Abstract
Purpose
The purpose of this paper is to present the problem of three‐dimensional flow of a fluid of constant density forced through the porous bottom of a circular porous slider moving laterally on a flat plate.
Design/methodology/approach
The transformed nonlinear ordinary differential equations are solved via the homotopy perturbation method (HPM) for small as well as moderately large Reynolds numbers. The convergence of the obtained HPM solution is carefully analyzed. Finally, the validity of results is verified by comparing with numerical methods and existing numerical results.
Findings
Close agreement of the two sets of results is observed, thus demonstrating the accuracy of the HPM approach for the particular problem considered.
Originality/value
Interesting conclusions which can be drawn from this study are that HPM is very effective and simple compared to the existing solution method, able to solve problems without using Padé approximants and can therefore be considered as a clear advantage over the N.M. Bujurke and Phan‐Thien techniques.
Keywords
Citation
Madani, M., Khan, Y., Mahmodi, G., Faraz, N., Yildirim, A. and Nasernejad, B. (2012), "Application of homotopy perturbation and numerical methods to the circular porous slider", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 22 No. 6, pp. 705-717. https://doi.org/10.1108/09615531211244844
Publisher
:Emerald Group Publishing Limited
Copyright © 2012, Emerald Group Publishing Limited
Nomemclature
ρ=density of fluid
∇=del operator
γ=kinematic viscosity
p=pressure
R=Reynolds number
U=velocity of slider in lateral direction
W=velocity of fluid injected through porous bottom of slider
η=z/d
q=(u, v, w) = velocity components of the fluid
d=distance
h=dimensionless vertical velocity
h′, f=dimensionless lateral velocities
1 Introduction
During the past few decades, there has been remarkable interest in the flow of Newtonian and non‐Newtonian fluids through channels with porous walls owing to their applications in various branches of engineering and technology. In this paper, we shall discuss the flow of a fluid forced through the porous bottom of the slider and in turn thus separating the slider from the ground. From a technological point of view, flows of this type correspond to porous sliders, which are becoming increasingly important due to their attractive performance and their application in fluid‐cushioned moving pads. It is a well‐known fact that fluid‐cushioned porous sliders are useful in reducing the frictional resistance between two solid surfaces moving relative to each other. For Newtonian fluids, previous studies include the porous circular slider (Wang, 1974), the porous flat slider (Skalak and Wang, 1975, 1978) and the porous elliptic slider (Wang, 1978; Watson et al., 1978). Later, for a second‐order viscoelastic fluid, the fluid dynamics of a porous flat slider was studied by Bhatt (1981) obtaining the first‐order perturbation solution for the case of a very low cross‐flow Reynolds number. However, Bhatt's results seem to be in error, as was also pointed out by Ariel (1993). Recently, Ariel (1993) has extended Skalak and Wang's (1975) analysis to a Walter's B′ viscoelastic fluid, which is characterized by two material constants. In his study, the perturbation and exact numerical solutions have been obtained. The flow between porous plates has been studied by many authors, notably Berman (1953), Proudman (1960), Terrill (1964), Elkouh (1968), Rasmussen (1970), Brady (1984), Cox (1991), Sellars (1955), White et al. (1958), Yuan (1956), Zaturska et al. (1988) and Shantha and Shanker (2010).
Most engineering problems, especially some fluid flow equations, are nonlinear, therefore some of them are solved using numerical methods and some are solved using the analytical method of perturbation (Rajabi et al., 2007; Ariel, 2009, 2010; Ariel et al., 2006). In the numerical method, stability and convergence should be considered, so as to avoid divergent or inappropriate results. In the analytical perturbation method, we should insert a small parameter into the equation (Ganji, 2006). Therefore, finding the small parameter and inserting it into the equation are deficiencies of the common perturbation methods. The Perturbation method is one of the well‐known methods used in solving the nonlinear equations, having been studied by a large number of researchers such as Bellman (1964) and Cole (1968). In fact, these scientists paid more attention to the mathematical aspects of the subject which included a loss of physical verification. This loss in the physical verification of the subject was recovered by Nayfeh (1973) and Van Dyke (1975).
In recent years, there has appeared an ever increasing interest among scientists and engineers in analytical techniques for studying nonlinear problems. Such techniques have been dominated by the perturbation methods and have found many applications in science, engineering and technology. However, like other analytical techniques, perturbation methods have their own limitations. For example, all perturbation methods require the presence of a small parameter in the nonlinear equation and that the approximate solutions of the equation containing this parameter be expressed as a series of expansions in the small parameter. The selection of small parameter requires a special skill and is very important. Therefore, a welcomed analytical method is one which does not require a small parameter in the equation modeling the phenomena. Since there are some limitations with the common perturbation method, and also because the basis of the common perturbation method was based upon the existence of a small parameter, developing the method for different applications is very difficult. Therefore, many different new methods have recently been introduced allowing ways to eliminate the small parameter, such as the artificial parameter method (Liu, 1997), the homotopy analysis method (Liao, 1995), the variational iteration method (He, 1998a, b, 1999b; He et al., 2010; Faraz et al., 2011), the Adomian decomposition method (Adomian, 1994; Wazwaz, 2006; Ganji and Ganji, 2010), the Laplace decomposition method (Khan, 2009), the homotopy perturbation transform method (Khan and Wu, 2011) and the differential transform method (Rashidi, 2009). One of the semi‐exact methods is the homotopy perturbation method (HPM). He (1999, 2000, 2004, 2006a, b, 2008) developed and formulated HPM by merging the standard homotopy and perturbation. He's HPM proved to be compatible with the versatile nature of the physical problems and has been used in a wide class of functional equations (see Kimiaeifar et al., 2011; Ganji et al., 2009; Fathizadeh and Rashidi, 2009; Kelleci and Yıldırım, 2010; Shadloo and Kimiaeifar, 2011; Raftari and Yıldırım, 2010; Hesameddini and Latifizadeh, 2009; Siddiqui et al., 2009; Xu, 2007; Madani and Fathizadeh, 2010 and the references therein).
The present paper studies a three‐dimensional problem using the HPM. The method is very well suited to physical problems since it does not require unnecessary linearization, perturbation and other restrictive methods and assumptions which may affect the problem being solved, sometimes seriously. Unlike the method of separation of variables that requires initial and boundary conditions, the HPM provides an analytical solution by using the initial conditions only. The fact that HPM solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this technique over the decomposition method. The basic motivation of the present analysis is primarily concerned with the possibility of extending Wang (1974) low Reynolds number perturbation series by HPM and then by comparing those results with other numerical methods. To the best of the author's knowledge, no attempt has been made to exploit this method to solve circular porous slider problems. Lastly, this article aims to also compare the results with solutions from the existing ones (Phan‐Thien and Bush, 1984; Bujurke and Achar, 1993).
2 Governing equations
The flow field due to a circular porous slider (Bujurke and Achar, 1993) is governed by the constant density Navier‐Stokes equation and equation of continuity are: Equation 1 Equation 2 where q=(u, v, w) is the velocity components in the directions x, y, z, respectively. The relevant boundary conditions (Bujurke and Achar, 1993) are (Figure 1): Equation 3 Upon making use of the following substitutions (Bujurke and Achar, 1993): Equation 4 where η=z/d and K, A are constants to be determined. The slider is circular, therefore the isobars must be concentric circles about the z‐axis, and the resulting nonlinear differential equations is of the following form (Bujurke and Achar, 1993): Equation 5 or after differentiating: Equation 6 Equation 7 The boundary conditions (3) become (Bujurke and Achar, 1993): Equation 8 Where R=Wd/γ is the cross‐flow Reynolds number.
3 HPM solution for circular porous slider problem
In this section, we apply the HPM (He, 1999a, 2000, 2004, 2006a, b, 2008) to nonlinear ordinary differential equations (6) and (7). According to the HPM (He, 1999a, 2000, 2004, 2006a, b, 2008), the homotopy construction of equations (6) and (7) can be expressed in the form: Equation 9 We consider f and h as following: Equation 10 Substituting equation (10) into equation (9) and rearranging based on powers of p‐terms, equation (9) is reduced to: Equation 11 Equation 12 Equation 13 Equation 14 Equation 15 Solving equations (11)‐(15) yields: Equation 16 Equation 17 Equation 18 Equation 19 Equation 20 The solutions of equations (6) and (7) when p→1 will be: Equation 21
4 Convergence of the series solution
The HPM provides an analytical solution in terms of an infinite power series. The analytical solution given in equations (16)‐(20) can be expressed in the following series forms: Equation 22 In which N is number of iteration. Re'paci (1990) provides rigorous proof of the convergence of the series solution for a more general form of the problem. The series hN and fN both consist of positive and negative terms and also zero coefficient, although not in a regular alternating fashion. The ratio test was applied to the absolute values of the series coefficient. This provides a sufficient condition for convergence of the series in the form: Equation 23 However, the approach which was preferred in this study to demonstrate the convergence of the series was to replace limm→∞ with limm→M in equation (23) where M is largest coefficient index of series, it is 4N+3 and 4N+1 for the hN and fN series, respectively. The results of the evaluated value of |aM/aM−1| for hN and fN series corresponding to different values of N are shown in Figure 2, respectively. It is clear from these figures that the ratio decays as N increases, obviously indicating that the series equations (16)‐(20) is convergent.
5 Results and discussion
In this paper, HPM and some numerical methods (finite difference method and fourth‐order Runge‐Kutta method) are used to find solutions of the circular porous slider problem. Validity of the HPM is shown in Tables I and II and Figures 3 and 4 for different values of the cross‐flow Reynolds numbers R. Excellent agreement was found between the numerical, existing (Wang, 1974; Phan‐Thien and Bush, 1984; Bujurke and Achar, 1993) and analytical solutions in Tables I and II and in these figures as well. This accuracy gives us high confidence in the validity of this problem and reveals an excellent agreement of engineering accuracy.
After validating the HPM solution and comparing it with the results of numerical methods and existing solution results (Wang, 1974; Phan‐Thien and Bush, 1984; Bujurke and Achar, 1993), the behaviour of the vertical velocity profile h(η) and lateral velocity profiles h′(η), f(η) for different values of Reynolds numbers R are shown in Figures 3 and 4. The vertical velocity profile h(η) and lateral velocity profile h′(η) is plotted for small as well as moderately large Reynolds numbers in Figure 3. It is seen that the velocities h(η) and h′(η) increase by increasing the value of R, however the opposite behavior was seen for f in Figure 4.
For a porous slider, the important physical quantities are lift and drag. The dimensionless expression for the lift and drag is given by: Equation 24 Equation 25
For a Newtonian fluid, the lift is of order R−3 whereas the drag components are of order R−1 (equations (24) and (25)). Tables I and II illustrates the non‐dimensional lift and drag components for various values of Reynolds numbers. From this table, we arrive at the conclusion that for a Newtonian fluid both lift and drag increase rapidly, although at different rates as the cross‐flow Reynolds number decreases. Physically this can be explained as follows: if everything else is held fixed, the decrease in the value of the cross‐flow Reynolds number (R=Wd/γ) results only from the decrease in the gap width. Thus, the lift is independent of translation. However, h(η) does effect f(η) through equation (7) and the drag is quite dependent on the cross‐flow.
6 Conclusion
In this study, the HPM was used for finding the analytic solutions of the set of nonlinear ordinary differential equations derived from the similarity transform for the steady three‐dimensional problem of the flow field due to a circular porous slider. Comparison of the present solution is made with the existing solution (Wang, 1974; Phan‐Thien and Bush, 1984; Bujurke and Achar, 1993) and some numerical methods for different values of Reynolds numbers. An excellent agreement is noted. We analyzed the convergence of the obtained series solutions, carefully. The convergence analysis elucidates that the HPM gives accurate results. The graphical and tabular presentation of the results reveals the reliability and the efficiency of the above method. Some of the quantitatively interesting conclusions which can be drawn from this study are summarized as follows:
- •
HPM is very effective and simple as compared to the existing solution method (Wang, 1974).
- •
The fact that HPM solves the above problems without using Padé approximants can be considered as a clear advantage of this technique over the N. M. Bujurke and Phan‐Thien techniques (Phan‐Thien and Bush, 1984; Bujurke and Achar, 1993).
Corresponding author
Yasir Khan can be contacted at: [email protected]
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