Approximate Analysis of MHD Squeeze Flow between Two Parallel Disks with Suction or Injection by Homotopy Perturbation Method

Abstract

An analysis has been performed to study magneto-hydrodynamic (MHD) squeeze flow between two parallel infinite disks where one disk is impermeable and the other is porous with either suction or injection of the fluid. We investigate the combined effect of inertia, electromagnetic forces, and suction or injection. With the introduction of a similarity transformation, the continuity and momentum equations governing the squeeze flow are reduced to a single, nonlinear, ordinary differential equation. An approximate solution of the equation subject to the appropriate boundary conditions is derived using the homotopy perturbation method (HPM) and compared with the direct numerical solution (NS). Results showing the effect of squeeze Reynolds number, Hartmann number and the suction/injection parameter on the axial and radial velocity distributions are presented and discussed. The approximate solution is found to be highly accurate for the ranges of parameters investigated. Because of its simplicity, versatility and high accuracy, the method can be applied to study linear and nonlinear boundary value problems arising in other engineering applications.

1. Introduction

This paper deals with the study of magneto-hydrodynamic (MHD) squeeze flow of an electrically conducting fluid between two infinite, parallel disks. The lower disk is stationary and permeable with (suction or injection). The upper disk is impermeable and moves toward the lower disk with a specified time dependent velocity. The use of a MHD fluid in lubrication prevents the adverse impact of temperature on the fluid viscosity when the system operates under extreme conditions. The problem considered is of general interest in the theory of magneto-hydrodynamic lubrication and other related applications. In particular, the results of the present investigation are directly applicable to the hydrodynamics of high temperature bearings lubricated with liquid metals. A number of theoretical and experimental investigations into magneto-hydrodynamic effects in lubrication have been reported. These include among other works of Hughes and Elco [1], Kuzma et al. [2] and Krieger et al. [3]. These authors considered the electromagnetic force term in the Navier-Stokes equations but neglected some or all the inertia terms. When low viscosity lubricants are used to reduce energy losses in lubrication devices, inertial effects become important and must be included especially if the squeeze Reynolds number is not small.

In the present work we investigate the combined effect of inertia, electromagnetic forces, and surface suction or injection in a squeeze film between two parallel disks. This combination of effects in squeezing flow has not been studied previously. The plates are made of a nonconducting material. There is no externally applied electric field and the induced electric field is negligible. Since the problem defies an exact analytical solution, special techniques must be used to derive approximate analytical solution.

One such technique is the homotopy perturbation method (HPM) proposed and applied by He [4–6]. His pioneering work has prompted many authors [7–20] to apply HPM to solve a wide variety of homogeneous and nonhomogeneous linear and nonlinear problems. Over the past several years the accuracy of HPM has been repeatedly verified. In view of the success of HPM reported by many researchers, we were tempted to adopt this method to solve the present problem.

2. The Homotopy Perturbation Method

The basic idea embodied in the HPM and a brief summary of the method can be found in [4–6]. The convergence and stability of this method have been established in [10]. The latest developments of the method can be found in a series of papers by pioneering researchers such as He [21–23] and Yıldırım [24–30]. Other papers that bear close relevance to the present work are those of Mahmood et al. [31], Mehmood and Ali [32], and Z. Z. Ganji and D. D. Ganji [33].

3. Solution Procedure

We consider axisymmetric incompressible flow between two parallel infinite disks, which at time 𝑡^∗, are spaced a distance 𝐻(1−𝑎𝑡^∗)^1/2 apart and a magnetic field proportional to 𝐵₀(1−𝛼𝑡^∗)^−1/2 is applied perpendicular to the disks. The upper disk at 𝑧=𝐻(1−𝛼𝑡^∗)^1/2is moving with velocity −𝛼𝐻(1−𝛼𝑡^∗)^−1/2/2 toward the stationary lower disk at 𝑧=0. The axial coordinate is denoted by𝑧^∗ and the radial coordinate by 𝑟^∗. With the axial and radial velocities denoted by 𝑤^∗and 𝑢^∗, respectively, we introduce the following quantities:

𝑢^∗=𝛼𝑟^∗2(1−𝛼𝑡^∗)𝑓^′(𝜂),𝑤^∗=−𝛼𝐻√1−𝛼𝑡^∗𝐵𝑓(𝜂),𝐵=⁰√1−𝛼𝑡^∗𝑧𝜂=^∗𝐻√1−𝛼𝑡^∗,𝑟=𝑟^∗,𝑡=𝑡^∗ (3.1) into the continuity and momentum equations governing the flow. The continuity equation is automatically satisfied and the momentum equation is reduced to the following fourth-order, nonlinear, ordinary differential equation: 𝑓^′′′′(𝜂)−𝑆(𝜂𝑓^′′′(𝜂)+3𝑓^′′(𝜂)−2𝑓(𝜂)𝑓^′′′(𝜂))−𝑀²𝑓^′′(𝜂)=0, (3.2) where primes denote differentiation with respect to the similarity variable 𝜂, (1/𝑟)(𝜕𝑝^∗/𝜕𝑟)=𝑝₁(𝑡), 𝑆=(𝜌𝐻²)/2𝜈 and 𝑀=𝜎𝐵²₀/𝜇. Here 𝜌 denotes the density, 𝜇 is the absolute viscosity, 𝜈 is kinematic viscosity, 𝜎 is the fluid electrical conductivity, 𝐵₀ is the magnetic field acting normal to the plates, 𝐻 is the distance separating the plates at time 𝑡=𝑡^∗=0 and 𝛼 a constant and has the units of 𝑠⁻¹.

The boundary conditions are given by 𝑓(0)=𝐴,𝑓^′1(0)=0,𝑓(1)=2,𝑓^′(1)=0, (3.3) where A is a constant such that 𝐴>0 corresponds to suction and 𝐴<0 to injection at the lower stationary disk.

According to the HPM, we can construct a homotopy of (2) as follows: 𝐻(𝑓,𝑝)=(1−𝑝)(𝑓^′′′′−𝑓₀^′′′′)+𝑝(𝑓^′′′′−𝑆(𝜂𝑓^′′′+3𝑓^′′−2𝑓𝑓^′′′)−𝑀²𝑓^′′), (3.4) where primes denote differentiation with respect to η and asterisks have been omitted for convenience.

We consider a three term-solution for 𝑓 as follows:

𝑓=𝑓₀+𝑝𝑓₁+𝑝²𝑓₂. (3.5)

Assuming 𝑓₀^′′′′=0and substituting f from (5) into (4) and after some algebraic manipulation, we obtain the following set of equations: 𝑝⁰∶𝑓₀^′′′′𝑓=0,₀(0)=𝐴,𝑓^′₀(0)=0,𝑓₀(1)=0.5,𝑓^′₀𝑝(1)=0₁∶−𝑆𝜂𝑓₀^′′′+𝑓₁^′′′′−3𝑆𝑓₀^′′+2𝑆𝑓₀𝑓₀^′′′−𝑀²𝑓₀^′′𝑓=0,₁(0)=0,𝑓^′₁(0)=0,𝑓₁(1)=0,𝑓^′₁𝑝(1)=0.₂∶−𝑀²𝑓₁^′′−𝑆𝜂𝑓₁^′′′−3𝑆𝑓₁^′′+2𝑆𝑓₁𝑓₀^′′′+𝑓₂^′′′′+2𝑆𝑓₀𝑓₁^′′′𝑓=0,₂(0)=0,𝑓^′₂(0)=0,𝑓₂(1)=0,𝑓^′₂(1)=0. (3.6) We utilized the symbolic algebra package Maple 11 to solve (6) with the appropriate boundary conditions and obtained the following analytical solutions 𝑓₀1(𝜂)=6(−6+12𝐴)𝜂³1+(2(3−6𝐴)𝜂²𝑓+𝐴),₁1(𝜂)=−3(−1+2𝐴)(105𝑆𝜂⁷1𝐴−30𝑆𝜂⁶1𝐴+6𝑆𝜂⁴1𝐴−15𝑆𝜂⁵+1𝑀24²𝜂⁴+160𝑆𝜂⁶−1210𝑆𝜂⁷+18𝑆𝜂⁴−1𝑀60²𝜂⁵)+16(−3935𝑆𝐴−117370𝑆−5𝑀²+31235𝑆𝐴²+65𝑀²𝐴)𝜂³+12(514𝑆𝐴+191140𝑆+𝑀20²−4435𝑆𝐴²−1𝑀10²𝐴)𝜂²,𝑓₂1(𝜂)=170(−1+2𝐴)(7920(1728𝑆²𝐴²−1728𝑆²𝐴+432𝑆²))𝜂¹¹+15040(−604𝑆²𝐴²+6048𝑆²𝐴−1512𝑆²)𝜂¹⁰+13024(5040𝑆²𝐴²−1008𝑀²𝑆𝐴+3444𝑆²+504𝑀²𝑆−9408𝑆²𝐴)𝜂⁹+11680(2520𝑀²𝑆𝐴−1260𝑀²𝑆+5040𝑆²𝐴²+7140𝑆²𝐴−4830𝑆²)𝜂⁸+1840(70𝑀⁴+4038𝑆²−9216𝑆²𝐴²+1498𝑀²𝑆−1596𝑀²𝑆𝐴−108𝑆²𝐴)𝜂⁷+1360(−105𝑀⁴−546𝑀²𝑆𝐴−4164𝑆²𝐴+2400𝑆²𝐴²−987𝑀²𝑆−1983𝑆²)𝜂⁶[phone]𝑆²𝐴²+468𝑆²+732𝑀²𝑆𝐴+285𝑀²𝑆+2508𝑆²𝐴+42𝑀⁴)𝜂⁵+124(−624𝑆35²𝐴³−128𝑀35²𝑆𝐴²+44𝑀35²1𝑆𝐴−𝑀10⁴2𝐴+7𝑀²𝑆−54𝑆35²𝐴²+57𝑆140²+309𝑆70²1𝐴+𝑀20⁴)𝜂⁴+16(−17𝑀140²9𝑆𝐴+𝑀1400²𝑆−32773𝑆40425²𝐴²+779𝑆64680²−50339𝑆161700²𝐴+38𝑀175²𝑆𝐴²+7466𝑆2695²𝐴³+1𝑀1400⁴−1𝑀700⁴𝐴)𝜂³+121(−𝑀525²𝑆𝐴+5414𝑆40425²𝐴²−2047𝑆646800²−3253𝑆323400²1𝐴−𝑀2800⁴+13𝑀1050²𝑆𝐴²−2726𝑆13475²𝐴³+1𝑀1400⁴3𝐴−𝑀1400²𝑆)𝜂²,𝑓(𝜂)=𝑓₀(𝜂)+𝑓₁(𝜂)+𝑓₂(𝜂). (3.7)

Once 𝑓(𝜂) is known in an analytical form, the definition of 𝑢^∗in (1) can be used to calculate the axial component of the velocity, 𝑢^∗. By differentiating 𝑓(𝜂) with respect to 𝜂, which is straightforward and can be easily performed by Maple 11, we use the definition of 𝑤^∗in (1) to compute the radial component of the velocity.

The axial and radial velocities are each functions of parameters S, M and A. Since the axial and radial velocities are proportional to 𝑓and 𝑓^′, respectively, they suffices to present the results for 𝑓. These results are displayed in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 for twelve different combinations of parameters S, M, and A. Figures 1–8 show that for both the cases of no injection (𝐴=0) and injection (𝐴<0), the axial component of the velocity increases monotonically as 𝜂increases. The velocity profile is not significantly affected by the increase in the fluid electrical conductivity and/or the magnetic field, that is, the increase in M from 0 to 5. A careful examination of Figures 1–8 also reveals that the effect of squeeze Reynolds number S on the axial velocity profiles is minimal.

Figure 1 

Variation of axial velocity for 𝑀=0, 𝑆=0.01, 𝐴=−1.

Figure 2 

Variation of axial velocity for 𝑀=0, 𝑆=0.01, 𝐴=0.

Figure 3 

𝑀=0, 𝑆=1, 𝐴=−1.

Figure 4 

𝑀=0, 𝑆=1, 𝐴=0.

Figure 5 

Variation of axial velocity for 𝑀=5, 𝑆=0.01, 𝐴=−1.

Figure 6 

Variation of axial velocity for 𝑀=5, 𝑆=0.01, 𝐴=0.

Figure 7 

Variation of axial velocity for 𝑀=5, 𝑆=1, 𝐴=−1.

Figure 8 

Variation of axial velocity for 𝑀=5, 𝑆=1, 𝐴=0.

Figure 9 

Variation of axial velocity for 𝑀=0, 𝑆=0.01, 𝐴=1.

Figure 10 

Variation of axial velocity for 𝑀=0, 𝑆=1, 𝐴=1.

Figure 11 

Variation of axial velocity for 𝑀=5, 𝑆=0.01, 𝐴=1.

Figure 12 

Variation of axial velocity for 𝑀=5, 𝑆=1, 𝐴=1.

Figures 9–12 show the effect of suction (𝐴>0) on the axial velocity distributions. In each case, the axial velocity decreases monotonically as 𝜂 increases. Siddiqui et al. [20] have observed the same pattern for the axial (normal in their case) velocity in the study of two-dimensional MHD squeeze flow between parallel plates without injection or suction. As in the case for injection (Figures 1–8), the velocity patterns are only minimally influenced by the changes in parameters M and S.

The results for 𝑓^′which represents the radial component of the velocity appear in Figures 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24. Figures 13–20 show the results for the radial velocity with no injection (𝐴=0) and injection (𝐴<0) for different combination values of M and S. In each case, the radial velocity increases as 𝜂increases, reaches a peak value in the neighborhood of 𝜂=0.5 and then decays to zero around 𝜂=1. For the case of suction (𝐴>0), which is illustrated in Figures 21–24, the radial velocity becomes negative and decreases as 𝜂increases. The trough in radial velocity is attained in the neighborhood of 𝜂=0.5. After that the velocity increases and reaches the zero value around 𝜂=1.

Figure 13 

Variation of radial velocity for 𝑀=0, 𝑆=0.01, 𝐴=−1.

Figure 14 

Variation of radial velocity for 𝑀=0, 𝑆=0.01, 𝐴=0.

Figure 15 

Variation of radial velocity for 𝑀=0, 𝑆=1, 𝐴=−1.

Figure 16 

Variation of radial velocity for 𝑀=0, 𝑆=1, 𝐴=0.

Figure 17 

Variation of radial velocity for 𝑀=5, 𝑆=0.01, 𝐴=−1.

Figure 18 

Variation of radial velocity for 𝑀=5, 𝑆=0.01, 𝐴=0.

Figure 19 

Variation of radial velocity for 𝑀=5, 𝑆=1, 𝐴=−1.

Figure 20 

Variation of radial velocity for 𝑀=5, 𝑆=1, 𝐴=0.

Figure 21 

Variation of radial velocity for 𝑀=0, 𝑆=0.01, 𝐴=1.

Figure 22 

Variation of radial velocity for 𝑀=0, 𝑆=1, 𝐴=1.

Figure 23 

Variation of radial velocity for 𝑀=5, 𝑆=0.01, 𝐴=1.

Figure 24 

Variation of radial velocity for 𝑀=5, 𝑆=1, 𝐴=1.

Figure 25 has been prepared to illustrate the effect of varying the imposed magnetic field (Hartmann number M ) on the radial distribution of velocity for suction (𝐴=1) and injection (𝐴=−1). As the strength of the magnetic field is increased, that is, as the Hartmann number increases, the impact is felt most on the maximum radial velocity, more so for injection than for suction. The effect of squeeze Reynolds number S on the radial velocity distribution is shown in Figure 26 for a fixed value of 𝑀=5. It can be seen that even with a hundred-fold increase in S, the radial velocity profile is not affected significantly.

Figure 25 

Variation of radial velocity with 𝑀 and 𝐴 for 𝑆=1.

Figure 26 

Variation of radial velocity with 𝑆 and 𝐴 for 𝑀=5.

For every case investigated (Figures 1–26), the HPM predictions have been compared with the corresponding direct numerical solutions (NSs) obtained by using Maple 11 software. This software uses a Fehlberg fourth-fifth order Runge-Kutta finite-difference method for the numerical solution of the boundary value problem [34]. In each case, the HPM and NS are found to be consistently close. This consistent closeness strongly vouches for the accuracy of HPM. Equation (9) therefore provides simple and highly accurate analytical solutions for the problem studied in this paper and is useful for rapid engineering calculations.

4. Conclusions

He’s homotopy perturbation method (HPM) has been utilized to derive approximate analytical solutions for the radial and axial velocity distributions in magneto-hydrodynamic (MHD) squeeze flow between two parallel infinite disks where one disk is impermeable and the other is porous with either suction or injection of the fluid. The approximate solutions have been compared with the direct numerical solutions generated by the symbolic algebra package Maple 11 which uses a Fehlberg fourth-fifth order Runge-Kutta finite-difference method for solving nonlinear boundary value problems. The comparison showed that the HPM solutions are highly accurate and provide a rapid means of computing the flow velocities between the plates.

For both the cases of no injection and injection, the axial component of the velocity increases monotonically as the similarity variable increases. The velocity profiles are not significantly affected by the increase in the fluid electrical conductivity and/or the magnetic field. Similarly, the effect of squeeze Reynolds number S on the axial velocity profiles is minimal. The effect of imposed magnetic field on the maximum radial velocity is more pronounced for injection than for suction.

Acknowledgment

The authors express their gratitude to the reviewers for their valuable suggestions which were incorporated to enhance the value of the paper.