# Numerical study for the BVP of the liquid film flow over an unsteady stretching sheet with thermal radiation and magnetic field

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## Abstract

In this paper, we introduce a method based on replacement of the unknown function by truncated series of the well-known shifted Chebyshev (of third-kind) expansion of functions. We give an approximate formula for the integer derivative of this expansion. We state and prove some theorems on the convergence analysis. By means of collocation points the introduced method converts the proposed problem to solving a system of algebraic equations with shifted Chebyshev coefficients. As an application for this efficient numerical method, we employ it in solving the system of ordinary differential equation that describes the thin film flow and heat transfer with the effect of thermal radiation, magnetic field, and slip velocity.

## Keywords

Liquid film Thermal radiation Unsteady stretching sheet Chebyshev collocation method Convergence analysis## MSC

41A04 65N12 76S02## 1 Introduction

The thin film fluid flow has become dependent on many theoretical and experimental studies in recent years due to its widespread applications in industry and engineering such as continuous casting, crystal growing, tinning of copper wires, chemical processing equipment, and wire and fiber coating. Many authors studied the thin film fluid flow and heat transfer under different cases; see, for example, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. In [11], the authors studied the flow of an incompressible liquid film down a wavy incline and applied the Galerkin method with only one ansatz function to the Navier–Stokes equations. They derived a second-order weighted residual integral boundary layer equation to describe eddies in the troughs of the wavy bottom. Marin [12] considered a cylinder made of a microstretch thermoelastic material for which one plane end is subjected to plane boundary data varying harmonically in time. On the lateral surface and other bases, we have zero body force and heat supply. Finally, Melvin and Herman [13] presented algorithmic matters of a computer code to solve linear two-point boundary-value problems. The proposed method used a superposition coupled with an orthonormalization procedure and a variable-step Runge–Kutta–Fehlberg integration scheme. Just and Stempien [14] studied the Pareto optimal control system for a nonlinear one-dimensional extensible beam equation and its Galerkin approximation. In [15], the authors proposed a modified and simple algorithm for fractional modeling arising in unidirectional propagation of long wave in dispersive media by using the fractional homotopy analysis transform method. The proposed technique can be used to solve nonlinear problems without using the Adomian and He’s polynomials, which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method. This modified method yields an analytical and approximate solution in terms of a rapidly convergent series with easily computable terms. Also, exploiting variational methods and the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the *p*-biharmonic operator are investigated in [16]. Finally, the radiative effects for some bidimensional thermoelectric problems are investigated in [17].

After these previous publications, a number of researchers have successfully applied several numerical methods in this field. Among these numerical methods, the Chebyshev collocation method is a general approximate analytical method used to get the solutions for some of nonlinear differential equations. The Chebyshev collocation method has some advantages for handling this class of problems, in which the Chebyshev coefficients for the solution can be calculated very easily by numerical programs. For this reason, this method is much faster than the other methods. Chebyshev polynomials are a well-known family of orthogonal polynomials on the interval \([-1,1]\) with many applications. They are widely used because of their good properties in the approximation of functions [18, 19, 20]. Some of these properties take a very concise form in the case of the Chebyshev polynomials, making them of leading importance among orthogonal polynomials. The Chebyshev polynomials belong to an exclusive class of orthogonal polynomials, known as Jacobi polynomials, which correspond to weight functions of the form \((1-x)^{\alpha}(1 +x)^{\beta}\) and which are solutions of Sturm–Liouville equations. The Chebyshev collocation method is used to solve many problems in many papers, for example, [18, 19, 20].

In this work, we use the properties of Chebyshev polynomials to derive an approximate formula of the integer derivative of the approximate solution and estimate an error upper bound of this formula. Due to a high accuracy of this method, it is inevitable to use it to solve numerically the resulting nonlinear system of ordinary differential equations, which describe a flow and heat transfer of thin liquid film affected by the presence of thermal radiation and magnetic field.

*η*,

*S*is the unsteadiness parameter,

*M*is the magnetic parameter,

*R*is the radiation parameter,

*δ*is the slip velocity parameter,

*γ*is the dimensionless film thickness, and Pr is the Prandtl number. Here, we must refer that the previous system of equations is a generalization of the pioneering research of Wang [21]. Our problem can be reduced to the Wang problem by taking \(M=R=\delta=0\).

## 2 Procedure of solution

### 2.1 Approximate the solution and its convergence analysis

*n*are defined as

#### Theorem 1

*Suppose that the function*\(\Omega(t)\)

*satisfies the following conditions*:

- 1.
*The second derivative*\(\Omega''(t)\)*is a square*-*integrable function on*\([0,1]\),*i*.*e*., \(\Omega''(t)\in L_{2}[0,1]\); - 2.
*The second derivative*\(\Omega''(t)\)*is bounded on*\([0,1]\),*i*.*e*., \(\vert \Omega^{\prime\prime}(t) \vert \leq\ell\)*for some constant**ℓ*.

*Then the infinite series*(5)

*of the shifted Chebyshev expansion is uniformly convergent*,

*and*

#### Proof

*i*, we can get the required formula (8), and hence it uniformly converges to \(\Omega(t)\), which completes the proof. □

#### Theorem 2

*Suppose that*\(\Omega(t)\in C^{m}[0,1]\).

*Then the error*\(E_{m}= \Vert \Omega(t)-\Omega_{m}(t) \Vert \)

*of approximation of the function*\(\Omega(t)\)

*by*\(\Omega_{m}(t)\)

*using formula*(7)

*can be estimated as follows*:

*where*\(\hbar=\operatorname{max}_{t\in[0,1]}\Omega^{(m+1)}(t)\)

*and*\(\Delta=\operatorname {max}[t_{0}, t-t_{0}]\),

*and*\(C^{m}[0,1]\)

*is the space of all*

*m*

*times continuously differentiable functions in the interval*\([0,1]\).

#### Proof

*m*defined by

In the following theorem,we give the main approximate formula for the integer derivative \(D^{(n)}\Omega_{m}(t)\).

#### Theorem 3

*Suppose that we approximate the function*\(\Omega(t)\)

*in the form*(7).

*Then*\(D^{(n)}(\Omega_{m}(t))\)

*can be defined as*

#### Proof

The proof of this theorem can be done directly with the help of formula (7) and some properties of the third-kind shifted Chebyshev polynomials. □

### 2.2 Procedure solution

Equations (15)–(16), together with five equations of the boundary conditions (17), give a system of \((2m+2)\) algebraic equations, which can be solved, for the unknowns \(f_{i}\), \(\theta_{i}, i =0,1,\ldots,m\), using the Newton iteration method. In our numerical study, we take \(m=5\), that is, five terms of the truncated series solution (12) at \(\eta=1\).

## 3 Results and discussion

Comparison of *γ* and \(-f^{\prime\prime}(0)\) with \(\delta=M=0\) using the previous work and the Chebyshev collocation method

| Data of [25] | Present results | ||
---|---|---|---|---|

| \(-f^{\prime\prime}(0)\) | | \(-f^{\prime\prime}(0)\) | |

1.4 | 0.674089 | 1.012781 | 0.6739267 | 1.0126853 |

1.6 | 0.331976 | 0.642412 | 0.309138 | 0.6423921 |

1.8 | 0.127013 | 0.3320138 | 0.1270089 | 0.3091378 |

*S*on the dimensionless velocity \(f'(\eta)\). These plots reveal the fact that increasing values of the parameter

*S*results in enhancing the velocity distribution along the thin film region, but the reverse is observed for the film thickness. The dimensionless temperature for different values of unsteadiness parameter

*S*is displayed in Fig. 2. Note that both the free surface temperature \(\theta(\gamma)\) and the dimensionless temperature increases with the increase in

*S*.

*M*on the dimensionless velocity is presented in Fig. 3. Observed that, along the sheet, the dimensionless velocity increases with an increase in the

*M*parameter, but the reverse trend is away from the sheet. Also, we further find that the increasing value of the parameter

*M*leads to decreasing the film thickness. In Fig. 4, we depict the effect of the same parameter

*M*on the dimensionless temperature. Note that, with the increasing values of magnetic parameter

*M*, both the free surface temperature \(\theta(\gamma)\) and the dimensionless temperature increase inside the film region.

*δ*. From this plot it is evident that, along the sheet, the effect of increasing values of the parameter

*δ*is responsible for thinning the film thickness and decreasing the dimensionless velocity. Figure 6 displays the temperature \(\theta(\eta)\) profiles versus

*η*for various values of the same parameter

*δ*. It is elucidating that both the dimensionless temperature distribution and the free surface temperature \(\theta(\gamma)\) increase with an increase in parameter

*δ*. Figure 7 depicts the effect of the radiation parameter

*R*on the temperature profile. It is interesting to note that the dimensionless temperature distribution increases as the radiation parameter

*R*increases. Likewise, some qualitative behaviors for the thin film flow and heat transfer characteristics are demonstrated in the same figure, in which the thickness for the film is fixed for changing the values of the radiation parameter

*R*.

## 4 Conclusion and remarks

- (1)
The effect of increasing both the values of the unsteadiness parameter and the magnetic parameter increases both the dimensionless velocity and the dimensionless temperature throughout the film layer.

- (2)
The temperature distribution can be affected by changing the values of the velocity slip parameter and the thermal radiation parameter.

- (3)
Because of the presence of slip velocity parameter, there may be a lower velocity distribution near the stretching sheet and also thinning the film thickness.

## Notes

#### Acknowledgements

The author is very grateful to the editor and referees for carefully reading the paper and for their comments and suggestions, which have improved the paper.

#### Availability of data and materials

Not applicable.

### Authors’ contributions

The paper by one author. Author read and approved the final manuscript.

### Funding

Not applicable.

#### Competing interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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