Constant Accelerated Flow for a Third-Grade Fluid in a Porous Medium and a Rotating Frame with the Homotopy Analysis Method

Abstract

The homotopy analysis method (HAM) is applied to obtain the approximate analytic solution of a constant accelerated flow for a third-grade fluid in a porous medium and a rotating frame. HAM is an analytic technique which provides us with a new way to obtain series solutions of such nonlinear problems. The approximate analytic solution for constant accelerated flow is obtained by using HAM. HAM contains the auxiliary parameter ℏ, which provides us with a straightforward way to obtain the convergence region of the series solution. Graphical results are plotted and the consequences discussed. The obtained solutions clearly satisfy the governing equations and all the imposed initial and boundary conditions. Many interesting results can be obtained as the special cases of the presented analysis. The influence of the material parameters of a third-grade fluid and rotation upon the velocity field is finally deliberated.

1. Introduction

It is difficult to solve nonlinear problems, especially by an analytic technique. The homotopy analysis method (HAM) [1, 2] is an analytic technique for nonlinear problems, which was initially introduced by Liao in 1992. This method has been successfully applied to many nonlinear problems in engineering and science, such as the magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet [3], the boundary-layer flows over an impermeable stretched plate [4], the nonlinear model of the combined convective and radiative cooling of a spherical body [5], the exponentially decaying boundary layers [6], and the unsteady boundary layer flows over a stretching flat plate [7]. Thus, the validity, effectiveness, and flexibility of the HAM are verified via all of the successful applications. Also, many types of nonlinear problems were solved with HAM by others [8–22]. The equations governing the flow of a viscous fluid, namely, the Navier-Stokes equations, are nonlinear. But there are several complicated fluids which are not well described by these equations. Due to this reason, many constitutive equations have been proposed for the non-Newtonian fluids. The equations for non-Newtonian fluids are much complicated and of higher order than the Navier-Stokes equations. Even the various investigators are presently engaged in finding the solutions for such flow problems. Some recent attempts relevant to the flows of non-Newtonian fluids in nonrotating frame are given in [23–30]. The study of rotating flows has gained considerable importance due to their applications in cosmical and geophysical fluid dynamics. Recently, there are a few works in this area such as an oscillating hydromagnetic non-Newtonian flow in a rotating system [31], a hydromagnetic Couette flow of an Oldroyd-B fluid in a rotating system [32], and Stokes’ first problem for the rotating flow of a third-grade fluid [33]. In all of these above-mentioned studies, the rotating flows of non-Newtonian fluids have been studied as a boundary value problem. Therefore, all the mentioned studies lack the features of unsteadiness. This study fills the gap in this area. Thus, the main objective of the present study is to obtain an approximate analytic solution for unsteady third-grade fluid in a rotating frame. The flow in the fluid is induced by a constant accelerated plate. In addition the graphical results are plotted and discussed, where the effect of the material parameters of third-grade fluid and rotation upon the velocity field is deliberated.

2. Governing Equations

Consider an incompressible third-grade fluid occupying the space 𝑧>0. The plate at 𝑧=0 is moved with a constant acceleration 𝐴 in the 𝑥-direction for 𝑡>0 and induced the motion in the fluid. Both the fluid and plate are in a solid body rotation. Initially the fluid and plate are at rest. The laws which govern the flow are [33] 𝜌div𝐕=0,𝜕𝐕+()(𝜕𝑡𝐕⋅∇𝐕+2Ω×𝐕+Ω×Ω×𝑟)=−∇𝑝+div𝐓, (2.1)(2.2) in which 𝐕 is the velocity, 𝜌 the fluid density, 𝑡 the time, 𝑝 the hydrostatic pressure, 𝐓 the extra stress tensor, Ω the constant angular velocity, and 𝑟 the radial coordinate with 𝑟²=𝑥²+𝑦².

The extra stress tensor 𝐓 in a third-grade fluid is [33] 𝐓=𝜇𝐴₁+𝛼₁𝐴₂+𝛼₂𝐴₁²+𝛽₁𝐴₃+𝛽₂𝐴₂𝐴₁+𝐴₁𝐴₂+𝛽₃tr𝐴₁²𝐴₁. (2.3) Here 𝜇 is the dynamic viscosity; 𝛼_𝑖(𝑖=1,2) and 𝛽_𝑗(𝑗=1,2,3) are the material constants. The kinematical tensors 𝐴_𝑛 are 𝐴₁=grad𝐕+grad𝐕^𝑇,𝐴_𝑛+1=𝜕+(𝐴𝜕𝑡𝐕⋅∇)_𝑛+𝐴_𝑛grad𝐕+grad𝐕^𝑇𝐴_𝑛,𝑛>1. (2.4) The thermodynamics of the fluid requires that [34] 𝜇≥0,𝛼₁≥0,𝛼₁+𝛼₂≤24𝜇𝛽₃,𝛽₁=𝛽₁=0,𝛽₃≥0. (2.5) Therefore, (2.3) can be written as 𝐓=𝜇+𝛽₃tr𝐴₁²𝐴₁+𝛼₁𝐴₂+𝛼₂𝐴₁². (2.6) Since the plate is infinite, so the velocity field 𝑉 for the present flow is 𝐕=[𝑢(𝑧,𝑡),𝑣(𝑧,𝑡),𝑤(𝑧,𝑡)], (2.7) which together with the incompressibility condition yields 𝑤=0  (𝑢, 𝑣, and 𝑤 are the velocities in the 𝑥, 𝑦, 𝑧 directions, resp.).

Substituting (2.6) and (2.7) into (2.2), one obtains 𝜕𝑢1𝜕𝑡−2Ω𝑣=−𝜌𝜕𝑝+1𝜕𝑥𝜌𝜇𝜕²𝑢𝜕𝑧²+𝛼₁𝜕³𝑢𝜕𝑧²𝜕𝑡+2𝛽₃𝜕𝜕𝑧𝜕𝑢𝜕𝑧𝜕𝑢𝜕𝑧²+𝜕𝑣𝜕𝑧²,𝜕𝑣1𝜕𝑡+2Ω𝑢=−𝜌𝜕𝑝+1𝜕𝑦𝜌𝜇𝜕²𝑣𝜕𝑧²+𝛼₁𝜕³𝑣𝜕𝑧²𝜕𝑡+2𝛽₃𝜕𝜕𝑧𝜕𝑣𝜕𝑧𝜕𝑢𝜕𝑧²+𝜕𝑣𝜕𝑧²,10=−𝜌𝜕𝑝,𝜕𝑧 (2.8)(2.9)(2.10) where the modified pressure 𝜌𝑝=𝑝−2Ω²𝑥²+𝑦² (2.11) and (2.10) shows that 𝑝≠𝑝(𝑧).

The boundary and initial conditions corresponding to constant accelerated plate are 𝑢()()𝑢=𝐴𝑡,𝑣=0at𝑧=0,𝑡>0,𝑢→0,𝑣→0as𝑧→∞∀𝑡,𝑧,0=0,𝑣𝑧,0=0,𝑧>0, (2.12) where 𝐴 is constant accelerated.

Combining (2.8) and (2.9) and then neglecting the pressure gradient, we have 𝜕𝐹𝜕𝜕𝑡+2𝑖Ω𝐹=𝑣²𝐹𝜕𝑧²+𝛼₁𝜌𝜕³𝐹𝜕𝑧²+𝜕𝑡2𝛽₃𝜌𝜕𝜕𝑧𝜕𝐹𝜕𝑧²𝜕𝐹,𝜕𝑧 (2.13) in which 𝑣 is the kinematic viscosity and 𝐹=𝑢+𝑖𝑣,𝐹=𝑢−𝑖𝑣. (2.14) The boundary and initial conditions now are 𝐹(0,𝑡)=𝐴𝑡,𝐹(𝑧,𝑡)→0as𝑧→∞,𝐹(𝑧,0)=0. (2.15) The above equation can be normalized using the following dimensionless parameters: 𝐹𝑓=(𝑣𝐴)^1/3𝐴,𝜂=𝑧𝑣^21/3𝐴,𝜏=𝑡²𝑣^1/3𝑣,Ω𝐴^21/3=Ω₁. (2.16) Accordingly, the above equations, after dropping the asterisks, take the form 𝜕𝑓𝜕𝜏+2𝑖Ω₁𝜕𝑓=²𝑓𝜕𝜂²𝜕+𝑎³𝑓𝜕𝜂²𝜕𝜕𝜏+2𝑏𝜕𝜂𝜕𝑓𝜕𝜂²𝜕𝑓,𝑓()𝜕𝜂0,𝜏=𝜏,𝑓𝜂,𝜏→0as𝜂→∞,𝑓𝜂,0=0, (2.17)(2.18) in which 𝛼𝑎=¹𝜌𝐴²𝑣^41/3𝛽,𝑏=³𝜌𝐴⁴𝑣^51/3. (2.19)

3. Essential Ideas Related to the Homotopy Analysis Method (HAM)

Consider a nonlinear equation in a general form: 𝒩[𝑢(𝑟,𝑡)]=0, (3.1) where 𝒩 is a nonlinear operator and 𝑢(𝑟,𝑡) is unknown function. Let 𝑢₀(𝑟,𝑡) denote an initial guess of the exact solution 𝑢(𝑟,𝑡), ℏ≠0 an auxiliary parameter, ℋ(𝑟,𝑡)≠0 an auxiliary function, and ℒ an auxiliary linear operator, 𝒬∈[0,1] as an embedding parameter, and by means of homotopy analysis method, we construct the so-called zeroth-order deformation equation (1−𝒬)ℒ𝜙(𝑟,𝑡;𝒬)−𝑢₀(𝑟,𝑡)=𝒬ℏℋ(𝑟,𝑡)𝒩𝜙(𝑟,𝑡;𝒬). (3.2) It is very significant that one has great freedom to choose auxiliary objects in HAM in accordance to the rule of its solution expression.

In many cases, by means of analyzing its physical background, its initial/boundary conditions, and/or its type of nonlinearity, we might know what kinds of base functions are proper to represent the solution, even without solving a given nonlinear problem. Furthermore, it is important to obey the rule of solution expression denoted by Liao [1], and thus the auxiliary function 𝐻(𝑟,𝑡) should be chosen so that the particular solution of the high-order deformation equations (e.g., (3.8)) must be expressed by a sum of the base functions. Note that we have established the initial and base functions founded on boundary conditions.

Clearly, when 𝒬=0,1 it holds 𝜙(𝑟,𝑡;0)=𝑢₀(𝑟,𝑡),𝜙(𝑟,𝑡;1)=𝑢(𝑟,𝑡), (3.3) respectively. Then as long as 𝒬 increases from 0 to 1, the solution 𝜙(𝑟,𝑡;𝒬) varies from the initial guess 𝑢₀(𝑟,𝑡) to the exact solution 𝑢(𝑟,𝑡). Liao [2] by the Taylor theorem expanded 𝜙(𝑟,𝑡;𝒬) in a power series of 𝒬 as follows: 𝜙(𝑟,𝑡;𝒬)=𝜙(𝑟,𝑡;0)+^∞_𝑚=1𝑢_𝑚(𝑟,𝑡)𝒬^𝑚, (3.4) where 𝑢_𝑚()=1𝑟,𝑡𝜕𝑚!^𝑚𝜙(𝑟,𝑡;𝒬)𝜕𝒬^𝑚_𝒬=0. (3.5) The convergence of the series (3.4) depends upon the auxiliary parameter ℏ, auxiliary function ℋ(𝑟,𝑡), initial guess 𝑢₀(𝑟,𝑡), and auxiliary linear operator ℒ. If they are chosen properly, the series (3.4) is convergent at 𝒬=1, one has 𝑢(𝑟,𝑡)=𝑢₀(𝑟,𝑡)+^∞_𝑚=1𝑢_𝑚(𝑟,𝑡). (3.6) According to definition (3.5), the governing equation can be inferred from the zeroth-order deformation equation (3.2). We define the vector 𝑢_𝑛(𝑟,𝑡)=𝑢₀(𝑟,𝑡),𝑢₁(𝑟,𝑡),…,𝑢_𝑛(𝑟,𝑡). (3.7) Differentiating the zeroth-order deformation equation (3.2) 𝑚-times with respect to 𝒬, dividing them by 𝑚!, and finally setting 𝒬=0, we obtain the so-called 𝑚th-order deformation equation: ℒ𝑢_𝑚(𝑟,𝑡)−𝜒_𝑚𝑢_𝑚−1(𝑟,𝑡)=ℏℋ(𝑟,𝑡)ℛ_𝑚𝑢_𝑚−1,𝑟,𝑡, (3.8) where 𝜒_𝑚=ℛ0,𝑚≤1,1,𝑚>1,_𝑚𝑢_𝑚−1=1,𝑟,𝑡()!𝜕𝑚−1𝜕𝒬^{𝑚−1𝑚−1}𝒩^∞_𝑚=0𝑢_𝑚(𝑟,𝑡)𝒬^𝑚_𝒬=0. (3.9)

Theorem 3.1 (Liao [2]). As long as the series (3.6) is convergent, it is convergent to the exact solution of (3.1).

Note that homotopy analysis method contains the auxiliary parameter ℏ, which provides us with the control and adjustment for the convergence of the series solution (3.6).

4. HAM Solution

For HAM solution of (2.17), we choose 𝑓₀𝜂,𝜏=𝜏𝑒^−𝜂 (4.1) as the initial guess and ℒ𝑓=𝜕𝜂,𝜏;𝒬²𝑓𝜂,𝜏;𝒬𝜕𝜂²+𝜕𝑓𝜂,𝜏;𝒬𝜕𝜂 (4.2) as the auxiliary linear operator satisfying ℒ𝑐₁𝜏+𝑐₂𝜏𝑒^−𝜂=0. (4.3) We consider the auxiliary function ℋ(𝑟,𝑡)=1 (4.4) a zeroth-order deformation problem (1−𝒬)ℒ𝑓𝜂,𝜏;𝒬−𝑓₀𝑓,𝑓𝜂,𝜏;𝒬=𝒬ℏ𝒩𝜂,𝜏;𝒬₀𝜂,𝜏=𝜏𝑒^−𝜂,𝒩𝑓𝜂,𝜏;𝒬=𝜕𝑓𝜂,𝜏;𝒬𝜕𝜏+2𝑖Ω₁𝑓−𝜕𝜂,𝜏;𝒬²𝑓𝜂,𝜏;𝒬𝜕𝜂²𝜕−𝑎³𝑓𝜂,𝜏;𝒬𝜕𝜂²𝜕𝜕𝜏−2𝑏𝜕𝜂𝜕𝑓𝜂,𝜏;𝒬𝜕𝜂²𝜕𝑓^⋆𝜂,𝜏;𝒬,𝜕𝜂 (4.5) in which 𝑓^⋆𝜂,𝜏;𝒬=𝑓𝜂,𝜏;𝒬 (4.6) The 𝑚th-order deformation problem is given by ℒ𝑓_𝑚𝜂,𝜏−𝜒_𝑚𝑓_𝑚−1𝜂,𝜏=ℏ𝜕𝑓_𝑚−1𝜕𝜏−𝑓^′′_𝑚−1−𝑎𝜕𝑓^′′_𝑚−1𝜕𝜏+2𝑖Ω₁𝑓_𝑚−1−2𝑏^𝑚−1_𝑛=0𝑓^′𝑛_{𝑚−1−𝑛𝑖=0}𝑓^′_𝑛−𝑖𝑓_𝑖^′′⋆+2𝑓^′′_𝑛−𝑖𝑓^′_𝑖^⋆,𝑓_𝑚(0,𝜏)=0,𝑓_𝑚(∞,𝜏)=0,𝑓_𝑚𝜂,0=0,(𝑚≥1). (4.7)(4.8) We can use MATHEMATICA for solving the set of linear equations (4.7) with condition (4.8). It is found that the solution in a series form is given by 𝑓𝜂,𝜏=(𝜏)𝑒^−𝜂+𝑒^−5𝜂5.9216×10⁹𝑒^4𝜂ℏ−5.9216×10⁹𝑒^4𝜂ℏ−5.9216×10⁹𝑒^4𝜂ℏ𝜏+5.9216×10⁹𝑒^4𝜂ℏ𝜏−5.9216×10⁹𝑒^4𝜂ℏ𝑎+5.9216×10⁹𝑒^4𝜂ℏ𝑎−𝑒^2𝜂ℏ𝜏³𝑏+𝑒^4𝜂ℏ𝜏³𝑏+1.18432×10¹⁰𝑖𝑒^4𝜂ℏ𝜏Ω₁−1.18432×10¹⁰𝑖𝑒^4𝜂ℏ𝜏Ω₁+⋯. (4.9) The analytic solution given by (4.9) contains the auxiliary parameter ℏ, which influences the convergence region and the rate of approximation for the HAM solution. In Figures 1(a) and 1(b), the ℏ-curves are plotted for 𝑓(𝜂,𝜏), 𝑓(̇𝜂,𝜏) when 𝜂=𝜏=0.1, 𝑎=0.1, 𝑏=0, and Ω₁=1 at 4th-order approximation for real and imaginary part of 𝑓(𝜂,𝜏), respectively.

(a)

(b)

Figure 1 

The ℏ-curve at 4th-order approximation with tiny dashes: 𝑓(0.1,0.1) with large dashes: 𝑓(̇0.1,0.1).

As pointed out by Liao, the valid region of ℏ is a horizontal line segment on the ℏ-curve graph, and this is obviously shown in Figures 1(a) and 1(b). It is clear that the valid region for this case is −1<ℏ<0.5; that is, both Figures 1(a) and 1(b) indicate that the convergence of the HAM solution is valid for values of ℏ between −1 and 0.5. In this case for ℏ=−0.1, the obtained results are summarized in Figures 2–6.

(a)  

(b)

Figure 2 

Influence of the third grade parameter on the velocity distribution for 𝜏=1; 𝑎=0.1; Ω1=1: red color 𝑏=0.001, green color 𝑏=0.002, blue color 𝑏=0.003.

(a)

(b)

Figure 3 

The influence of the angular velocity on the velocity distribution for 𝜏=1; 𝑎=0.1; 𝑏=0.001; red color Ω1=1; green color Ω1=0.5, blue color Ω1=1.

(a)  

(b)

Figure 4 

Influence of the various values of the second-grade parameter on the velocity distribution for 𝜏=1. 𝑏=0; Ω1=1: red color 𝑎=0.1, green color 𝑎=0.5, blue color 𝑎=1.

(a)

(b)

Figure 5 

Influence of various values of 𝜏 on the velocity distribution. 𝑎=0.1; 𝑏=0.005; Ω1=1: red color 𝜏=0.5, green color 𝜏=0.75, blue color 𝜏=1.

(a)

(b)

Figure 6 

Influence of the angular velocity on the velocity distribution for the Newtonian case for 𝜏=1; 𝑎=0; 𝑏=0: red color Ω1=0.1, green color Ω1=0.5, blue color Ω1=1.

5. HAM Results and Discussions

The aim of this section is to address the influence of several pertinent parameters on the dimensionless velocity field components. In this paper, the homotopy analysis method (HAM) [2] is applied to obtain the solution of the nonlinear differential equation (2.17) with conditions (2.18). HAM provides us with a convenient way to control the convergence of the approximation series, which is a fundamental qualitative difference in analysis between HAM and other methods. Solutions for the non-Newtonian fluid models are obtained for some values of 𝜏. The HAM solution 𝑓 is used to express the nondimensional velocity profile. Graphical results for the flow are obtained for various values of the parameters 𝑎, 𝑏, Ω₁, and 𝜏. The insets (a) and (b) in each plot represent the real and imaginary parts of the derived velocity profile, respectively.

Figures 2(a) and 2(b) present the velocity profile 𝑓 for various values of the material constant, third-grade parameter 𝑏. These figures indicate that increasing the parameter 𝑏 would increase the real part of the velocity profile, whiles the imaginary part of the velocity profile decreases for large values of 𝑏. Figures 3(a) and 3(b) show the influence of the angular velocity, that is, the rotational parameter Ω₁ on the velocity profile 𝑓. It is clear from the figures that the increase in Ω₁ results in the decrease in the real and imaginary parts of the velocity profile. In Figures 4(a) and 4(b), it is noted that the velocity profile increases in the real part and the imaginary part by increasing the second-grade parameter 𝑎. Figures 5(a) and 5(b) show how the velocity profile changes for various values of time 𝜏. It is found that here the real part of the velocity profile increases whereas the imaginary part of the velocity profile decreases by increasing 𝜏. In Figures 6(a) and 6(b), the velocity distribution is presented in the Newtonian case (𝑎=𝑏=0) for the various values of Ω₁. It is observed that the effect of Ω₁ in a Newtonian fluid and a third-grade fluid is similar.

6. Concluding Remarks

In this paper, the unsteady rotating flow engendered by a constant accelerated plate has been studied via the use of the homotopy analysis method. From the presented analysis, results for the real and imaginary parts of the velocity field are presented. It is observed that at 𝜏=1 and different values of Ω₁, the flow characteristics in a third-grade fluid are similar to that of Newtonian fluid.

Thus, these examples show the flexibility and potential of the homotopy analysis method for solving complicated nonlinear problems in engineering.

Acknowledgments

This research is partially funded by the MOHE FRGS Vote no. 78675 and the UTM RUG Vote no. 05J13. M. Nazari is thankful to the UTM for the International Doctoral Fellowship (IDF).