### Articles

Kyungpook Mathematical Journal 2017; 57(4): 651-665

**Published online** December 23, 2017 https://doi.org/10.5666/KMJ.2017.57.4.651

Copyright © Kyungpook Mathematical Journal.

### Numerical Solutions of Third-Order Boundary Value Problems associated with Draining and Coating Flows

Jishan Ahmed

Department of Mathematics, Faculty of Science and Engineering, University of Barisal, Bangladesh

**Received**: February 5, 2015; **Revised**: November 17, 2015; **Accepted**: November 16, 2017

### Abstract

Some computational fluid dynamics problems concerning the thin films flow of viscous fluid with a free surface and draining or coating fluid-flow problems can be delineated by third-order ordinary differential equations. In this paper, the aim is to introduce the numerical solutions of the boundary value problems of such equations by variational iteration method. In this paper, it is shown that the third-order boundary value problems can be written as a system of integral equations, which can be solved by using the variational iteration method. These solutions are gleaned in terms of convergent series. Numerical examples are given to depict the method and their convergence.

**Keywords**: Variational iteration method (VIM), Boundary value problems, Draining and coating Flows, Numerical solutions

### 1. Introduction

The numerical solution of third-order boundary value problems (BVPs) is of great importance due to its wide application in scientific research. The third-order differential equations arise in many physical problems such as electromagnetic waves, thin film flow, and gravity-driven flows [6, 11, 25, 26]. In this paper, variational iteration method (VIM) is used to obtain a numerical solution to the third-order boundary value problems associated with draining and coating flows of the following form:

with boundary conditions

where _{i}

using quintic polynomial spline functions respectively. Caglar et al. [8] introduced fourth-degree B-splines for solving third-order BVPs. All these techniques have their inbuilt deficiencies. So we may reasonably infer that a large number of authors have solved third-order BVPs using spline functions which can be exploited easily but the numerical results converge slowly. Recently, Noor and Mohyud-Din [21] have employed homotopy perturbation for solving higher-order boundary value problems. He [12–15] developed the variational iteration method for solving nonlinear initial and boundary value problems. It is worth mentioning that the method was first considered by Inokuti et al. [17]. The main objective of this paper is to apply the variational iteration method to solve a system of integral equations. This technique provides a sequence of functions which converges to the exact solution of the problem. This technique solves the problem without any need to discretization of the variables. Therefore, it is not affected by computation round off errors and one is not faced with necessity of large computer memory and time. The idea outlined in this paper can be applied to computational fluid dynamics (CFD) problems as well. For example, the two dimensional steady state laminar viscous flow over a semi-infinite flat plate is modeled by the nonlinear two-point boundary value Blasius problem [7]

with boundary conditions

where a

for some sufficiently large

By using VIM method, it would be possible to obtain a solution of Blasius equation in the form of a power series for small

### 2. Differential Equations Relevent to Draining and Coating Flows

Coating flows involve covering a surface with one or more thin layers of fluid. They range from rain running down a window to manufacturing processes, such as the production of videotapes. Difficulties in modelling coating flows arise for a number of reasons. For example, operating conditions may require a running speed which leads to instabilities, such as air entrainment and ribbing. Some draining or coating fluid-flow problems, in which surface tension forces are important, can be described by third-order ordinary differential equations. Tuck and Schwartz [26] discussed a series of third-order ordinary differential equations (ODEs) arising in the study of the flow of a thin film of viscous fluid over a solid surface. When such a film drains down a vertical wall and the effects of surface tension and gravity as well as viscosity are taken into account, one is led to an equation of the form

for the film profile

If the surface is prewetted by a very thin film of thickness

In [26] the authors formulate a series of well-posed mathematical problems arising from the study of these draining flows.

Despite the seeming simplicity of

where ^{−2} arises in the study of draining and coating flows on a dry surface.

In this case, it becomes convenient to apply the VIM for solving the

### 3. Physical Descriptions of the Third-Order Differential Equations for Draining and Coating Flows

Fluid dynamic problems involving surface tension forces are described in general by partial differential equations in space and time, with rather high, typically fourth-order, spatial differentiations. For example, the thickness

Typically, ^{−2} the viscous shearing forces. The special draining flow of interest is assumed steady in a frame of reference that is falling with the layer; hence there is an apparent upward movement of the wall in this frame. Tuck and Schwartz [26] used a boundary condition of the form

Numerical solution of (_{0}, then we could call upon VIM to solve this problem numerically.

and

When the surface is dry, insight into the shape of the film close to the tip may be obtained by studying the limit of solutions of _{0}, obtained by setting _{0} +

Tanner [24] conducted a series of experiments on the spreading of droplets of silicone oil, from which he obtained a relationship between the speed of the three-phase contact line and the maximum slope of the droplet, now usually referred to as Tanner’s Law. Motivated by these experimental results, Tanner [24] solved

### 4. Variational Iteration Method

Consider the following differential equation

where

where λ is a Lagrange multiplier [12–15], which can be determined by imposing the stationary conditions. The subscripts _{n}_{n}

subject to the boundary conditions,

The system (

subject to the boundary conditions (

where λ_{i}_{1}_{2}, …, _{n}

then the approximations can be completely determined; finally we approximate the solution

by the

### 5. Numerical Results

The third-order boundary value problems in an infinite interval has been widely used to describe the evolution of physical phenomena, for example some draining or coating fluid-flow problems, see [6,11,25,26]. We refer the reader to [4,5,8,18,20,25] for the study of the finite interval problems of third-order differential equations, and to [6, 11, 25, 26] for the study of the infinite interval problems. The third-order boundary value problems in an infinite interval is a model for a viscous fluid draining over a wet surface for a thin film flowing on an inclined plane with an opening at the bottom of the plane. In the study of draining and coating flows, the thin films flowing on an inclined plane with an opening (a gap) at the bottom of the plane, representing an outlet, can be modeled as a third-order ordinary differential equation. In the present paper, we have considered the problem of draining and coating flows, which can be modeled as the third-order ordinary differential equation of the type (

### Example 1

Consider the boundary value problem [10]:

subject to the boundary conditions

The analytical solution of this problem is

Using the transformation,

with _{i}

with

Thus, we get the resulting series solution as

Imposing the boundary conditions at ^{−13}

Table 1 shows the comparison between exact solution and the numerical solution obtained using the proposed VIM. The maximum absolute error obtained by the proposed method is compared with that of obtained by [1,3,9,10,18,23,27] in Table 2.

### Example 2

Consider the boundary value problem [10]:

subject to the boundary conditions

The analytical solution of this problem is

Using the transformation,

with _{i}

with

Thus, we get the resulting series solution as

Imposing the boundary conditions at ^{−22}

Table 3 shows the comparison between exact solution and the numerical solution obtained using the proposed VIM. The maximum absolute error obtained by the proposed method is compared with that of obtained by [1, 10, 23] in Table 4.

### 6. Conclusion

In this paper, the variational iteration method has been successfully implemented to get the numerical solutions of third-order BVPs associated with draining and coating flows. The given problems have been converted into a system of first-order differential equations, which leads to the system of integral equations. The mehod provides analytical results to a rather wide class of nonlinear equations without linearization, perturbation, or discretization, which can lead to complex numerical computations. The numerical results obtained by the present method are in good agreement with the exact solutions and is confirmation with great accuracy than the results obtained by the previous methods so far. The results obtained here can easily be extended to third-order ordinary differential equations of the form (

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