Solution of the Boundary Layer Equation of the Power-Law Pseudoplastic Fluid Using Differential Transform Method

Abstract

The boundary layer equation of the pseudoplastic fluid over a flat plate is considered. This equation is a boundary value problem (BVP) with the high nonlinearity and a boundary condition at infinity. To solve such problems, powerful numerical techniques are usually used. Here, through using differential transform method (DTM), the BVP is replaced by two initial value problems (IVP) and then multi-step differential transform method (MDTM) is applied to solve them. The differential equation and its boundary conditions are transformed to a set of algebraic equations, and the Taylor series of solution is calculated in every sub domain. In this solution, there is no need for restrictive assumptions or linearization. Finally, DTM results are compared with the numerical solution of the problem, and a good accuracy of the proposed method is observed.

1. Introduction

Boundary layer equation of the Newtonian fluid over a flat plate is one of the classic issues of mechanical engineering. Because of high level of application of non-Newtonian fluids in industry, the boundary layer equation of the non-Newtonian fluids is being noted by engineers. On the other hand, nonlinearity of these equations attracted the interest of mathematicians to evaluate the power and accuracy of the approximate and numerical methods. One of the most significant non Newtonian models is the power law one. In this model, shear thinning or shear thickening property of the fluid is considered and the shear stress has a nonlinear relation with shear rate, while shear stress has a linear relation with shear rate for the Newtonian fluid.

As we pointed out above, boundary layer problems are among the nonlinear ones and most of them do not have an exact analytical solution. So, numerical and approximate methods are used by researchers to solve such equations. The most known numerical method used to solve boundary layer problems is the shooting one. Based on this method, boundary value problem is transformed to an initial value problem with unknown initial values. After that, the problem is replaced with a system of first-order ordinary differential equations and usually is solved through Runge-Kutta method. Shooting method is appropriate to solve many boundary value problems, but this one is not useful for solving some BVPs because of the instability of the solution.

Approximate techniques like decomposition method (DM), homotopy analysis method (HAM), homotopy perturbation method (HPM), and variational iteration method (VIM) are good substitutes for numerical methods. During the recent years, boundary layer problems have been solved using some of these methods, such as HAM [1–9], HPM [10–14], VIM [15–17], and DM [18–21]. In most of the researches, some modifications were introduced to overcome the nonlinearity and the boundary condition at infinity.

Differential transform method (DTM) is also one of the other approximate methods to solve differential equations. Here, DTM and multi-step DTM are used to solve the boundary layer equation of the pseudoplastic fluid. This method was introduced by Zhou [22] to solve initial value problems in analysis of the electrical circuits. After that, DTM is applied to differential algebraic equation [23, 24], partial differential equation [25–30], integral equation [31–33], ordinary differential equation [34–38], and fractional differential equation [39–42]. The method is an iterative technique to find the Taylor series solution of the problem. There is no need for the high calculation cost to determine the coefficients of Taylor series, which is the reverse of the standard Taylor series method.

This paper is organized as follows. In Section 2, DTM and multi-step DTM are introduced. In Section 3, the boundary layer equation of the pseudoplastic fluid is extracted from continuity and momentum equations as presented in [43]. In Section 4, the problem is solved and the results are illustrated as some figures and tables.

2. Differential Transform Method

The differential transform is defined as follows: 𝑋(𝑘)=1𝑑𝑘!^𝑘𝑥(𝑡)𝑑𝑡^𝑘_𝑡=𝑡0, (1) where𝑥(𝑡)is an arbitrary function and𝑋(𝑘)is the transformed function. The inverse transformation is as follows: 𝑥(𝑡)=^∞_𝑘=0𝑋(𝑘)𝑡−𝑡₀^𝑘. (2) Substituting (1) into (2), we have 𝑥(𝑡)=^∞_𝑘=0𝑡−𝑡₀^𝑘𝑑𝑘!^𝑘𝑥(𝑡)𝑑𝑡^𝑘_𝑡=𝑡0. (3) The function𝑥(𝑡)is usually considered as a series with limited terms and (2) can be rewritten as 𝑥(𝑡)≈^𝑚_𝑘=0𝑋(𝑘)𝑡−𝑡₀^𝑘, (4) where𝑚represents the number of Taylor series’ components. Usually, through elevating this value, we can increase the accuracy of the solution.

Although the DTM series solution is a good approximate of the exact solution, but this series is diverged for greater areas. There are two ways to overcome this problem. One is the using pade approximate technique. Usually, Pade approximate gives us more exact information about the behavior of the solution (see [44]). Another one is to use multi-step DTM. Based on this one, solution domain is divided to some sub domains.

To solve a differential equation in the domain[0,𝑇]using multi-step DTM, this domain is divided to𝑁sections. We suppose the sub domains are equal and length of each sub domain is𝐻=𝑇/𝑁. So, there is a separate function for every sub domain as follows: 𝑥(𝑡)=𝑥₁(𝑡),𝑡∈𝑡₁,𝑡₂⋮𝑥_𝑖(𝑡),𝑡∈𝑡_𝑖,𝑡_𝑖+1⋮𝑥_𝑁(𝑡),𝑡∈𝑡_𝑁,𝑡_𝑁+1, (5) where𝑡_𝑖=(𝑖−1)𝐻. Multi-step DTM for every sub domain is defined as 𝑋_𝑖(𝑘)=𝐻^𝑘𝑑𝑘!^𝑘𝑥_𝑖(𝑡)𝑑𝑡^𝑘_{𝑡=𝑡𝑖}. (6) The inverse multi-step DTM is 𝑥_𝑖(𝑡)=^∞_𝑘=0𝑋_𝑖(𝑘)𝑡−𝑡_𝑖𝐻^𝑘. (7) Some of the properties of DTM and multi-step DTM are shown in Table 1. These properties are extracted from (1) and (6).

Table 1 

The properties of the DTM and multi-step DTM.

3. Mathematical Formulation

Two-dimensional boundary layer equations for an incompressible fluid are 𝑈𝜕𝑈𝜕𝑋+𝑉𝜕𝑈=1𝜕𝑌𝜌𝜕𝜏𝜕𝑌𝜕𝑈+𝜕𝑋𝜕𝑉𝜕𝑌=0, (8)(9) where𝜌is the fluid density,𝜏is the shear stress, and𝑈and𝑉are the velocities in direction of𝑥and𝑦, respectively. For the non-Newtonian power-law fluid, the shear stress is calculated through the following relation: 𝜏=𝑚𝜕𝑈𝜕𝑌^𝑛−1𝜕𝑈,𝜕𝑌 (10) where𝑚and𝑛are the flow consistency and power-law index, respectively.

Here, the following dimensionless parameters are used: Re=𝜌𝑈_∞^2−𝑛𝐿^𝑛𝑚𝑋,𝑥=𝐿𝑌,𝑦=𝐿Re^1/(𝑛+1),𝑈𝑢=𝑈_∞𝑉,𝑣=𝑈_∞Re^1/(𝑛+1), (11) where𝐿is the plate length,𝑈_∞is the far field velocity, andReis generalized Reynolds number.

Combining (8), (10), and (11), the dimensionless momentum equation can be obtained: 𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢=𝜕𝜕𝑦𝜕𝑦𝜕𝑢𝜕𝑦^𝑛−1𝜕𝑢𝜕𝑦 (12) with the boundary conditions 𝑢(𝑥,0)=𝑣(𝑥,0)=0,𝑢(𝑥,∞)=1,𝑢0,𝑦=1. (13) Similarity variable and stream function for the problem is defined as [45] 𝑦𝜂=(𝑛(𝑛+1)𝑥)^1/(𝑛+1)Ψ=(𝑛(𝑛+1)𝑥)^1/(𝑛+1)𝑓𝜂. (14) Regarding (14), the velocities can be obtained as a function of the similarity variable: 𝑢=𝜕Ψ𝜕𝑦=𝑓^′𝜂𝑣=−𝜕Ψ=1𝜕𝑥𝑛()𝑛+1𝑛+1𝑥^{𝑛1/(𝑛+1)}𝜂𝑓^′𝜂−𝑓𝜂. (15) Substituting (15) in (12), we have 𝑓^{′′𝑛−1}𝑓^′′′+𝑛𝑓𝑓^′′=0. (16) Equation (16) is the boundary layer equation of the power-law fluid as a function of index𝑛. For pseudoplastic fluid (𝑛<1) the value of the𝑓^′′(𝜂)is more than zero [43, 45–48]. So, (16) can be simplified as 𝑓^′′′𝜂+𝑓𝜂𝑓^′′𝜂^2−𝑛𝑓(0)=0for𝑛≤1,=0,𝑓^′(0)=0,𝑓^′(∞)=1. (17)

4. Solution of the Problem

In this section, we try to solve (17) using DTM and multi-step DTM. The solution consists of two stages; first through mathematical relations and applying DTM, the value of𝑓^′′(0)is found. After that, boundary layer equation of the pseudoplastic fluid is solved as an initial value problem (IVP) using multi-step DTM.

4.1. Applying DTM (Obtaining𝑓^′′(0))

The BVP (17) can be transformed to an initial value problem with the replacement of the unknown initial condition𝑓^′′(0)=𝜆instead of the boundary condition at infinity𝑓^′(∞)=1: 𝑓^′′′𝜂𝜂𝑠𝜂𝑓(0)+𝑓=0for𝑛≤1=0,𝑓^′(0)=0,𝑓^′′(0)=𝜆, (18) where 𝑠𝜂=𝑓^′′𝜂^2−𝑛. (19) By applying the DTM on (18) at𝜂=0, the recursive relation is derived: 𝐹(𝑘+3)=−1(𝑘+1)(𝑘+2)(𝑘+3)^𝑘_𝑟=0𝐹(𝑟)𝑆()𝐹(0)(1)(2)=𝜆𝑘−𝑟=0,𝐹=0,𝐹2. (20) Regarding (19), 𝑠^′𝜂𝑓^′′𝜂=(2−𝑛)𝑓^′′′𝜂𝑠𝜂. (21) Taking differential transform on (21), we have ^𝑘_𝑟=0()𝑆()𝐹()=()𝑟+1𝑟+1)(𝑘−𝑟+1)(𝑘−𝑟+2𝑘−𝑟+22−𝑛^𝑘_𝑟=0(𝑟+1)(𝑟+2)(𝑟+3)𝐹(𝑟+3)𝑆(𝑘−𝑟). (22) From (19), the value of𝑆(0)can be obtained: 𝑆(0)=𝑠(0)=𝑓^′′(0)^2−𝑛=𝜆^2−𝑛. (23) Combining (20), (22), and (23), the coefficients of the Taylor series solution of (18) can be calculated: 𝐹(0)(1)(2)=𝜆=0,𝐹=0,𝐹2,𝐹(3)=0,𝐹(4)=0,𝐹(5)=−1𝜆120^3−𝑛,𝐹(6)(7)𝐹(8)=1=0,𝐹=0,𝜆1920^5−2𝑛−1𝜆4032^5−2𝑛𝐹(9)()𝐹()=𝑛,=0,𝐹10=0,11−89𝜆1900800^7−3𝑛+1027𝜆19958400^7−3𝑛1𝑛−𝜆71280^7−3𝑛𝑛²,⋮ (24) Now, consider a similar equation (18) in which𝜆=1: 𝑔^′′′𝜂+𝑔𝜂𝑔^′′𝜂^2−𝑛𝑔(0)=0for𝑛≤1=0,𝑔^′(0)=0,𝑔^′′(0)=1. (25) In the same manner, the coefficients𝐺(𝑘)are 𝐺(0)(1)(2)=1=0,𝐺=0,𝐺2,𝐺(3)=0,𝐺(4)=0,𝐺(5)=−1,𝐺(6)(7)(8)=1120=0,𝐺=0,𝐺−11920𝐺(9)()𝐺()=4032𝑛,=0,𝐺10=0,11−89+19008001027119958400𝑛−𝑛71280²,⋮ (26) Substituting (24) and (26) in (2), the𝑓(𝜂)and𝑔(𝜂)are obtained. These functions have a relation as follows: 𝑓𝜂=𝜆^𝑞𝑔𝜆^𝑝𝜂,𝑝=2−𝑛3,𝑞=2𝑛−13. (27) For𝑛=1, both values of𝑝and𝑞are 1/3 for the Blasius equation. From (27), we have 𝑓^′𝜂=𝜆^𝑞+𝑝𝑔^′𝜆^𝑝𝜂⟶𝑓^′(∞)=𝜆^𝑞+𝑝𝑔^′(∞). (28) Regarding (28) and the boundary condition at infinity in (17), the value of𝜆can be obtained: 1𝜆=𝑔^′(∞)^3/(1+𝑛). (29)

4.2. Applying Multi-Step DTM (Solution of IVPs)

As was noted, multi-step DTM usually is used for solving problems in which Taylor series is diverged in the solution domain. This technique was already used to solve engineering and computational problems [49–57].

To solve boundary layer problems, the domain[0,∞)is replaced by[0,𝜂_∞). But𝜂_∞should be great enough that the solution is not to be dependent on. The solution domain should be divided to𝑁equal parts (𝐻=𝜂_∞/𝑁). So, we have 𝑔_𝑖^′′′𝜂_𝑖+𝑔_𝑖𝜂_𝑖𝑠_𝑖𝜂_𝑖()=0,𝑖−1𝐻≤𝜂_𝑖<𝑖𝐻,for1≤𝑖≤𝑁, (30) where 𝑠_𝑖𝜂=𝑔_𝑖^′′𝜂^2−𝑛. (31) Applying multi-step DTM on (30) and (31), we have 𝐺_𝑖(𝑘+3)=−𝐻³(𝑘+1)(𝑘+2)(𝑘+3)^𝑘_𝑟=0𝐺_𝑖(𝑟)𝑆_𝑖(𝑘−𝑟),for1≤𝑖≤𝑁_𝑘𝑟=0(𝑟+1)𝐻𝑆_𝑖()()𝑟+1𝑘−𝑟+1)(𝑘−𝑟+2𝐻²𝐺_𝑖()=()𝑘−𝑟+22−𝑛^𝑘_𝑟=0(𝑟+1)(𝑟+2)(𝑟+3)𝐻³𝐺_𝑖(𝑟+3)𝑆_𝑖(𝑘−𝑟),for1≤𝑖≤𝑁. (32) So 𝑆_𝑖(1)=2−𝑛𝐺_𝑖(2)3𝐺_𝑖(3)𝑆_𝑖(0). (33) And for𝑘≥1𝑆_𝑖()=1𝑘+12(𝑘+1)𝐺_𝑖(2)−^𝑘−1_𝑟=0(𝑟+1)𝑆_𝑖()×()𝐺𝑟+1)(𝑘−𝑟+1𝑘−𝑟+2_𝑖()+()𝑘−𝑟+22−𝑛^𝑘_𝑟=0(𝑟+1)(𝑟+2)(𝑟+3)𝐺_𝑖×(𝑟+3)𝑆_𝑖(𝑘−𝑟). (34) The initial conditions of the problem are considered for first sub domain (𝑖=1). Regarding (6), (25), and (31), initial conditions transformed as follows: 𝐺₁(0)=𝑔₁(0)𝐺=0₁(1)=𝐻𝑔^′₁(0)𝐺=0₁(2)=𝐻²2𝑔₁^′′(0)=𝐻²2𝑆₁(0)=𝑠₁(0)=𝑔₁^′′(0)^2−𝑛=1. (35) The boundary conditions of each sub domain are the continuity of the𝑔(𝜂_𝑖),𝑔^′(𝜂_𝑖), and𝑔^′′(𝜂_𝑖). These boundary conditions can be obtained from (7):𝑔_𝑖𝜂_𝑖+1=^𝑚_𝑘=0𝐺_𝑖(𝑘)𝑔_𝑖+1𝜂_𝑖+1=𝐺_𝑖+1(0)⟶𝐺_𝑖+1(0)=^𝑚_𝑘=0𝐺_𝑖(𝑘)𝑔^′_𝑖𝜂_𝑖+1=^𝑚_𝑘=1𝑘𝐻𝐺_𝑖(𝑘)𝑔^′_𝑖+1𝜂_𝑖+1=𝐺_𝑖+1(1)𝐻⟶𝐺_𝑖+1(1)=^𝑚_𝑘=1𝑘𝐺_𝑖(𝑘)𝑔_𝑖^′′𝜂_𝑖+1=^𝑚_𝑘=2𝑘(𝑘−1)𝐻²𝐺_𝑖(𝑘)𝑔^′′_𝑖+1𝜂_𝑖+1=2𝐻²𝐺_𝑖+1(2)⟶𝐺_𝑖+1(2)=12^𝑚_𝑘=2𝑘(𝑘−1)𝐺_𝑖(𝑘). (36)(37) And we have for𝑆𝑆_𝑖+1(0)=𝑠_𝑖+1𝜂_𝑖+1=𝑔^′′_𝑖+1𝜂_𝑖+1^2−𝑛=^𝑚_𝑘=2𝑘(𝑘−1)𝐻²𝐺_𝑖(𝑘)^2−𝑛. (38) The value of the𝑔^′(𝜂_∞)can be calculated by differentiating from (7): 𝑔^′(∞)≃𝑔^′𝜂_∞=𝑔^′_𝑁𝜂_𝑁+1=^𝑚_𝑘=1𝑘𝐻𝐺_𝑁(𝑘). (39) The unknown parameter (𝜆) is calculated from (29). Now, (18) is solved with a similar process like (25) using multi-step DTM. The only difference is that the condition𝑓^′′(0)=𝜆is replaced by the condition𝑔^′′(0)=1.

In Figure 1  𝑓^′(𝜂)is illustrated for different values of𝑛and compared with the numerical solution and results of Lemieux et al. [43]. The numerical solution of the problem is done by the MATLAB software. Figures 2 and 3 show the variation of𝑓(𝜂)and𝑓^′′(𝜂), respectively. The most important step of this scheme is to choose the appreciate finite value of𝜂_∞. Thus to estimate this value, we start with an initial guess for𝜂_∞and solve the problem. The solution process is repeated with another larger value of𝜂_∞until the values of𝑓^′′(0)and𝑔^′′(0)differ regarding the desired accuracy.

Figure 1 

Variation of𝑓′(𝜂)and comparison with the numerical solution and results of Lemieux et al. [43].

Figure 2 

Variation of𝑓(𝜂)for different values of𝑛.

Figure 3 

Variation of𝑓′′(𝜂)for different values of𝑛.

In Table 2 the values of the𝑓^′′(0)=𝜆are compared with the numerical solution and results of Lemieux et al. [43]. The approximate solution of the problem is presented in Table 3 for a special case (𝑛=0.7).

Table 2 

Comparison of the DTM results, numerical solution, and results of Lemieux et al. [43].

Table 3 

The solution of 𝑓(𝜂) for 𝑛=0.7 when 𝜂_∞=10 and 𝑁=10.

5. Conclusion

In this paper, the boundary layer equation of the pseudoplastic fluid over a flat plate was extracted from the boundary layer theory as presented in [43]. This equation is a boundary value problem with high nonlinearity and boundary condition at infinity. Using DTM, the BVP transformed to a pair of initial value problems and multi-step DTM is used to solve them.

Using the method, the differential equation and boundary conditions are transformed into a recurrence set of equations. Finally, the coefficients of power series are obtained based on the solution of this set of equations. The results validated with the numerical solution, and a good accuracy is observed. The proposed method overcame nonlinearity without using linearization or restrictive assumptions.