Analytic approximate solutions to the chemically reactive solute transfer problem with partial slip in the flow of a viscous fluid over an exponentially stretching sheet with suction/blowing

1 Introduction

In all hydraulic machines, the liquid fluids have chemical reactions with the materials from control structures and adjuster devices. Beyond the viscous flow, with all its implications on the energy transfer, the chemical reaction of the liquid fluid affects this transfer in a negative sense. Therefore, the establishment of some relations for the viscous flow, taking into account the chemical reaction, allows for the energetic evaluation of the flow, as more appropriate to each situation.

Hydraulic system liquids are used primarily to transmit and distribute forces to various units to be actuated. Liquids are able to do this because they are almost incompressible. Pascal’s Law states that pressure applied to any part of a confined liquid is transmitted with undiminished intensity to every other part. Manufacturers of hydraulic devices usually specify the type of liquid best suited for use with their equipment in view of the working conditions, the service required, temperatures expected inside and outside the systems, the pressures the liquid must withstand, the possibilities of corrosion, and other conditions that must be considered. If incompressibility and fluidity were the only qualities required, any liquid that is not too thick could be used in a hydraulic system. One of the most important properties of any hydraulic fluid is its viscosity. Viscosity is internal resistance to flow. A liquid such as gasoline that has a low viscosity flows easily, while a liquid such as tar that has a high viscosity flows slowly. Regardless of its function and design, every hydraulic system has a minimum number of basic components, in addition to a conduit through which the fluid is transmitted. A basic system consists of a pump, reservoir, directional valve, check valve, pressure relieve valve, selector valve, actuator, and filter [1].

In the last few decades, several researchers have studied boundary layer flow, with its practical applications in engineering, electrochemistry and polymer processing technology. Lomen et al. [2] applied a perturbation method to the transport equation for a single reactive chemical with nonlinear (quadratic) rate loss relevant to a soil and water system.

Several researchers applied the shooting method to study the asymptotic behavior of the nonlinear ordinary differential equations that describe the distribution of a solute undergoing a chemical reaction, such as: Zhao et al. [3], Bhattacharyya and Layek [4], Bhattacharyya [5], and Mukhopadhyay [6, 7, 8, 9, 10].

An approximate solution to a differential equation in the form of an analytic expression can be found by the method of series, the method of small parameters, the method of successive approximations, the perturbation method, the Ritz and Galerkin method, among others. Each of these methods defines one or more infinite processes that under certain conditions can be used to obtain an exact solution to a problem. Termination of the process after a finite number of steps yields an approximate solution. In fact, an analytical method gives a solution in the form of symbols, i.e. closed form solution. A numerical method gives solution at certain points only. Hence, the analytical solution denotes an exact solution that can be used to study the behavior of the system with varying properties.

In the last years, two interesting methods to obtain an approximate analytical solution were used for solving more sophisticated PDEs. The Homotopy Analysis Method (HAM) was developed by Liao [11] who utilized the idea of homotopy in topology. The Optimal Homotopy Asymptotic Method (OHAM) was introduced by Marinca and Herisanu [12]. An advantage of OHAM is that it does not need to identify the ħ– curve, and the control and adjustment of the convergence region is provided in a conventional way. Furthermore, the OHAM has a built in convergence criteria similar to HAM but with a greater degree of flexibility.

Along with the analytical methods, some computational methods to solve nonlinear problems from engineering and computer sciences have been developed by Li et al. [17, 18, 19, 20, 21, 22, 23, 24], Guo et al. [25], Korda et al. [26].

In this paper, the optimal homotopy asymptotic method (OHAM) [12, 13, 14, 15, 16] is applied to obtain accurate, effective analytic approximate solutions. The quality of the approximate solutions is investigated using two important statistical tests: the Bartlett test and the Durbin-Wattson test. Our procedure does not depend upon small or large parameters, and provides us with a simple way to optimally control the convergence of the approximate solutions.

2 Equations of motion

We analyze the two-dimensional flow of an incompressible viscous fluid past a flat sheet in the half-plane y > 0. As in [9], two equal and opposite forces are applied along the x- and y- axis so that the wall is stretched keeping the origin fixed. The schematic diagram of the physical model is presented in Fig. 1. Also, the continuity, momentum and concentration equations governing such a type of the flow are given as [9]:

where k=12k0ex/L.

Figure 1

Schematic diagram of the physical model.

The physical initial/boundary conditions for this case are similar as in [9], [27, 28]:

where U = U₀e^x/L, C_w = C_∞ + C₀e^x/(2L), N = N₁e^{- x/(2L)} is the velocity slip factor, which changes with x (at N = 0, no-slip case is observed), N₁ is the initial value of the velocity slip factor, V(x) > 0 is the velocity of suction, V(x) < 0 is the velocity of blowing, V(x) = V₀e^x/(2L) is the velocity on the wall, V₀ is the initial strength of suction, and k₀ is a constant having the same dimension as k.

Using the similarity transformations:

and substituting Eq. (4) into Eqs. (1) and (2), the governing equations become:

where Sc = ν/D and β = k₀L/U₀.

The initial/boundary conditions Eq. (3) become:

where S=V02LU0ν>0(<0) is the suction (or blowing) parameter, λ=N1U0ν2L and the prime denotes differentiation with respect to η.

3 Basic ideas of the optimal homotopy asymptotic method

In Marinca and Herisanu [12], the optimal homotopy asymptotic method is employed to compute analytical approximate solutions for equations of the general form:

subject to the boundary / initial conditions of the type

For the flow of viscous fluid determined by Eqs. (5) and (6) with initial/boundary conditions (7), the corresponding operators L, N and B will be introduced in the next section.

Using the principle of OHAM [12, 13, 14, 15, 16], we are going to construct the following homotopy:

where p ∈ [0, 1] is the embedding parameter, L is a linear operator, and H(y, C_i) ≠ 0 is an auxiliary convergence-control function. H is a function of the variable y and of the parameters C₁, C₂, …, C_s.

The homotopy (10) satisfies the following properties:

If the function F has the expression:

by substituting this expression in the homotopy Eq. (10) we obtain the following relation:

If we choose the following particular expression of the operator 𝓗:

we can obtain the governing equations of F₀(y) and F₁(y, C_i) by equating the coefficients of p⁰ and p¹, respectively:

F₀(y) can be readily found by solving the linear Eq. (16). In order to compute F₁(y, C_i) by Eq. (17), we take into account the fact that the nonlinear operator N has the general form:

where n is a positive integer, and h_i(y) and g_i(y) are known functions that depend on F₀(y) and on N.

The general solution of the nonhomogeneous linear equation (17) is obtained by summing the general solution of the corresponding homogeneous equation and a particular solution of the nonhomogeneous equation. Unfortunately, the computation of such a particular solution is not possible in most cases, so the computation of the function F₁(y, C_i), introduced as a third modified version of Optimal Homotopy Asymptotic Method in [12], consists of the following steps:

1. We consider the F₁(y, C_i) of the form:

   F1(y,Ci)=∑i=1mHi(y,hj(y),Cj)gi(y),j=1,…,s,(19)

   or

   F1(y,Ci)=∑i=1mHi(y,gj(y),Cj)hi(y),j=1,…,s,BF1(y,Ci),dF1(y,Ci)dy=0.(20)

   The above expressions of H_i(y, h_j(y), C_j) contain linear combinations of the functions h_j, j = 1, …, s. They also contain the parameters C_j, j = 1, …, s. The summation limit m is an arbitrary positive integer number.

2. Taking into account Eq. (13), the first-order analytical approximate solution of Eqs. (8) - (9) is:

   F¯(y,Ci)=F(y,1)=F0(y)+F1(y,Ci).(21)

3. Finally, the convergence-control parameters C₁, C₂, …, C_s can be optimally computed by means of various methods, such as: the least square method, the Galerkin method, the collocation method, the Kantorowich method, or the weighted residual method.

With these parameters known, the first-order approximate solution (21) is well-determined.

4 Application of the OHAM to the chemically reactive solute transfer problem

Now we are going to apply our procedure to obtain approximate solutions of Eqs. (5) and (6) with the initial / boundary conditions Eq. (7).

For this purpose, in the case of the nonlinear equation Eq. (5), we choose the linear operator of the form:

where K is an unknown positive parameter and will be determined later.

As in Marinca and Herisanu [12], it is easy to show that the linear operator is not unique.

The initial approximation f₀(η) can be obtained from the following problem:

which has the solution

The nonlinear operator N_f(η), corresponding to nonlinear differential Eq. (5), is defined by:

For the initial approximation f₀(η) given by Eq. (24), the nonlinear operator Eq. (25) becomes:

Comparing Eqs. (26) and (18), one can write:

The function f₁(η) given by Eq. (19) becomes:

where we have freedom to choose a lot of possibilities for the unknown functions H_i, i = 1, 2, as follows (see Marinca and Herisanu [12]):

Substituting Eq. (29) into Eq. (28) we have:

where

The first-order approximate solution given by Eq. (21) is obtained from Eqs. (24) and (30):

In this way, we can find other solutions as well.

For Eq. (6) with initial/boundary condition given by Eq. (7) (for the unknown function φ), the expression for the linear operator L_φ(η) is chosen as:

where K₁ > 0 is an unknown parameter at this moment.

Eq. (16) can be written in the form:

and has the solution

The nonlinear operator N_φ corresponding to the unknown function φ is obtained from the expression Eq. (6) in the form:

For the initial approximation φ₀(η) given by Eq. (34), the nonlinear operator Eq. (35) becomes:

By comparing the Eqs. (18) and (36) one can get:

The first approximation φ₁(η, D_i), given by Eq. (19), becomes:

where D_i are unknown parameters, and the unknown auxiliary functions H1∗(η,Di),...,H5∗(η,Di) can be chosen in the form:

where M1=−∑i=17Di−∑i=911Di.

Substituting Eq. (39) into Eq. (38) one can get:

The first-order approximate solution given by Eq. (21) is obtained from Eqs. (34) and (40):

5 Results and discussions

In order to prove the accuracy of the obtained results, we will determine the convergence-control parameters K, C_i, K₁ and D_i, which appear in Eqs. (31) and (41), by the least square means method. In this way, the convergence-control parameters are optimally determined, and the first-order approximate solutions become known for different values of the known parameters S, λ, β and Sc. In what follows, we illustrate the accuracy of the OHAM, comparing previously obtained approximate solutions with the numerical integration results, computed by means of the shooting method combined with fourth-order Runge-Kutta method using Wolfram Mathematica 6.0 software. For some values of the parameters S, λ, β and Sc we are going to determine the approximate solutions.

Tables 2 - 4 show the comparisons between the OHAM approximate solutions (f̄_OHAM, f̄′_OHAM) given by Eqs. (42), (49), (50), and numerical integration, for λ = 0.1 and different values of the suction/blowing coefficient S. Table 5 provides a comparison between the OHAM approximate solutions φ̄_OHAM (concentration) given by Eqs. (43), (46), (47) and (48) for the flow with blowing, i.e. S = -0.5, and numerical results for λ = 0.1, β = 0.1 and different values of the Schmidt coefficient Sc.

On the other hand, in Tables 6 and 7 the comparisons between the skin-friction coefficient f″(0) and the limit value f(∞), respectively, are presented, obtained from Eqs. (42), (49) and (50) and numerical results, for λ = 0.1 and different values of the suction/blowing coefficient S.

A comparison between the mass transfer coefficient φ′(0) obtained from Eqs. (43), (46), (47) and (48) with numerical results, for the flow with blowing (S = -0.5), and for the flow with suction (S = 0.5), respectively, are presented in Tables 8 and 9, respectively.

If the analytical approximate solutions f̄ and φ̄ are given by the Eqs. (31) and (41) respectively, then the residuals from Eqs. (5) and (6) respectively are:

and

An error analysis by computing the integral of the square residuals given by Eqs. (44) and (45) is presented in Table 10.

The profiles of the stream function f̄(η) given by Eqs. (42), (49), (50) are depicted in Fig. 1 for λ = 0.1. We observe that the stream function f̄(η) increases with an increase of the suction/blowing coefficient S.

From Fig. 3 the velocity profiles f̄′(η) decreases with an increasing of the suction/blowing coefficient S.

Fig. 4 presents the variation of the shear-stress function f̄″(η) obtained from Eqs. (42), (48) and (50). It is noticed that the shear-stress function is increasing against an increase of the suction/blowing coefficient.

On the other hand, in Fig. 5 the profiles of the concentration function φ̄(η) for the flow with blowing (S = -0.5) and suction (S = 0.5) are plotted. It is noticed that the concentration function is decreasing with respect to an increase of the Schmidt coefficient Sc. For fixed value of the slip parameter λ and the reaction rate parameter β, from Fig. 5 we can notice that the concentration function φ̄(η) decreases against an increase of the suction/blowing coefficient S, for all the values of the Schmidt coefficient Sc.

The effect of the slip parameter λ on the chemically reactive solute flow is presented in Fig. 6 for stream function f̄(η), Fig. 7 for horizontal velocity f̄′(η) and Fig. 8 for shear-stress f̄″(η). Thus, for the flow with blowing / suction, the stream function f̄(η) and horizontal velocity f̄′(η) decrease with an increase of the slip parameter λ. Hence, we conclude that the shear-stress function f̄″(η) increases with an increase of the slip parameter λ, for all the values of the suction/blowing parameter S.

Also, from Fig. 9 we can notice that the variation of the concentration φ̄(η) decreases with the increasing of the reaction rate parameter β.

From the Tables 2 - 10 and Figs. 2 - 9 we can summarize that the approximate solutions obtained by means of the OHAM technique are effective and very accurate. This comparisons proved the accuracy, validity and flexibility of the Optimal Homotopy Asymptotic Method.

Figure 2

Variation of the stream function f̄(η) with increasing of the suction/blowing parameter S for λ = 0.1:

— OHAM solution; • • • • • • • • numerical solution

The obtained results are in agreement with the fluid flow scenario from a hydraulic installation for different values of the characteristic quantities: U₀ = 0.1 [m/s], C₀ = 0.1, ν = 46 ⋅ 10^-6 [m²/s] and C_∞ = 0.1, respectively.

Figure 3

Variation of the horizontal velocity f̄′(η) with increasing of the suction/blowing parameter S for λ = 0.1:

— OHAM solution; • • • • • • • • numerical solution

Figure 4

Variation of the shear stress f̄″(η) with increasing of the suction/blowing parameter S for λ = 0.1: — OHAM solution; • • • • • • • • numerical solution

Figure 5

Variation of the concentration φ(η) with increasing of the suction/blowing parameter S for λ = 0.1:

— OHAM solution ; • • • • • • • • numerical solution ; OHAM solution (with lines and dashing lines respectively)

Figure 6

Variation of the stream function f̄(η) with increasing of the slip parameter λ for different values of the suction/blowing parameter S: • • • • • • • • numerical solution ; OHAM solution (with lines and dashing lines respectively)

Figure 7

Variation of the horizontal velocity f̄′(η) with increasing of the slip parameter λ for different values of the suction/blowing parameter S: • • • • • • • • numerical solution ; OHAM solution (with lines and dashing lines respectively)

Figure 8

Variation of the shear stress f̄″(η) with increasing of the slip parameter λ for different values of the suction/blowing parameter S: • • • • • • • • numerical solution ; OHAM solution (with lines and dashing lines respectively)

Figure 9

Variation of the concentration φ(η) given by Eqs. (53), (54) and (55) with increasing of the reaction rate parameter β for S = - 0.5, λ = 0.1 and Sc = 0.7: — OHAM solution; • • • • • • • • numerical solution

Remarks

1. From Fluid Mechanics [30] it is known that the stream function decreases with respect to the increase of the slip parameter λ (this signifies the loss of energy). Following most studies in this area, there exists a correlation between the loss coefficient λ (corresponding to the slip parameter), slowing of the velocity f̄′ and the stream function f̄, with the increases of the energy losses. The obtained data in the present paper correspond to the real situation from hydraulic machines (turbines, pumps, naval propellers), where the efficiency energy of the machinery is affected by the level of turbulence.

2. In the case of the hydraulic oil with a large application at the hydraulic drive systems, the analytical results obtained in our paper (using the specific physical sizes), are presented in Figs. 10 - 13.

Figure 10

Variation of the velocity component u from Eq. (4) for λ = 0.1, S = - 0.3, for hydraulic oil at a temperature of 40° C.

Figure 11

Variation of the velocity component v from Eq. (4) for λ = 0.1, S = - 0.3, for hydraulic oil at a temperature of 40° C.

Figure 12

The vector field (u, v) from Eq. (4) for λ = 0.1, S = - 0.3, for hydraulic oil at a temperature of 40° C.

Figure 13

The concentration C from Eq. (4) for λ = 0.1, S = - 0.3, Sc = 2, β = 0.1 for hydraulic oil at a temperature of 40° C.

6 Conclusions

In aerospace, hydraulic processes occur very often, with strongly nonlinear behaviors and even situations with singularities. Therefore, a numerical solution can capture all such situations and an approximate analytical solution is a more realistic option. The OHAM method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions.

The analytical treatment related to the chemically reactive solute transfer problem with partial slip in the flow of a viscous fluid over an exponentially stretching sheet with suction/blowing is presented. The governing nonlinear partial differential equations (for the mass transfer and concentration) are reduced to nonlinear ordinary differential equations using some similarity transformations. The obtained nonlinear ordinary differential equations are analytically solved using the OHAM method. Some numerical examples are given for different values of the suction/blowing coefficient S. In the case of the flow with blowing (S < 0) and suction (S > 0), the effects of the Schmidt parameter Sc on the concentration φ(η) are studied. The quality of the approximate solutions are made by means of the two important statistical tests: the Bartlett test and the Durbin-Wattson test. Also, the Akaike Informational Criterion (AIC) is used to make an optimal expression of the analytical solution (from our best knowledge, this tool was not used for this reason) and this simplifies the form of the solution, being easiest in physical applications.

The obtained analytical results are compared with the corresponding numerical results obtained using the fourth-order explicit Runge-Kutta method in comparison with the shooting method. The validity, flexibility, accuracy and convergence of the approximate solutions are demonstrated by means of the auxiliary functions H_k(η, C_i) and Hk∗(η,Dj), which involves the presence of the some convergence-control parameters C_i and D_j that are optimally identified by means of the least-squares method using the Wolfram Mathematica software 6.0.

These comparisons proved that the OHAM technique is effective and practical.