^′_𝑚,𝑛𝑟^𝑛𝜃,𝑚≥0,_𝑚(𝑟)=^3𝑚+1_𝑛=0𝑑^′_𝑚,𝑛𝑟^𝑛,𝑚≥0. (28)

6. Graphical Results and Discussion

In order to report the convergence of the obtained series solutions and the effects of sundry parameters in the present investigation we plotted Figures 1–13. Figures 1–4 are prepared to see the convergence region. Figures 1 and 2 correspond to Reynolds’ model whereas Figures 3 and 4 relate to Vogel’s model. Figure 5 shows the temperature variation for different values of 𝐵 for Reynolds’ model. It can be seen that temperature decreases as 𝐵 increases. Figure 6 depicts the velocity variation for Reynolds’ model for different values of 𝐵. Velocity also decreases as 𝐵 increases. Figure 7 shows the velocity variation for different values of 𝑃 for Reynolds’ model. It can be seen that velocity increases as 𝑃 increases. Figure 8 is plotted in order to see the temperature variation for Reynolds’ model for different values of Γ; it is depicted that temperature increases as Γ increases. Figures 9–13 correspond to Vogel’s model. Figure 9 depicts temperature variation for Vogel’s model for different values of 𝑡. It is seen that temperature increases as 𝑡 increases. Figure 10 shows the velocity variation for Vogel’s model for different values of 𝑡. It is observed that velocity decreases as 𝑡 increases. Figure 11 is prepared to observe the temperature variation for Vogel’s model for different values of Γ. It is observed that temperature decreases as Γ increases. Figure 12 is plotted to see the the velocity variation for Vogel’s model for different values of 𝑠. It is observed that velocity decreases as 𝑠 increases. Figure 13 depicts the velocity variation for Vogel’s model for different values of 𝐵. It is observed that velocity decreases as 𝐵 increases.

Figure 1 

ℎ-curve for Reynolds’ model for velocity profile.

Figure 2 

ℎ-curve for Reynolds’ model for temperature profile.

Figure 3 

ℎ-curve for Vogel’s model for velocity profile.

Figure 4 

ℎ-curve for Vogel’s model for temperature profile.

Figure 5 

Temperature profile for Reynolds’ model for different values of 𝐵.

Figure 6 

Velocity profile for Reynolds’ model for different values of 𝐵.

Figure 7 

Velocity profile for Reynolds’ model for different values of 𝑃.

Figure 8 

Temperature profile for Reynolds’ model for different values of Γ.

Figure 9 

Temperature profile for Vogel’s model for different values of 𝑡.

Figure 10 

Velocity profile for Vogel’s model for different values of 𝑡.

Figure 11 

Temperature profile for Vogel’s model for different values of Γ.

Figure 12 

Velocity profile for Vogel’s model for different values of 𝑠.

Figure 13 

Velocity profile for Vogel’s model for different values of 𝐵.

7. Conclusions

In this paper, we consider the flow of Eyring-Powell model fluid in the annulus between two cylinders whose viscosity depends upon the temperature. We discussed the steady flow in the annulus due to the motion of inner cylinder and constant pressure gradient. In the problem considered the flow is found to be remarkedly different from that for the incompressible Navier-Stokes fluid with constant viscosity. The behavior of pertinent parameters is analyzed and depicted through graphs. Using usual similarity transformations the governing equations have been transformed into non-linear ordinary differential equations. The highly non-linear problem is then solved by homotopy analysis method. Effects of the various parameters on velocity and temperature profiles are examined.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.