The Solution of a Coupled Nonlinear System Arising in a Three-Dimensional Rotating Flow Using Spline Method

Abstract

The behavior of the non-linear-coupled systems arising in axially symmetric hydromagnetics flow between two horizontal plates in a rotating system is analyzed, where the lower is a stretching sheet and upper is a porous solid plate. The equations of conservation of mass and momentum are transformed to a system of coupled nonlinear ordinary differential equations. These equations for the velocity field are solved numerically by using quintic spline collocation method. To solve the nonlinear equation, quasilinearization technique has been used. The numerical results are presented through graphs, in which the effects of viscosity, through flow, magnetic flux, and rotational velocity on velocity field are discussed.

1. Introduction

In fluid mechanics, the problems associated with the flow that occurs due to the rotation of a single disk or that between two rotating disks have been found of interest of many researchers. Flows between finite disks were studied by Dijkstra and van Heijst [1], Adams and Szeri [2] and Szeri et al. [3]. Berker [4] showed that when the two disks are rotating with the same angular speed, there exists a one parameter family of solutions all but one of which is not rotationally symmetric. This result has been extended by Parter and Rajagopal [5], to disks rotating with differing angular speeds; they prove that the rotationally symmetric solutions are never isolated when considered within the full scope of the Navier-Stokes equations. The numerical study of the asymmetric flow has been carried out by Lai et al. [6, 7].

Recently, hydromagnetic flow and heat transfer problems have become more important industrially. In view of these, Chakrabarti and Gupta [8] studied the hydromagnetic flow and heat transfer in a fluid, initially at rest and at uniform temperature, over a stretching sheet at a different uniform temperature. Banerjee [9] studied the effect of rotation on the hydromagnetic flow between two parallel plates where the upper plate is porous and solid, and the lower plate is a stretching sheet.

In this paper, we analyze the behavior of the solution of the nonlinear coupled systems arising in axially symmetric hydromagnetic flow between two horizontal plates in a rotating system, where the lower plate is a stretching sheet. The governing coupled ordinary differential equations are solved by quintic spline collocation method.

In Section 2, the mathematical model of the problem given by Vajravelu and Kumar [10] is presented. The quintic spline collocation method is explained in Section 3. The results are displayed in graphical manner in Section 4, and the discussion of results is drawn in Section 5.

2. Formulation of the Problem

We consider the steady flow of an electrically conducting fluid between two horizontal parallel plates when the fluid and the plates rotate in unison about an axis normal to the plates with an angular velocity Ω. A Cartesian coordinate system is considered in such a way that the 𝑥-axis is along the plate, the 𝑦-axis is perpendicular to it, and the 𝑧-axis is normal to the 𝑥𝑦-plane as shown in Figure 1.

Figure 1 

Flow configuration.

The origin is located at the centre of the channel, and the plates are located at 𝑦=−ℎ and ℎ. The lower plate is is stretched by introducing two equal and opposite forces so that the position of the point (0,−ℎ,0) remains unchanged. A uniform magnetic flux with density 𝐵_𝑜 is acting along 𝑦-axis about which the system is rotating. The upper plate is subjected to a constant wall injection with a velocity 𝑉_𝑜. The equations of motion in a rotating frame of reference are 𝑢𝜕𝑢𝜕𝑥+𝜐𝜕𝑢1𝜕𝑦+2Ω𝜔=−𝜌𝜕𝑝^∗𝜕𝜕𝑥+𝜈²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²−𝜎𝐵²_𝑜𝜌𝑢𝑢,𝜕𝜐1𝜕𝑦=−𝜌𝜕𝑝^∗𝜕𝜕𝑦+𝜈²𝜐𝜕𝑥²+𝜕²𝜐𝜕𝑦²,𝑢𝜕𝜔𝜕𝑥+𝜐𝜕𝜔𝜕𝜕𝑦−2Ω𝑢=𝜈²𝜔𝜕𝑥²+𝜕²𝜔𝜕𝑦²−𝜎𝐵²_𝑜𝜌𝜔,𝜕𝑢+𝜕𝑥𝜕𝜐𝜕𝑦=0, (2.1)(2.2)(2.3)(2.4) where 𝑢,𝜐, and 𝜔 denote the fluid velocity components along the 𝑥,𝑦, and 𝑧 directions, 𝜈 is the kinematics coefficient of viscosity, 𝜌 is the fluid density, and 𝑝^∗ is the modified fluid pressure. The velocity is independent of 𝑧 and that all derivatives towards 𝑧 therefore do not appear in the equations of motion. This leads to the absence of 𝜕𝑝^∗/𝜕𝑧 in (2.3) which implies that there is a net cross-flow along the 𝑧-axis.

The boundary conditions are 𝑢=𝐸𝑥,𝜐=0,𝜔=0at𝑦=−ℎ,𝑢=0,𝜐=−𝜐₀,𝜔=0at𝑦=ℎ. (2.5)

We introduce nondimensional variables 𝑦𝜂=ℎ,𝑢=𝐸𝑥𝑓^′𝜂,𝜐=−𝐸ℎ𝑓𝜂,𝜔=𝐸𝑥𝑔𝜂, (2.6) where a prime denotes differentiation with respect to 𝜂.

Substituting (2.6) in (2.1) to (2.4), we have −1𝜌𝜕𝑝^∗𝜕𝑥=𝐸²𝑥𝑓^′2−𝑓𝑓^′′−𝑓^′′′𝑅+𝑀²𝑅𝑓^′+2𝐾²𝑅𝑔,−1𝜌ℎ𝜕𝑝^∗𝜕𝜂=𝐸²ℎ𝑓𝑓^′+1𝑅𝑓^′′,𝑔^′′−𝑅𝑓^′𝑔−𝑓𝑔^′+2𝐾²𝑓^′−𝑀²𝑔=0, (2.7)(2.8)(2.9) where 𝑅=𝐸ℎ²𝜈𝑀,viscosityparameter,₂=𝜎𝐵²₀ℎ²𝐾𝜌𝜈,magneticparameter,₂=Ωℎ²𝜈,therotationparameter. (2.10) Equation (2.7) with the help of (2.8) can be written as 𝑓^′′′−𝑅𝑓^′2−𝑓𝑓^′′−2𝐾²𝑔−𝑀²𝑓^′=𝐴, (2.11) where A is a constant.

Differentiation of (2.11) with respect to η gives 𝑓^′′′′−𝑅𝑓^′𝑓^′′−𝑓𝑓^′′′−2𝐾²𝑔^′−𝑀²𝑓^′′=0. (2.12) Thus we solve the following nonlinear system numerically for several sets of values of the parameters: 𝑓^′′′′−𝑅𝑓^′𝑓^′′−𝑓𝑓^′′′−2𝐾²𝑔^′−𝑀²𝑓^′′𝑔=0,_′′−𝑅𝑓^′𝑔−𝑓𝑔^′+2𝑘²𝑓^′−𝑀²𝑔=0, (2.13) subject to the boundary conditions 𝑓=0,𝑓^′=1,𝑔=0at𝜂=−1,𝑓=𝜆,𝑓^′=0,𝑔=0at𝜂=1, (2.14) where 𝜆=𝜐₀/𝐸ℎ a parameter depending on the y component of velocity at the upper plate.

3. Quintic Spline Collocation Method

The fifth degree spline is used to find numerical solutions to the boundary value problems discussed in (2.13) together with (2.14). A detailed description of spline functions generated by subdivision is given by de Boor [11].

Consider equally spaced knots of a partition π: 𝑎=𝑥_𝑜<𝑥₁<𝑥₂<⋯<𝑥_𝑛=𝑏 on [𝑎,𝑏]. Let 𝑆₅[𝜋] be the space of continuously differentiable, piecewise, Quintic polynomials on 𝜋. That is, 𝑆₅[𝜋] is the space of Quintic polynomials on 𝜋. The Quintic spline is given by Bickley [12] and by G. Micula and S. Micula [13] 𝑠(𝑥)=𝑎₀+𝑏₀𝑥−𝑥₀+12𝑐₀𝑥−𝑥₀²+16𝑑₀𝑥−𝑥₀³+1𝑒24⁰𝑥−𝑥₀⁴+1120^𝑛−1_𝑘=0𝐹_𝑘𝑥−𝑥_𝑘⁵₊, (3.1) where the power function (𝑥−𝑥_𝑘)₊ is defined as 𝑥−𝑥_𝑘+=𝑥−𝑥_𝑘,if𝑥>𝑥_𝑘,0,if𝑥≤𝑥_𝑘, (3.2)

Consider a fourth-order linear boundary value problem of the form 𝑦^𝑖𝑣(𝑥)+𝑝(𝑥)𝑦^′′′(𝑥)+𝑞(𝑥)𝑦^′′(𝑥)+𝑟(𝑥)𝑦^′(𝑥)+𝑡(𝑥)𝑦(𝑥)=𝑚(𝑥),𝑎≤𝑥≤𝑏, (3.3) subject to the boundary conditions 𝛼₀𝑦₀+𝛽₀𝑦_𝑛^′+𝛾₀𝑦_𝑛^′′+𝛿₀𝑦_𝑛^′′′=𝜂₀,𝛼₁𝑦₀^′+𝛽₁𝑦_𝑛+𝛾₁𝑦_𝑛^′′+𝛿₁𝑦_𝑛^′′′=𝜂₁,𝛼₂𝑦₀^′′+𝛽₂𝑦_𝑛+𝛾₂𝑦_𝑛^′′+𝛿₂𝑦_𝑛^′′′=𝜂₂,𝛼₃𝑦₀^′′′+𝛽₃𝑦_𝑛+𝛾₃𝑦_𝑛^′′+𝛿₃𝑦_𝑛^′′′=𝜂₃, (3.4) where 𝑦(𝑥),𝑝(𝑥),𝑞(𝑥),𝑟(𝑥),𝑡(𝑥),  and 𝑚(𝑥) are continuous functions defined in the interval 𝑥∈[𝑎,𝑏];𝜂₀,𝜂₁,𝜂₂,𝜂₃ are finite real constants.

Let (3.1) be an approximate solution of (3.3), where 𝑎₀, 𝑏₀, 𝑐₀, 𝑑₀, 𝑒₀, 𝐹₀, 𝐹₁,…,𝐹_𝑛−1 are real coefficients to be determined.

Let 𝑥₀,𝑥₁,…,𝑥_𝑛 be 𝑛+1 grid points in the interval [𝑎,𝑏], so that 𝑥_𝑖=𝑎+𝑖ℎ,𝑖=0,1,…𝑛;𝑥₀=𝑎,𝑥_𝑛=𝑏,ℎ=𝑏−𝑎𝑛. (3.5) It is required that the approximate solution (3.1) satisfies the differential equation at the point 𝑥=𝑥_𝑖. Putting (3.1) with its successive derivatives in (3.3), we obtain the collocation equations as follows: ^𝑛−1_𝑘=0𝐹_𝑘𝑥_𝑖−𝑥_𝑘++12𝑝𝑥_𝑖𝑥_𝑖−𝑥_𝑘²₊+16𝑞𝑥_𝑖𝑥_𝑖−𝑥_𝑘³₊+1𝑟𝑥24^𝑖𝑥_𝑖−𝑥_𝑘⁴₊+1𝑡𝑥120^𝑖𝑥_𝑖−𝑥_𝑘⁵₊+𝑒₀1+𝑝𝑥_𝑖𝑥_𝑖−𝑥₀+12𝑞𝑥_𝑖𝑥_𝑖−𝑥₀²+16𝑟𝑥_𝑖𝑥_𝑖−𝑥₀³+1𝑡𝑥24^𝑖𝑥_𝑖−𝑥₀⁴+𝑑₀𝑝𝑥_𝑖+𝑞𝑥_𝑖𝑥_𝑖−𝑥₀+12𝑟𝑥_𝑖𝑥_𝑖−𝑥₀²+16𝑡𝑥_𝑖𝑥_𝑖−𝑥₀³+𝑐₀𝑞𝑥_𝑖+𝑟𝑥_𝑖𝑥_𝑖−𝑥₀+12𝑡𝑥_𝑖𝑥_𝑖−𝑥₀²+𝑏₀𝑟𝑥_𝑖+𝑡𝑥_𝑖𝑥_𝑖−𝑥₀+𝑎₀𝑡𝑥_𝑖=𝑚𝑥_𝑖,𝑖=0,1,2,…,𝑛. (3.6) From boundary conditions, ^𝑛−1_𝑘=0𝐹_𝑘𝛿₀2𝑏−𝑥_𝑘²₊+𝛾₀6𝑏−𝑥_𝑘³₊+𝑒₀𝛿₀()+𝛾𝑏−𝑎⁰2(𝑏−𝑎)²+𝑑₀𝛿₀+𝛾₀(𝑏−𝑎)+𝑐₀𝛾₀+𝑏₀𝛽₀+𝑎₀𝛼₀=𝜂₀,_{𝑛−1𝑘=0}𝐹_𝑘𝛿₁2𝑏−𝑥_𝑘²₊+𝛾₁6𝑏−𝑥_𝑘³₊+𝛽₁120𝑏−𝑥_𝑘⁵₊+𝑒₀𝛿₁()+𝛾𝑏−𝑎¹2(𝑏−𝑎)²+𝛽₁()24𝑏−𝑎⁴+𝑑₀𝛿₁+𝛾₁()+𝛽𝑏−𝑎¹6(𝑏−𝑎)³+𝑐₀𝛾₁+𝛽₁2(𝑏−𝑎)²+𝑏₀𝛽₁(𝑏−𝑎)+𝛼₁+𝑎₀𝛽₁=𝜂₁,_{𝑛−1𝑘=0}𝐹_𝑘𝛿₂2𝑏−𝑥_𝑘²₊+𝛾₂6𝑏−𝑥_𝑘³₊+𝛽₂120𝑏−𝑥_𝑘⁵₊+𝑒₀𝛿₂()+𝛾𝑏−𝑎²2(𝑏−𝑎)²+𝛽₂()24𝑏−𝑎⁴+𝑑₀𝛿₂+𝛾₂()+𝛽𝑏−𝑎²6(𝑏−𝑎)³+𝑐₀𝛾₂+𝛽₂2(𝑏−𝑎)²+𝛾₂(𝑏−𝑎)+𝛼₂+𝑏₀𝛽₂(𝑏−𝑎)+𝑎₀𝛽₂=𝜂₂,_{𝑛−1𝑘=0}𝐹_𝑘𝛿₃2𝑏−𝑥_𝑘²₊+𝛾₃6𝑏−𝑥_𝑘³₊+𝛽₃120𝑏−𝑥_𝑘⁵₊+𝑒₀𝛿₃()+𝛾𝑏−𝑎³2(𝑏−𝑎)²+𝛽₃()24𝑏−𝑎⁴+𝑑₀𝛿₃+𝛾₃()+𝛽𝑏−𝑎³6(𝑏−𝑎)³+𝛼₃+𝑐₀𝛾₃+𝛽₃2(𝑏−𝑎)²+𝑏₀𝛽₃(𝑏−𝑎)+𝑎₀𝛽₃=𝜂₃. (3.7) Using the power function (𝑥−𝑥_𝑘)₊ in the above equations, a system of 𝑛+5 linear equations in 𝑛+5 unknowns 𝑎₀, 𝑏₀, 𝑐₀, 𝑑₀, 𝑒₀, 𝐹₀, 𝐹₁,…,𝐹_𝑛−1 is thus obtained. This system can be written in matrix-vector form as follows: 𝐴𝑋=𝐵, (3.8) where 𝑋=[𝐹_𝑛−1, 𝐹_𝑛−2,…,𝐹₂, 𝐹₁, 𝐹₀, 𝑒₀, 𝑑₀, 𝑐₀, 𝑏₀, 𝑎₀]^𝑇,  𝐵=[𝜂₃, 𝜂₂, 𝜂₁, 𝜂₀, 𝑚(𝑥_𝑛), 𝑚(𝑥_𝑛−1),…,𝑚(𝑥₁), 𝑚(𝑥₀)]^𝑇.

The coefficient matrix 𝐴 is an upper triangular Hessenberg matrix with a single lower subdiagonal, principal and upper diagonal having nonzero elements. Because of this nature of matrix 𝐴, the determination of the required quantities becomes simple and consumes less time. The values of these constants ultimately yield the quintic spline 𝑠(𝑥) in (3.1).

In case of nonlinear boundary value problem, the equations can be converted into linear form using quasilinearization method (Bellman and Kalaba [14]), and hence this method can be used as iterative method. The procedure to obtain a spline approximation of 𝑦_𝑖(𝑖=0,1,2,…,𝑗, where 𝑗 denotes the number of iteration) by an iterative method starts with fitting a curve satisfying the end conditions and this curve is designated as 𝑦_𝑖. We obtain the successive iterations 𝑦_𝑖’s with the help of an algorithm described as above till desired accuracy.

4. Quintic Spline Solution

We solve following nonlinear system numerically for several sets of values of the parameters: 𝑓^′′′′−𝑅𝑓^′𝑓^′′−𝑓𝑓^′′′−2𝐾²𝑔^′−𝑀²𝑓^′′𝑔=0,_′′−𝑅𝑓^′𝑔−𝑓𝑔^′+2𝐾²𝑓^′−𝑀²𝑔=0. (4.1)(4.2) subject to the boundary conditions 𝑓=0,𝑓^′=1,𝑔=0at𝜂=−1,𝑓=𝜆,𝑓^′=0,𝑔=0at𝜂=1. (4.3)

The spline collocation method is used to solve the differentiation system (4.1) to (4.3). Equation (4.2) is a linear equation of order two, whereas (4.1) is a nonlinear equation of order four. For solving nonlinear equation by spline collocation method, we require a linear form of differentiation equation. The quasilinearization technique transforms (4.1) into linearized form as𝑓^𝑖𝑣_𝑖+1+𝑅𝑓_𝑖𝑓^′′′_𝑖+1−𝑅𝑓_𝑖^′+𝑀²𝑓^′′_𝑖+1−𝑅𝑓_𝑖^′′𝑓^′_𝑖+1+𝑅𝑓_𝑖^′′′𝑓_𝑖+1=2𝐾²𝑔^′_𝑖+𝑅𝑓_𝑖𝑓_𝑖^′′′−𝑓_𝑖^′𝑓_𝑖^′′, (4.4) with boundary conditions 𝑓_𝑖+1(−1)=0,𝑓^′_𝑖+1()𝑓−1=1,_𝑖+1(1)=𝜆,𝑓^′_𝑖+1(1)=0. (4.5) The linear equation (4.2) can be written as 𝑔^′′+𝑅𝑓𝑔^′−𝑅𝑓^′+𝑀²𝑔=−2𝐾²𝑓^′ (4.6) with boundary conditions 𝑔(−1)=0,𝑔(1)=0. (4.7) We use the quintic spline method to find solutions of (4.4) of fourth order with the conditions (4.5), whereas the cubic spline method is used for finding solutions of second-order equation (4.6) and (4.7).

The quintic spline given by 𝑠𝜂_𝑖=𝑎₀+𝑏₀𝜂_𝑖−𝜂₀+12𝑐₀𝜂_𝑖−𝜂₀²+16𝑑₀𝜂_𝑖−𝜂₀³+1𝑒24⁰𝜂_𝑖−𝜂₀⁴+1120^𝑛−1_𝑘=0𝐹_𝑘𝜂_𝑖−𝜂_𝑘⁵₊ (4.8) is an approximate solution of the problem given by (4.4) and (4.5). Substituting 𝑠(𝜂) in both of above equations, we obtain the collocation as ^𝑁−1_𝑘=0𝐹_𝑘𝜂_𝑖−𝜂_𝑘++𝑅𝑓_𝑖2𝜂_𝑖−𝜂_𝑘²₊−𝑅𝑓^′_𝑖+𝑀²6𝜂_𝑖−𝜂_𝑘³₊−𝑅𝑓_𝑖^′′2𝜂_𝑖−𝜂_𝑘⁴₊+𝑅𝑓^′′′𝜂120^𝑖−𝜂_𝑘⁵₊+𝑒₀1+𝑅𝑓_𝑖𝜂_𝑖−𝜂₀−𝑅𝑓^′_𝑖+𝑀²2𝜂_𝑖−𝜂₀²−𝑅𝑓_𝑖^′′6𝜂_𝑖−𝜂₀³+𝑅𝑓_𝑖^′′′𝜂24^𝑖−𝜂₀⁴+𝑑₀𝑅𝑓_𝑖−𝑅𝑓^′_𝑖+𝑀²𝜂_𝑖−𝜂₀−𝑅𝑓_𝑖^′′2𝜂_𝑖−𝜂₀²+𝑅𝑓_𝑖^′′′6𝜂_𝑖−𝜂₀³+𝑐₀−𝑅𝑓^′_𝑖+𝑀²−𝑅𝑓_𝑖^′′𝜂_𝑖−𝜂₀+𝑅𝑓_𝑖^′′′2𝜂_𝑖−𝜂₀²+𝑏₀𝑅𝑓_𝑖^′′′𝜂_𝑖−𝜂₀−𝑅𝑓_𝑖^′′+𝑎₀𝑅𝑓_𝑖^′′′=2𝐾²𝑔^′_𝑖+𝑅𝑓_𝑖𝑓_𝑖^′′′+𝑓_𝑖𝑓_𝑖^′′,𝑖=0(1)𝑁. (4.9) First two boundary conditions in (4.5) immediately give 𝑎₀=0,𝑏₀=1, (4.10) and other two boundary conditions are 1120^𝑁−1_𝑘=0𝐹_𝑘𝜂_𝑁−𝜂_𝑘⁵₊+1𝑒24⁰𝜂_𝑁−𝜂₀⁴+16𝑑₀𝜂_𝑁−𝜂₀³+12𝑐₀𝜂_𝑁−𝜂₀²+𝑏₀𝜂_𝑁−𝜂₀+𝑎₀1=𝜆,24^𝑁−1_𝑘=0𝐹_𝑘𝜂_𝑁−𝜂_𝑘⁵₊+16𝑒₀𝜂_𝑁−𝜂₀³+12𝑑₀𝜂_𝑁−𝜂₀²+𝑐₀𝜂_𝑁−𝜂₀+𝑏₀=0. (4.11) First of all, initial assumptions regarding 𝑓_𝑖,𝑓^′_𝑖,𝑓_𝑖^′′,𝑓_𝑖^′′′ are necessary. Let a curve 𝑎𝜂³+𝑏𝜂²+𝑐𝜂+𝑑 be fitted through the points 𝜂=−1 and 𝜂=1 satisfying the boundary conditions (4.5). This requirement yields 𝑎=(1−𝜆)/4, 𝑏=−1/4, 𝑐=(3𝜆−1)/4 and 𝑑=(1+2𝜆)/4 so that 𝑓(𝜂)=((1−𝜆)/4)𝜂³−(1/4)𝜂²+((3𝜆−1)/4)𝜂+((1+2𝜆)/4).

Further, using the cubic spline 𝑠𝜂_𝑖=𝑎₀+𝑏₀𝜂_𝑖−𝜂₀+12𝑐₀𝜂_𝑖−𝜂₀²+16^𝑛−1_𝑘=0𝑑_𝑘𝜂_𝑖−𝜂₀³₊. (4.12) In (4.6) and (4.7), the following set of equations is obtained: ^𝑁−1_𝑘=0𝑑_𝑘𝜂_𝑖−𝜂_𝑘++0.1𝑓_𝑖2𝜂_𝑖−𝜂_𝑘²₊−0.1𝑓_𝑖′+16𝜂_𝑖−𝜂_𝑘³₊+𝑐₀1+0.1𝑓_𝑖𝜂_𝑖−𝜂₀−0.1𝑓^′_𝑖+12𝑛_𝑖−𝜂₀²+𝑏₀0.1𝑓_𝑖−0.1𝑓^′_𝑖+1𝜂_𝑖−𝜂₀+𝑎₀−0.1𝑓^′_𝑖+1=−2𝐾²𝑓^′_𝑖(1)𝑎,𝑖=0𝑁,₀=0 (4.13) and the last equation is 16^𝑁−1_𝑘=0𝑑_𝑘𝜂_𝑁−𝜂_𝑘³₊+12𝑐₀𝜂_𝑁−𝜂₀²+𝑏₀𝜂_𝑁−𝜂₀+𝑎₀=0. (4.14)

A straight line 𝑔(𝜂)=𝑎𝜂+𝑏 can be fitted to the boundary conditions for 𝑔. A line 𝑔(𝜂)=0 satisfied the boundary conditions (4.7). Therefore, initial values of 𝑔 and 𝑔^′ are also zero.

For 𝑁=10, 𝑅=0.1 and several sets of values of the parameters 𝜆,𝑀² and 𝐾², the velocity components 𝑓,𝑓^′, and 𝑔 are obtained in graphical form. The behavior of velocity components are shown in Figures 2, 3, and 4. The results obtained by the spline collocation method are compared with the graphical solutions obtained by Vajravelu and Kumar [10], which are tested by comparing the analytical solutions (for small value of 𝑅) with the exact solutions 𝑓 and 𝑔. The numerical results thus obtained by the spline method to find 𝑓,𝑓^′, and 𝑔 are in close agreement with the available data. In Figures 2, 3, and 4, 𝜂 equals a dimensionless injection parameter, 𝑀² is a magnetic parameter, and 𝐾² is a rotational parameter.

Figure 2 

Velocity profile for (𝑅=0.1,𝑁=10).

Figure 3 

Velocity profile for 𝑓′(𝑅=0.1,𝑁=10).

Figure 4 

Velocity profile for 𝑔 (𝑅=0.1, 𝑁=10).

With improper initial guess, convergence is not observed every time. As discussed by Vajravelu and Kumar [10], this is due to inherent instability of boundary-value problem. Because of this, integration from the starting point of the domain may produce rapidly increasing solutions, which may occasionally lead to overflow before the end-point is reached.

5. Discussion of the Results

The behavior of the nondimensional velocity component 𝑓 for 𝑅=0.1 is shown in Figure 2. From Figure 2, it is evident that the Lorenz force decreases the velocity component 𝑓 (see curves II and III), but the value of 𝑓 increases with an increase in the value of the parameter 𝜆 as seen in curves III and IV. Further, in Figure 2, curves IV and V show that 𝑓 decreases steadily for small 𝐾², whereas for large value of 𝐾², f increases near the lower plate and decreases near the upper plate.

Figure 3 describes the behavior of 𝑓^′(𝜂), which is proportional to velocity along parallel plates, for several sets of values of the parameters only 𝜆,𝑀² and 𝐾² (the rotation parameter) are varied while 𝑅 is kept constant at 0.1. From Figure 3, curves III and IV, it is evident that the velocity component 𝑓^′ increases with the parameter 𝜆(=𝜐₀/𝐸ℎ, 𝑦 component of velocity at the upper plate), and the increase is maximized near the stretching plate. However, the effect of the Lorenz force (magnetic parameter 𝑀²) of 𝑓^′ is to increase it near the upper plate and to decrease it near the lower (stretching) plate that is displayed from the curves II and III of Figure 3. Further, the effect of small 𝐾² is to increase 𝑓^′ at the upper plate and to decrease it near the lower plate is evident from curves I and II. But the effect of large 𝐾² is to increase 𝑓^′ near the plates and to decrease it at the centre of the channel. Also, the rotation of the channel brings humps near the plates, indicating the occurrence of a boundary layer near the plates. Furthermore, for large 𝐾² the coriolis force and the magnetic field that act against the pressure gradient cause reversal of the flow.

The transverse velocity 𝑔 increases as the parameter 𝜆 increases, which is shown in curves III and IV of Figure 4. But this is quite opposite to the phenomenon with the magnetic parameter 𝑀². However, the rotation parameter 𝐾² increases 𝑔 and the maximum of 𝑔 occurs near the stretching sheet for large 𝐾².

From above figures, it is observed that for large 𝐾² convergence in the results is not guaranteed. This fact is also observed by Vajravelu and Kumar [10].

We conclude from the above numerical experience that the spline collocation method can be successfully applied to solve the fourth-order nonlinear coupled equation, which governs the hydromagnetic fluid flow. This widens the field of applicability to the higher-order coupled differential equations. This encourages the applications to other types of problems.

Another useful conclusion is that the selection of the domain is not restricted to positive interval only. That is we can successfully apply the above method for the negative intervals as the domain of the problem.