Analytical solution for the flow of second grade fluid over a stretching sheet

Rafiq, Muhammad; Kamran, Muhammad; Ahmed, Naveed; Mohyud-Din, Syed Tauseef; Bashir, Yasir; Haider, Sajjad Ali; Farwa, Shabieh; Tahir, Madeeha

doi:10.1063/1.5093158

This article addresses the phenomenon of boundary layer flow and heat transfer in respect of the motion of second grade viscous fluid over an unsteady stretchable surface. Optimal variational iteration method (OVIM) is employed to solve the governing differential equation. OVIM is the modified version of variational iteration method (VIM). The convergence of the obtained solution as well as the effect of pertinent parameters on the velocity components are discussed, tabulated, and graphed. Behaviour of coefficient of skin-friction is also tabulated and explained for different parameters. Comparison of residual errors sought via use of VIM and OVIM is also presented.

I. INTRODUCTION

Study of boundary layer flow along with the phenomenon of heat transfer over stretching sheet fascinated many researchers for last two or three decades due to its manifold applications in engineering disciplines, like biomedical, metallurgy and chemical engineering. The accoutrements inured for surgical purposes also make use of this phenomenon. Specifically, we encounter with such type of flow while working in aerodynamics, inflation of balloons, drafting of plastic films etc. Concept of boundary layer flow over a stretching surface inaugurated in his innovative work by Sakiadis.^28,29 After the introduction of such type of flows with incomparable utilities, many renowned mathematicians and engineers investigated different rheological based on boundary layer flow, see Refs. 1, 7–9, 18, and 32.

As all physical phenomenons are time varying, so its natural to discuss unsteady flows, in this respect the hydrodynamic viscous thin fluid film flow over a sheet stretching was studied by Wang,⁴⁵ and by many others. Soon after,^2,3,6,19,45 made extensions to Wang’s analysis from different perspectives. Wang⁴³ and Wang et al.⁴⁴ also scrutinised the time–dependent flows of viscous as well as power law fluids over stretchable surface, they adopted homotopy analysis method to find the solution.⁴ provided an analytically similar solution for flow over a flat plate convective surface. Moreover, Makinde and Aziz²⁰ elongated the work contained in Ref. 4 by considering the flow over vertical stretchable sheet. Shahzad and Ali^30,31 acquiesced an approximate series solution for flow of non-Newtonian fluid by constant vertical stretching of sheet. Bachok et al.¹⁷ inquired unsteady boundary layer flow and transfer of heat over a surface that was stretched in vertical. Sheikholeslami et al.^35,37 made use of Adomian decomposition method. Further more, study of viscous fluid under the effects of magnetic field with the help of optimal homotopy asymptotic method has been carried out in Refs. 34 and 39. The homotopy perturbation and homotopy analysis methods have been used to investigate the flow of condensation film on tilted revolving disk and flow across parallel plates.^33,38 Sheikholeslami and Ganji³⁶ developed a systematic differential transform method for transfer of heat in conjunction with Brownian effects across parallel plates.

For second grade fluids, a number of attempts already have been made to determine analytic and closed form solutions of velocity field and stream functions against various geometrical and physical arrangements. Coscia and Galdi¹⁰ studied classical problems related to existence, stability, and uniqueness of solutions. Moreover, Dunn and Fosdick¹¹ mainly put focus on stability pertaining to flow problems. Ting⁴¹ found solutions for unsteady second grade fluid flowing in bounded regions. Hayat et al.¹³ discussed some unsteady solutions while considering stream functions of particular forms. Further more, Siddiqui and Kaloni⁴⁰ and Aziz et al.⁵ provided similar results by varying the form of stream function. Motivated by above mentioned contributions, in this paper we shall investigate the unsteady flow of a fluid film flowing in 2-dimensions over a surface with constant stretching. Precisely the current work has some significant differences with the prior work done so far. We deal with a no–linear problem that arose here analytically with the help of OVIM. The idea of OVIM for initial value problems was introduced by.^12,15,21,42 The OVIM for boundary value problems is an efficient technique developed by Tauseef et al.²² OVIM for boundary value problems is a modified form of VIM. In this technique, all initial and boundary conditions are consumed before constructing an iterative algorithm. Thus we construct an iterative algorithm which is free from undetermined coefficients. In most of the previous iterative algorithms for boundary value problems, each iteration contains the undetermined coefficients. Consuming all the boundary Conditions in the developed algorithm reduces the computational volume and enhances convergence. In OVIM we insert an auxiliary parameter h in the algorithm that facilitates fast convergence of the approximate solution in the given domain of the problem. The minimum of the norm 2 of the residual error is used to find the optimal choice of the parameter which is not in the VIM, see Refs. 12, 22, 23, and 26.

II. MATHEMATICAL FORMULATION

Consider an unsteady incompressible second grade fluid flowing in xy-plane in the form of thin film having uniform width h(t) that is placed on a horizontal sheet. The fluid is set into motion by stretching the sheet constantly along x-axis, and the y-axis is perpendicular to the direction of flow. By keeping in mind the geometry of the problem, we shall consider velocity field of the form

V (x, t) = (v_{1} (x, t), v_{2} (x, t)) .

(1)

The constitutive equation for a second grade incompressible homogenous fluid²⁷ is

T = - p I + μ A_{1} + α_{1} A_{2} + α_{2} A_{1}^{2},

(2)

where T is the Cauchy stress tensor, p is the pressure, I is the identity tensor, μ the dynamic viscosity, α₁ and α₂ are the material constants. The first two Rivlin-Ericksen tensors are

\begin{aligned} A_{1} & = \nabla . V + {(\nabla . V)}^{†}, \\ A_{2} & = \frac{d A_{1}}{d t} + A_{1} (\nabla . V) + {(\nabla . V)}^{†} A_{1}, \end{aligned}

(3)

where $\frac{d}{d t}$ is time derivative and superscript † denotes usual matrix transpose. Note that the Clausius-Duhemin equality (i.e. μ ≥ 0, α₁ ≥ 0, α₁ + α₂ = 0.) is also satisfied. For α₁ = α₂ = 0, the equation(3) gives the constitutive equation of viscous fluid. Using Equations (1), (2) and (3) lead to governing equations for the unsteady boundary layer flow that are given by

\frac{\partial v_{1}}{\partial x} + \frac{\partial v_{2}}{\partial y} = 0,

(4)

\begin{align} \frac{\partial v_{1}}{\partial t} + v_{1} \frac{\partial v_{1}}{\partial x} + v_{2} \frac{\partial v_{1}}{\partial y} & = ν \frac{\partial^{2} v_{1}}{\partial y^{2}} + \frac{α_{1}}{ρ} [) \frac{\partial^{3} v_{1}}{\partial t \partial y^{2}} + v_{1} \frac{\partial^{3} v_{1}}{\partial x \partial y^{2}} \\ + \frac{\partial v_{1}}{\partial x} \frac{\partial^{2} v_{1}}{\partial y^{2}} + \frac{\partial v_{1}}{\partial y} \frac{\partial^{2} v_{2}}{\partial y^{2}} + v_{2} \frac{\partial^{3} v_{1}}{\partial y^{3}}]), \end{align}

(5)

where ν and ρ denote the kinematic viscosity and density of the fluid respectively. We assume the corresponding boundary conditions are

\begin{aligned} v_{1} & = \frac{c_{1} x}{1 - c_{2} t}, & v_{2} & = 0 & a t y & = 0, \\ \frac{\partial v_{1}}{\partial y} & = 0, & v_{2} & = \frac{d h}{d t} & a t y & = h, \end{aligned}

(6)

where c₁ and c₂ are positive real numbers and 1 − c₂t > 0. Consider the dimensionless transformations¹⁹

η = \sqrt{\frac{c_{1}}{v_{2}}} {(1 - c_{2} t)}^{- \frac{1}{2}} y, ψ = \sqrt{c_{1} v_{2}} x {(1 - c_{2} t)}^{- \frac{1}{2}} f (η),

(7)

where η is the dimensionless local similarity variable and ψ(x, y) is the stream function defined by $(v_{1}, v_{2}) = (\frac{\partial ψ}{\partial y}, - \frac{\partial ψ}{\partial x})$ and ν is the kinematic viscosity of the second grade fluid. The continuity equation (4) is automatically satisfied and from (2)–(6), one can write

\begin{align} f f^{''} + f^{'''} - f^{' 2} - S () \frac{1}{2} η f^{''} + f^{'})) \\ + α \{) 2 f^{'''} f + S () \frac{1}{2} η f^{(i v)} + 2 f^{'''})) - f f^{(i v) - f^{'' 2}} \}) = 0, \end{align}

(8)

\begin{aligned} f (η) & = 0, & f^{'} (η) & = 1, a t η = 0, \\ f (χ) & = \frac{1}{2} χ S, & f^{''} (χ) & = 0, a t η = χ . \end{aligned}

(9)

Here $S = \frac{c_{2}}{c_{1}}$ denotes the unsteadiness parameter, $α = \frac{c_{1} α_{1}}{μ (1 - c_{2} t)}$ is the parameter of second grade with no dimension, the prime indicates the derivative with reference to η. Moreover, χ is the film thickness with no dimension and used to denote the value of η at the free surface so that (5) gives, see Ref. 19

χ = \sqrt{\frac{c_{1}}{ν (1 - c_{2} t)}} h (t), \frac{d h}{d t} = - \frac{c_{2} χ}{2} \sqrt{\frac{ν}{c_{1}}} {(1 - c_{2} t)}^{- \frac{1}{2}} .

(10)

Observe that h(t) get decreased monotonically with the increase in time and χ remains constant which depends only upon S, see Ref. 19. The detailed discussion on such cases can be found in the Refs. 2, 3, 6, 19, 43, and 44.

The shear stress τ_$w$ on the surface of the thin liquid film sheet is

τ_{w} = [) μ \frac{\partial v_{1}}{\partial y} + α_{1} () \frac{\partial^{2} v_{1}}{\partial y \partial t} + 2 \frac{\partial v_{1}}{\partial x} \frac{\partial v_{1}}{\partial y} + v_{1} \frac{\partial^{2} v_{1}}{\partial x \partial y} + v_{2} \frac{\partial^{2} v_{1}}{\partial y^{2}})) {])}_{y = 0}

(11)

and the local skin-friction coefficient or frictional drag coefficient is

R_{e x}^{\frac{1}{2}} C_{f} = \frac{τ_{w}}{ρ {(\frac{c_{1} x}{1 - c_{2} t})}^{2}} a n d α_{1} = \tilde{α_{1}} (1 - c_{2} t) .

(12)

In dimensionless form, we have

\begin{align} R_{e x}^{1 / 2} C_{f} & = [) f^{''} (η) + α () 3 f^{'} (η) f^{''} (η) - f (η) f^{'''} (η) \\ + \frac{S}{2} () 3 f^{''} (η) + η f^{'''} (η))) {])}_{η = 0}, \end{align}

(13)

where $R_{e x}^{1 / 2} = c_{1} x^{2} / ν (1 - c_{2} t)$ is the local Reynold’s number. We will solve (10) and (11) analytically using VIM and OVIM in the next section.

III. VARIATIONAL ITERATION METHOD (VIM)

The basic notion of the technique is illustrated by considering the problem of finding z(η) such that

L z (η) + N z (η) = g (η)

(14)

where L and N are linear and nonlinear operators respectively and g(η) is the source term. For a given z₀, approximate solution z_p+1 of equation (16) can be found as follows:

\begin{aligned} z_{p + 1} (η) & = z_{p} (η) + \int_{0}^{η} λ (s, η) [L z_{p} (s) + N z_{p} (s) - g (s)] d s, \\ p & = 0,1,2, \dots, \end{aligned}

(15)

where λ is named as the Lagrange multiplier. This Lagrange multiplier is obtained by taking variation δ on both sides of equation (17) with respect to the variable z_p.

δ z_{p + 1} (η) = δ z_{p} (η) + δ \int_{0}^{η} λ (s, η) [L z_{p} (s) + N \tilde{z_{p} (s)} - g (s)] d s,

where $\tilde{z_{p} (s)}$ is a term being restricted which in turn gives $δ \tilde{z_{p} (s)} = 0$ ⁠. An unknown parameter λ(s, η) is found by making use of the optimal conditions, see Refs. 24 and 25. This yield an exact solution z(η), where

z (η) = lim_{p \to \infty} z_{p} (η) .

(16)

This method of getting the approximate solution is known as the method of variational iteration abbreviated as (VIM). The VIM was introduced by Inokuti et al.¹⁶ However, He¹⁴ has established this method for giving solution to wider range of problems, emerging in several fields of pure and applied sciences. In VIM, proper selection initial approximation leads to the fast converging solution, see Refs. 24 and 25 and the references therein.

A. Optimal variational iteration method (OVIM)

Consider a second order nonlinear ordinary differential equation,

L z (η) + N z (η) = g (η)

(17)

subject to the boundary conditions

z (a) = α_{1}, z (b) = α_{2}, a, b, α_{1}, α_{2} \in R,

(18)

where $L (η) = \frac{d^{2}}{d η^{2}}$ is a linear differential operator, N(η) represents the nonlinear operator and g(η) is an inhomogeneous term and $R$ is set of real numbers. According to standard VIM, the correction functional used is given as

z_{p + 1} (η) = z_{p} (η) + \int_{0}^{η} λ (s, η) () L z_{p} (s) + N \tilde{z_{p} (s)} + g (s))) d s .

(19)

To make stable correction functional, the lagrange multiplier is defined as λ = s − η, see Refs. 24 and 25, we get the following iterative formula

\begin{aligned} z_{p + 1} (η) & = z_{p} (η) + \int_{0}^{η} (s - η) () L z_{p} (s) + N \tilde{z_{p} (s)} + g (s))) d s, \\ p & = 0,1,2,3, \dots . \end{aligned}

(20)

An unknown auxiliary parameter h is inserted into the iterative formula (20), for n = 0, (20) becomes

z_{p + 1} (η) = z_{p} (η) + h \int_{0}^{η} (s - η) () L z_{p} (s) + N z_{p} (s) + g (s))) d s .

(21)

For optimal variational iteration method, we will proceed as follows

z (η) = a_{1} + a_{0} η + h \int_{0}^{η} (s - η) () L z (s) + N z (s) + g (s))) d s,

(22)

where a₀ and a₁ are unknown parameters which can be determined by using the given boundary conditions. Substituting the values of a₀ and a₁ in (24), we obtain

\begin{matrix} z (η) & = & \frac{a α_{2} - b α_{1}}{a - b} + \frac{(α_{1} - α_{2}) η}{a - b} \\ + \frac{b h}{a - b} \int_{0}^{a} λ (s, a) () L z (s) + N z (s) + g (s))) d s \\ - \frac{a h}{a - b} \int_{0}^{a} λ (s, a) () L z (s) + N z (s) + g (s))) d s \\ - \frac{η h}{a - b} \int_{0}^{a} λ (s, a) () L z (s) + N z (s) + g (s))) d s \\ + \frac{η h}{a - b} \int_{0}^{b} λ (s, b) () L z (s) + N z (s) + g (s))) d s \\ + h \int_{0}^{η} (s - η) () L z (s) + N z (s) + g (s))) d s . \end{matrix}

From the above equation, we develop the following sequence

z_{0} (η) = \frac{a α_{2} - b α_{1}}{a - b} + \frac{(α_{1} - α_{2}) η}{a - b},

(23)

\begin{matrix} z_{1} (η, h) & = & \frac{b h}{a - b} \int_{0}^{a} λ (s, a) () L z_{0} (s) + N z_{0} (s) + g (s))) d s \\ - \frac{a h}{a - b} \int_{0}^{a} λ (s, a) () L z_{0} (s) + N z_{0} (s) + g (s))) d s \\ - \frac{η h}{a - b} \int_{0}^{a} λ (s, a) () L z_{0} (s) + N z_{0} (s) + g (s))) d s \\ + \frac{η h}{a - b} \int_{0}^{b} λ (s, b) () L z_{0} (s) + N z_{0} (s) + g (s))) d s \\ + h \int_{0}^{η} (s - η) () L z_{0} (s) + N z_{0} (s) + g (s))) d s, \end{matrix}

(24)

\begin{matrix} z_{p + 1} (η, h) & = & \frac{b h}{a - b} \int_{0}^{a} λ (s, a) () L z_{p} (s) + N z_{p} (s) + g (s))) d s \\ - \frac{a h}{a - b} \int_{0}^{a} λ (s, a) () L z_{p} (s) + N z_{p} (s) + g (s))) d s \\ - \frac{η h}{a - b} \int_{0}^{a} λ (s, a) () L z_{p} (s) + N z_{p} (s) + g (s))) d s \\ + \frac{η h}{a - b} \int_{0}^{b} λ (s, b) () L z_{p} (s) + N z_{p} (s) + g (s))) d s \\ + h \int_{0}^{η} (s - η) () L z_{p} (s) + N z_{p} (s) + g (s))) d s . \end{matrix}

(25)

The iterative algorithm (23)–(25) does not contain undetermined constants except an auxiliary parameter h which is used to control the convergence of approximate solution optimally by minimising the norm 2 of residual function over the domain of the given problem. For further details, see Ref. 22. This algorithm is called optimal variational iteration formula for solving boundary value problem (17).

IV. NUMERICAL EXAMPLE

In this section, we use VIM and OVIM to find an approximate solution of boundary value problem (7) and (8)for different values of S and α, while keeping β fixed.

A. Example 4.1

For S = 0.2, α = 0, β = 1, Equations (8) and (9) become

f^{'''} - f^{' 2} + f f^{''} - 0.2 () f^{'} + \frac{1}{2} η f^{''})) = 0,

(26)

\begin{aligned} f (0) & = 0, & f^{'} (0) & = 1, a t η = 0 \\ f (1) & = \frac{1}{10}, & f^{''} (1) & = 0, a t η = 1 . \end{aligned}

(27)

Using VIM, we construct the correction functional for (26) as follows

\begin{align} f_{p + 1} (η) & = f_{p} (η) + \int_{0}^{η} λ (s, η) () f_{p}^{'''} (s) - \tilde{f_{p}^{' 2}} (s) + \tilde{f_{p} (s)} \tilde{f_{p}^{''} (s)} \\ - 0.2 () \tilde{f_{p}^{'} (s)} + \frac{1}{2} s \tilde{f_{p}^{''} (s)})) d s . \end{align}

(28)

Using the optimality conditions,²⁵ one has

λ (s, η) = - \frac{{(s - η)}^{2}}{2!} .

(29)

Using (28) and (29), we have

\begin{align} f_{p + 1} (η) & = f_{p} (η) - \int_{0}^{η} \frac{{(s - η)}^{2}}{2!} () f_{p}^{'''} (s) - f_{p}^{' 2} (s) + f_{p} (s) f_{p}^{''} (s) \\ - 0.2 () f_{p}^{'} (s) + \frac{1}{2} s f_{p}^{''} (s))) d s \end{align}

(30)

We assume

f_{0} (η) = a_{3} + a_{2} (η) + a_{1} \frac{η^{2}}{2!} + a_{0} \frac{η^{3}}{3!},

(31)

as an initial approximation, where a_i, i = 0, 1, 2, 3, are unknowns and will be determined by boundary conditions (27). Using (31) in (30), we obtain the following iterations

\begin{matrix} f_{1} (η) & = & a_{3} + a_{2} η + 0.5 a_{1} η^{2} \\ + ((- 0.1666666 a_{3} + 0.003333333) a_{1} + 0.0000001 a_{0} + 0.1666666 {a_{2}}^{2} + 0.03333333 a_{2}) η^{3} \\ + ((0.0416667 a_{2} + 0.00833334) a_{1} + (- 0.0416667 a_{3} + 0.000833334) a_{0}) η^{4} \\ + (0.00833332 {a_{1}}^{2} + 0.00166666 a_{0}) η^{5} + 0.00277781 a_{1} a_{0} η^{6} + 0.000396825 {a_{0}}^{2} η^{7}, \\ ⋮ \end{matrix}

Applying the boundary conditions given by Eq. 27 on approximation f₃(η), we obtain

\begin{gathered} a_{0} = - 743.3768430, a_{1} = - 1.227521124, \\ a_{2} = 1.000000000, a_{3} = 0 . \end{gathered}

We define the following residual error function

r_{3} (η) = f_{3}^{'''} (η) - f_{3}^{' 2} (η) + f_{3} (η) f_{3}^{''} (η) - 0.2 () f_{3}^{'} (η) + \frac{1}{2} η f_{3}^{''} (η))) .

(32)

Now using OVIM, we proceed as follows

f (η, h) = a_{3} + a_{2} η + a_{1} \frac{η^{2}}{2!} + a_{0} \frac{η^{3}}{3!} - h \int_{0}^{η} \frac{{(s - η)}^{2}}{2!} () f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))) d s .

Using the boundary conditions, we have

\begin{matrix} a_{0} & = & - \frac{3}{10} + 3 - 3 h \int_{0}^{1} \frac{{(s - 1)}^{2}}{2!} () f_{p}^{'''} (s) - f_{p}^{' 2} + f_{p} (s) f_{p}^{''} (s) - 0.2 () f_{p}^{'} (s) + \frac{1}{2} η f_{p}^{''} (s))))) d s \\ + \frac{3}{2} h \int_{0}^{1} () f_{p}^{'''} (s) - f_{p}^{' 2} (s) + f_{p} (s) f_{p}^{''} (s) - 0.2 () f_{p}^{'} (s) + \frac{1}{2} η f_{p}^{''} (s))) d s, \end{matrix}

\begin{matrix} a_{1} & = & \frac{3}{10} - 3 + 3 h \int_{0}^{1} \frac{{(s - 1)}^{2}}{2!} () f_{p}^{'''} (s) - f_{p}^{' 2} + f_{p} (s) f_{p}^{''} (s) - 0.2 () f_{p}^{'} (s) + \frac{1}{2} η f_{p}^{''} (s))))) d s \\ - \frac{1}{2} h \int_{0}^{1} () f_{p}^{'''} (s) - f_{p}^{' 2} (s) + f_{p} (s) f_{p}^{''} (s) - 0.2 () f_{p}^{'} (s) + \frac{1}{2} η f_{p}^{''} (s))))) d s, \end{matrix}

a_{2} = 1, a_{3} = 0 .

The OVIM algorithm is given as follows

f_{0} (η) = η - 1.35 η^{2} + 0.45 η^{3},

\begin{matrix} f_{1} (η, h) & = & f_{0} (η) - h \int_{0}^{η} \frac{{(s - η)}^{2}}{2!} () f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) d s \\ + \frac{h η^{2}}{2} \{) \int_{0}^{1} \frac{3 {(s - 1)}^{2}}{2!} () f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) d s \\ + \frac{1}{2} \int_{0}^{1} () f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) d s \}) \\ + \frac{h η^{3}}{6} \{) \int_{0}^{1} \frac{- 3 {(s - 1)}^{2}}{2!} () f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) d s \\ + \frac{3}{2} \int_{0}^{1} () f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) d s \}) \end{matrix}

\begin{matrix} f_{p + 1} (η, h) & = & f_{p} (η, h) - h \int_{0}^{η} \frac{{(s - η)}^{2}}{2!} () f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))) d s \\ + \frac{h η^{2}}{2} \{) \int_{0}^{1} \frac{3 {(s - 1)}^{2}}{2!} () f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))))) d s \\ + \frac{1}{2} \int_{0}^{1} () f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))) d s \}) \\ + \frac{h η^{3}}{6} \{) \int_{0}^{1} \frac{- 3 {(s - 1)}^{2}}{2!} () f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))))) d s \\ + \frac{3}{2} \int_{0}^{1} () f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))) d s \}) . \end{matrix}

We obtain the following approximations

f_{0} (η) = η - 1.35 η^{2} + 0.45 η^{3},

\begin{matrix} f_{1} (η, h) & = & η + (- 1.350000000 - 0.06621428550 h) η^{2} + (0.1910000000 + 0.4100714285 h) η^{3} \\ - 0.132750000 η^{4} + 0.0652500000 η^{5} - 0.0202500000 η^{6} + 0.0028928571 η^{7}, \\ ⋮ \end{matrix}

We define the following residual function

\begin{align} r_{3} (η, h) & = f_{3}^{'''} (η, h) - f_{3}^{' 2} (η, h) + f_{3} (η, h) f_{3}^{''} (η, h) \\ - 0.2 () f_{3}^{'} (η, h) + \frac{1}{2} η f_{3}^{''} (η, h))) \end{align}

(33)

and the norm 2 of above residual error function is

e_{3} (h) = [) \frac{1}{101} \sum_{i = 0}^{100} \{) r_{3} (\frac{i}{100}, h) {\})}^{2} {])}^{\frac{1}{2}} .

The above residual function can be used to approximate e₃(h), and the optimal value of h can be determined by minimizing the e₃(h). The value of h is found to be 0.0027467499661031 when minimum value of e₃(h) is 0.0000609419536125150826. We may obtain more accuracy by increasing the number of iterations.

The graphs of residual errors r₃(η) and r₃(η, h) given by (32) and (33) are plotted in Figure 1. The graph of r₃(η) obtained by VIM diverges but the graph of r₃(η, h) obtained by OVIM converges to zero. It is concluded that OVIM is more reliable and accurate as compared to VIM.

FIG. 1.

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Residual Error comparison of VIM and OVIM Solutions when α = 0 and S = 0.2.

B. Example 4.2

For S = 0.2, α = 0.9, β = 1, (8) and (9) become

\begin{align} f^{'''} - f^{' 2} + f f^{''} - 0.2 () f^{'} + \frac{1}{2} η f^{''})) + 0.9 \{) 2 f f^{'''} + 0.2 () 2 f^{'''} \\ + \frac{1}{2} η f^{(i v)})) - f^{'' 2} - f f^{(i v)} \}) = 0, \end{align}

(34)

\begin{aligned} f (0) & = 0 & f^{'} (0) & = 1, a t η & = 0, \\ f (1) & = \frac{1}{10}, & f^{''} (1) & = 0 a t η & = 1 . \end{aligned}

(35)

We construct the correction functional for (34) as follows

\begin{matrix} f_{p + 1} (η) & = & f_{p} (η) + \int_{0}^{η} λ (s, η) [) f_{n}^{'''} (s) - {\tilde{f_{n}^{'} (s)}}^{2} + \tilde{f_{p} (s)} \tilde{f_{n}^{''} (s)} \\ - 0.2 () \tilde{f_{n}^{'} (s)} + \frac{1}{2} s \tilde{f_{n}^{''} (s)})) + 0.9 \{) 2 \tilde{f_{p} (s)} \tilde{f_{n}^{'''} (s)} \\ + 0.2 () 2 \tilde{f_{n}^{'''} (s)} + \frac{1}{2} s {\tilde{f_{p} (s)}}^{(i v)})) - {\tilde{f_{n}^{''} (s)}}^{2} \\ - \tilde{f_{p} (s)} {\tilde{f_{p} (s)}}^{(i v)} \})]) d s . \end{matrix}

(36)

Using the optimality conditions,²⁵ one has

λ (s, η) = - \frac{{(s - η)}^{2}}{2!} .

(37)

Using (37) in (36), We have the iterative scheme as

\begin{matrix} f_{p + 1} (η) & = & f_{p} (η) - \int_{0}^{η} \frac{{(s - η)}^{2}}{2} [) f_{p}^{'''} (s) - f_{p} {(s)}^{2} + f_{p} (s) f_{p}^{''} (s) \\ - 0.2 () f_{p} (s) + \frac{1}{2} s f_{p}^{''} (s))) + 0.9 \{) 2 f_{p} (s) f_{p}^{'''} (s) \\ + 0.2 () 2 f_{p}^{'''} (s) + \frac{1}{2} s f_{p} {(s)}^{(i v)})) - f_{p}^{''} {(s)}^{2} \\ - f_{p} (s) f_{p} {(s)}^{(i v)} \})]) d s . \end{matrix}

(38)

Using (31) and (38), we obtain the following iterations

f_{0} (η) = a_{3} + a_{2} (η) + a_{1} \frac{η^{2}}{2!} + a_{0} \frac{η^{3}}{3!},

\begin{aligned} f_{1} (η) & = a_{3} + a_{2} η + 0.50000 a_{1} η^{2} + ((- 0.05999 - 0.30000 a_{3}) a_{0} + 0.15000 {a_{1}}^{2}) η^{3} \\ + ((- 0.16666 a_{3} + 0.0033333) a_{1} + 0.16666 {a_{2}}^{2} + 0.033333 a_{2}) η^{3} \\ + ((- 0.04167 a_{3} - 0.07501 a_{2} + 0.07501 a_{1} + 0.0008334) a_{0} + (0.04167 a_{2} + 0.008334) a_{1}) η^{4} \\ + (0.01500 {a_{0}}^{2} + (- 0.01500 a_{1} + 0.001666) a_{0} + 0.008332 {a_{1}}^{2}) η^{5} \\ + (0.002778 a_{1} a_{0} - 0.002500 {a_{0}}^{2}) η^{6} + 0.0003968 {a_{0}}^{2} η^{7}, \\ ⋮ \end{aligned}

Applying the boundary conditions (35) on f₂(η), the unknowns are

a_{0} = 14.90349895, a_{1} = - 9.795246642, a_{2} = 1, a_{3} = 0 .

We define the following residual error function

\begin{matrix} r_{2} (η) & = & f_{2}^{'''} (η) - f_{2}^{' 2} (η) + f_{2} (η) f_{2}^{''} (η) - 0.2 () f_{2}^{'} (η) + \frac{1}{2} η f_{2}^{''} (e t a))) \\ + 0.9 \{) 2 f_{2} (η) f_{2}^{'''} (η) + 0.2 () 2 f_{2}^{'''} (η) + \frac{1}{2} η f_{2}^{(i v)} (η))) - f_{2}^{'' 2} (η) - f_{2} (η) f_{3}^{(i v)} (η) \}) . \end{matrix}

(39)

Now using OVIM, we proceed as follows

f_{0} (η) = η - 1.35 η^{2} + 0.45 η^{3},

\begin{matrix} f_{1} (η, h) & = & f_{0} (η) - h \int_{0}^{η} \frac{{(s - η)}^{2}}{2!} [) f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) \\ + 0.9 \{) 2 f_{0} (s) f_{0}^{'''} (s) + 0.2 () 2 f_{0}^{'''} (s) + \frac{1}{2} s f_{0}^{(i v)} (s))) - f_{0}^{'' 2} (s) - f_{0} (s) f_{0}^{(i v)} (s) \})]) d s \\ + \frac{h η^{2}}{2} \{) \int_{0}^{1} \frac{3 {(s - 1)}^{2}}{2!} [) f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) \\ + 0.9 \{) 2 f_{0} (s) f_{0}^{'''} (s) + 0.2 () 2 f_{0}^{'''} (s) + \frac{1}{2} s f_{0}^{(i v)} (s))) - f_{0}^{'' 2} (s) - f_{0} (s) f_{0}^{(i v)} (s) \})]) d s \\ + \frac{1}{2} \int_{0}^{1} [) f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) \\ + 0.9 \{) 2 f_{0} (s) f_{0}^{'''} (s) + 0.2 () 2 f_{0}^{'''} (s) + \frac{1}{2} s f_{0}^{(i v)} (s))) - f_{0}^{'' 2} (s) - f_{0} (s) f_{0}^{(i v)} (s) \})]) d s \}) \\ + \frac{h η^{3}}{6} \{) \int_{0}^{1} \frac{- 3 {(s - 1)}^{2}}{2!} [) f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) \\ + 0.9 \{) 2 f f_{0}^{'''} (s) + 0.2 () 2 f_{0}^{'''} (s) + \frac{1}{2} s f_{0}^{(i v)} (s))) - f_{0}^{'' 2} (s) - f_{0} (s) f_{0}^{(i v)} (s) \})]) d s \\ + \frac{3}{2} \int_{0}^{1} [) f_{0}^{'''} (s) - f_{0}^{' 2} (s) + f_{0} (s) f_{0}^{''} (s) - 0.2 () f_{0}^{'} (s) + \frac{1}{2} s f_{0}^{''} (s))) \\ + 0.9 \{) 2 f_{0} (s) f_{0}^{'''} (s) + 0.2 () 2 f_{0}^{'''} (s) + \frac{1}{2} s f_{0}^{(i v)} (s))) - f_{0}^{'' 2} (s) - f_{0} (s) f_{0}^{(i v)} (s) \})]) d s \}) \end{matrix}

\begin{matrix} f_{p + 1} (η, h) & = & f_{p} (η, h) - h \int_{0}^{η} \frac{{(s - η)}^{2}}{2!} [) f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))) \\ + 0.9 \{) 2 f_{p} (s, h) f_{p}^{'''} (s, h) + 0.2 () 2 f_{p}^{'''} (s, h) + \frac{1}{2} s f_{p}^{(i v)} (s, h))) - f_{p}^{'' 2} (s, h) - f_{p} (s, h) f_{p}^{(i v)} (s, h) \})]) d s \\ + \frac{h η^{2}}{2} \{) \int_{0}^{1} \frac{3 {(s - 1)}^{2}}{2!} [) f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))) \\ + 0.9 \{) 2 f_{p} (s, h) f_{p}^{'''} (s, h) + 0.2 () 2 f_{p}^{'''} (s, h) + \frac{1}{2} s f_{p}^{(i v)} (s, h))) - f_{p}^{'' 2} (s, h) - f_{p} (s, h) f_{p}^{(i v)} (s, h) \})]) d s \\ + \frac{1}{2} \int_{0}^{1} [) f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))) \\ + 0.9 \{) 2 f_{p} (s, h) f_{p}^{'''} (s, h) + 0.2 () 2 f_{p}^{'''} (s, h) + \frac{1}{2} s f_{p}^{(i v)} (s, h))) - f_{p}^{'' 2} (s, h) - f_{p} (s, h) f_{p}^{(i v)} (s, h) \})]) d s \}) \\ + \frac{h η^{3}}{6} \{) \int_{0}^{1} \frac{- 3 {(s - 1)}^{2}}{2!} [) f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))) \\ + 0.9 \{) 2 f f_{p}^{'''} (s, h) + 0.2 () 2 f_{p}^{'''} (s, h) + \frac{1}{2} s f_{p}^{(i v)} (s, h))) - f_{p}^{'' 2} (s, h) - f_{p} (s, h) f_{p}^{(i v)} (s, h) \})]) d s \\ + \frac{3}{2} \int_{0}^{1} [) f_{p}^{'''} (s, h) - f_{p}^{' 2} (s, h) + f_{p} (s, h) f_{p}^{''} (s, h) - 0.2 () f_{p}^{'} (s, h) + \frac{1}{2} s f_{p}^{''} (s, h))) \\ + 0.9 \{) 2 f_{p} (s, h) f_{p}^{'''} (s, h) + 0.2 () 2 f_{p}^{'''} (s, h) + \frac{1}{2} s f_{p}^{(i v)} (s, h))) - f_{p}^{'' 2} (s, h) - f_{p} (s, h) f_{p}^{(i v)} (s, h) \})]) d s \}), n \geq 1 . \end{matrix}

We obtain the following approximations

f_{0} (η) = η - 1.35 η^{2} + 0.45 η^{3},

f_{1} (η, h) = η + (- 1.3500 - 0.53400 h) η^{2} + (1.1225 + 0.49513 h) η^{3} - 0.8821 η^{4} + 0.2840 η^{5} - 0.03848 η^{6} + 0.00289 η^{7},

\begin{matrix} f_{2} (η, h) & = & η + (0.15250 h^{3} - 0.73511 h^{2} + 0.17100 h - 1.3500) η^{2} \\ + (0.29775 h + 1.4600 h^{2} - 0.24773 h^{3} + 0.94395) η^{3} \\ + (- 1.4155 h - 0.23810 h^{2} - 2.5001) η^{4} + (1.3910 h + 0.18950 h^{2} + 3.3560) η^{5} \\ + (- 1.0910 h - 0.030880 h^{2} - 2.7349) η^{6} + (0.46940 h + 0.0034970 h^{2} + 1.6760) η^{7} \\ + (- 0.11080 h - 0.77500) η^{8} + (0.013960 h + 0.25501) η^{9} \\ + (- 0.00073900 h - 0.057800) η^{10} + O {(η)}^{11}, \\ ⋮ \end{matrix}

We define the following residual error function

\begin{matrix} r_{2} (η, h) & = & f_{2}^{'''} (η, h) - f_{2}^{' 2} (η, h) + f_{2} (η, h) f_{2}^{''} (η, h) - 0.2 () f_{2}^{'} (η, h) + \frac{1}{2} η f_{2}^{''} (η, h))) \\ + 0.9 \{) 2 f_{2} (η, h) f_{2}^{'''} (η, h) + 0.2 () 2 f_{2}^{'''} (η, h) + \frac{1}{2} η f_{2}^{(i v)} (η, h))) - f_{2}^{'' 2} (η, h) - f_{2} (η, h) f_{2}^{(i v)} (η, h) \}) . \end{matrix}

(40)

and the norm 2 of residual function (40) is

e_{2} (h) = [) \frac{1}{101} \sum_{i = 0}^{100} \{) r_{2} (\frac{i}{100}, h) {\})}^{2} {])}^{\frac{1}{2}} .

The optimal value of h can be determined by minimizing e₂(h). At h = 0.810519126906262, the above norm 2 of residual error function is minimum.

The graphs of residual errors r₂(η) and r₂(η, h) given by (39) and (40) are plotted in Figure 2. The graph of r₂(η) obtained by VIM diverges but the graph of r₂(η, h) obtained by OVIM converges to zero. It is concluded that OVIM is more reliable and accurate as compared to VIM.

FIG. 2.

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Residual Error Comparison of VIM and OVIM Solutions when α = 0.9 and S = 0.2.

V. RESULTS AND DISCUSSIONS

Here we discuss the sway of different parameters like S and α on the components of velocity f, acceleration $\frac{d f}{d η}$ and skin-friction coefficients $R_{e x}^{1 / 2} C_{f}$ ⁠. The change in the coefficient of the skin-friction $R_{e x}^{1 / 2} C_{f}$ for α and S is shown in Table I.

TABLE I.

Values of skin-friction coefficient $R_{e x}^{1 / 2} C_{f}$ for different parameters at η = 0.

α	S	$R_{e x}^{1 / 2} C_{f}$
0.0	0.5	$-$ 2.24676880475314
0.2	0.5	$-$ 4.09820409923631
0.5	0.5	$-$ 6.93689203353080
0.7	0.5	$-$ 9.40436723010559
1.0	0.5	$-$ 13.8596936496045
2.0	0.5	$-$ 59.2327101256866
0.5	0	$-$ 6.15999848254210
0.5	0.2	$-$ 4.81951263596841
0.5	0.5	$-$ 7.11968336959140
0.5	1	$-$ 4.98044449104253
0.5	1.5	$-$ 2.83681213996240
0.5	2	$-$ 0.742432181172167

α	S	$R_{e x}^{1 / 2} C_{f}$
0.0	0.5	$-$ 2.24676880475314
0.2	0.5	$-$ 4.09820409923631
0.5	0.5	$-$ 6.93689203353080
0.7	0.5	$-$ 9.40436723010559
1.0	0.5	$-$ 13.8596936496045
2.0	0.5	$-$ 59.2327101256866
0.5	0	$-$ 6.15999848254210
0.5	0.2	$-$ 4.81951263596841
0.5	0.5	$-$ 7.11968336959140
0.5	1	$-$ 4.98044449104253
0.5	1.5	$-$ 2.83681213996240
0.5	2	$-$ 0.742432181172167

Figure 3 and Figure 4 are displaying the changes made by α on the component of velocity f and $\frac{d f}{d η}$ ⁠. These figures depict that f and $\frac{d f}{d η}$ the function which are increasing depending upon α. The variation in $\frac{d f}{d η}$ is considerably small as compared to f. The boundary layer thickness gets increased with increase in α. Fig. 5 and Fig. 6 are depicting the changes made by S on f and $\frac{d f}{d η}$ ⁠. It can be observed from the given figures that these velocities f and $\frac{d f}{d η}$ are the functions which are increasing depending upon S. The boundary layer thickness gets decreased as we take larger values of S. In Table I, the change of the coefficient of skin-friction $R_{e x}^{1 / 2} C_{f}$ for several values of α and S has been shown. It has been noted that the coefficient of skin-friction get increased by increasing α. In this table, it is also indicated that the coefficient of skin-friction $R_{e x}^{1 / 2} C_{f}$ get decreased when S < 2, for S = 2 the coefficient of skin-friction $R_{e x}^{1 / 2} C_{f}$ is almost zero and for S > 2 it is increased.

FIG. 3.

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Influence of second grade parameter α on f.

FIG. 4.

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Influence of second grade parameter α on $\frac{d f}{d η}$ ⁠.

FIG. 5.

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Influence of unsteadiness parameter S on $\frac{d f}{d η}$ ⁠.

FIG. 6.

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Influence of unsteadiness parameter S on $\frac{d f}{d η}$ ⁠.

VI. CONCLUSION

In this article we analysed the spectacle of boundary layer flow and heat transfer with respect to the motion of second grade fluid over a stretchable surface. Flow we considered was unsteady and fluid was viscous. Technique of OVIM is implemented to solve the governing differential equation. OVIM is the modified version of VIM. The convergence of solution we sought as well as the clout of pertinent parameters on the velocity profile are discussed, tabulated, and graphed. Behaviour of coefficient of skin-friction is also tabularized and elucidated for different parameters. Comparison of residual errors sought via use of VIM and OVIM is also presented. Amazingly OVIM gave much stable and reliable results as compared with VIM.

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2019

Author(s)

Analytical solution for the flow of second grade fluid over a stretching sheet

I. INTRODUCTION

II. MATHEMATICAL FORMULATION

III. VARIATIONAL ITERATION METHOD (VIM)

A. Optimal variational iteration method (OVIM)

IV. NUMERICAL EXAMPLE

A. Example 4.1

B. Example 4.2

V. RESULTS AND DISCUSSIONS

VI. CONCLUSION

REFERENCES

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Analytical solution for the flow of second grade fluid over a stretching sheet Open Access

I. INTRODUCTION

II. MATHEMATICAL FORMULATION

III. VARIATIONAL ITERATION METHOD (VIM)

A. Optimal variational iteration method (OVIM)

IV. NUMERICAL EXAMPLE

A. Example 4.1

B. Example 4.2

V. RESULTS AND DISCUSSIONS

VI. CONCLUSION

REFERENCES

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Analytical solution for the flow of second grade fluid over a stretching sheet