* Corresponding author: Department of Civil Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh E-mail addresses: [email protected] (A. S. M. Z. Hasan)
22
Homotopy Analysis of Two-Dimensional Laminar Jet A. S. M. Z. Hasan*, B. Ahmed and M. M. Rana
Department of Civil Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh
ARTICLE INFORMATION ABSTRACT
Received date: 22 Jan 2019 Revised date: 25 April 2019 Accepted date: 08 May 2019
This paper investigates an explicit solution of nonlinear boundary value problems of two-dimensional laminar jet. The powerful analytical method for nonlinear differential equations so-called homotopy analysis method (HAM) is applied to obtain multiple solutions in this study. A novel technique is to introduce for obtaining an unfamiliar new branches solution namely 2nd solution by means of orthogonalization of the initial approximation guess, taking same auxiliary linear operator and same auxiliary function of 1st solution. The obtained results of the HAM agree well with that of the existing closed-form solution both for 1st and 2nd form of solutions and the accurate homotopy-series converge with the different values of f′′(0) = - 2.4 × 10-14 and f′′(0) = - 8.0 × 10-11 indicating the multiform of same solution. The velocity profiles obtained from HAM also satisfy with the existing closed-form solution both for 1st and 2nd solutions and explore the physical application of the two- dimensional laminar jet.
Keywords
Boundary value problems HAM
Two-dimensional laminar jet Multiple solutions
1. Introduction
Most of the mathematical model of the physical phenomena in science and engineering such as in fluid dynamics, solid state physics, advanced mechanics and quantum mechanics etc. often formulated by nonlinear differential equations of boundary value problems. Several methods, for example variational iteration method (VIM) (Barari et al. 2008, Abbasbandy et al. 2009), iterative solutions (Ma and Silva 2004), He’s homotopy method (Beléndez et al.
2009), homotopy perturbation method (Jin 2009, Nayfeh 2000) and traditional methods such as Euler method, Runge- Kutta method, finite difference method etc. widely applied to obtain either analytical or numerical solutions. Among them, variational iteration method is the most suitable for obtaining closed form of solution. Though it is capable for boundary value problems, it is difficult to apply boundary value problems, which is extended to infinite limit. The numerical methods require large time to provide results due to it needs to solve the large size integration although it is available ultrafast digital computers. Moreover, it needs to define initial guess properly for obtaining real results. In addition, perturbation methods is suitable for solving equations with those have the weak nonlinearity. Unfortunately, these methods are unable to predict multiplicity of solutions of the nonlinear differential equations in an infinite interval of boundary value problems.
Journal of Engineering and Applied Science
Contents are available at www.jeas-ruet.ac.bd
23 Recently, it is the fundamental interest of new unfamiliar class of solutions so-called multiple solutions for all researchers in science and engineering for practical application of nonlinear boundary value problems. The powerful analytical method for solving nonlinear differential equations namely homotopy analysis method (HAM) was proposed by Liao (1992), which provides us a convenient way to guarantee the convergence of series solution of nonlinear problems by introducing an auxiliary parameter especially so-called convergence-control parameter c0. In the last two decades, this method is successfully applied to several nonlinear problems such as fractional partial differential equations (Mohyud-Din et al. 2011), viscous incompressible flow (Sajid 2009), similarity boundary equations (Liao and Pop 2004, Liao and Magyari 2006). In addition, this method is applied to obtain new branches of solutions of nonlinear problems (Liao and Tan 2007, Liao 2010, 2007, Shuicai and Liao 2005, Xu et al. 2008, 2010).
Therefore, this method is successfully applied to solve the nonlinear boundary value problems of two-dimensional laminar jet in this paper.
Nonlinear differential equations of boundary value problems obviously have multiple solutions (Graef et al. 2004, Karakostas and Tsamatos 2002). In the frame of HAM, Liao 2012 presented multiple solutions by introducing an unknown parameter in initial guess, called the multiple-solution-control parameter. Moreover, Abbasbandy et al.
(2009, 2010, 2011) predicted multiple solutions of some nonlinear differential equations by using the different values of convergence-control c0 with the same auxiliary linear operator ℒ, even the same initial guess and same auxiliary function. The main objective of this paper is to introduce a new unfamiliar class of solution. Hence, a novel technique is employed to orthogonalize the initial guess to obtain for new branches of solutions (2nd solution) of nonlinear boundary value problems and presented herein.
2. Basic Mathematical Equations
Assume that the velocity of u and v are the horizontal and vertical component in the direction of x and y coordinates of a two-dimensional laminar jet and ν is the kinematic viscosity. Let us consider the boundary-layer equations of this jet is described by the partial differential equations (PDEs):
𝑢𝜕𝑢
𝜕𝑥+ 𝑣𝜕𝑢
𝜕𝑦= 𝜈𝜕2𝑢
𝜕𝑦2, (1)
𝜕𝑢
𝜕𝑥+𝜕𝑣
𝜕𝑦= 0, (2) Subjected to boundary conditions
𝑢 → 0 𝑎𝑡 𝑦 → ±∞, (3) introducing the stream function ψ related to the velocities u and v respectively, the x momentum equation (1) assume in the form
𝜕𝜓
𝜕𝑦
𝜕2𝜓
𝜕𝑦𝑑𝑥−𝜕𝜓
𝜕𝑥
𝜕2𝜓
𝜕𝑦2 = 𝜈𝜕3𝜓
𝜕𝑦3, (4) with the boundary conditions
𝜓 → 0 𝑎𝑡 𝑦 → ±∞, (5) and
𝜌 ∫ (−∞∞ 𝜕𝜓𝜕𝑦)2𝑑𝑦 = 𝑀, (6) whereM and 𝜌 represent the mass and density of laminar jet.
Using the similarity transformation and introducing the base function 𝜂 which is defined in equation (7) 𝜓 = (𝑀𝜈𝑥
𝜌 )
1
3𝑓(𝜂), 𝜂 = ( 𝑀
𝜌𝜈2𝑥2)
1
3 , (7)
then
𝑢 = 𝜕𝜓
𝜕𝑦= (𝑀2
𝜌𝜈2𝑥)
1
3𝑓′(𝜂), (8)
𝑣 = −𝜕𝜓
𝜕𝑥=1
3(𝑀𝜈
𝜌𝑥2)
1
3(2𝜂𝑓′(𝜂) − 𝑓(𝜂)), (9)
24 the original PDEs become the following nonlinear ODE
3𝑓′′′(𝜂) + (𝑓′ (𝜂))2+ 𝑓(𝜂)𝑓′′(𝜂) = 0, (10) the boundary conditions become
𝑓′(±∞) = 0, 𝑓(0) = 𝑓′′(0) = 0, symmetry (11) and
1 = ∫ [𝑓−∞∞ ′(𝜂)]2𝑑𝜂, (12) For details, please refer (Mei 2002). The above equations have the closed-form solution in the following form
𝑓(𝜂) = √2𝑐tanh (3√2𝑐𝜂), (13) Where 𝑐3= 9
4√2.
3. Homotopy Analysis Method
Liao (2012), Liao and Pop (2004) showed that physically, most of boundary-layer flows exponentially tend a uniform flow at infinity and mathematically, this is confirmed by the closed-form solutions that can be gained by means of the homotopy analysis method. In the present study, equations (10) and (11) are solved by using BVPh 1.1 Liao (2012).
A nonlinear operator can be defined as
𝒩[𝑓(𝜂; 𝑝)] = 3𝜕3𝑓̅(𝜂;𝑝)
𝜕𝜂3 + (𝜕𝑓̅(𝜂;𝑝)
𝜕𝜂 )2+ 𝜕𝑓̅(𝜂;𝑝)
𝜕𝜂
𝜕2𝑓̅(𝜂;𝑝)
𝜕𝜂2 , (14) We construct the so-called zeroth-order deformation problem
(1 − 𝑝)ℒ[𝑓(𝜂; 𝑝)] − 𝑓0(𝜂)] = 𝑝𝑐0𝒩[𝑓(𝜂; 𝑝)], (15)
𝑓(0; 𝑝) = 𝑓′′(0; 𝑝) = 0, 𝑓′(±∞; 𝑝) = 0, (16) where the prime denotes the differentiation with respect to 𝜂, f0(𝜂) is an initial guess, ℒ is an auxiliary operator, and c0 is the convergence-control parameter, respectively. The above equation defines a kind of continuous variation 𝑓̅(𝜂;
p) from initial guess f0(𝜂) (at p = 0) to the solution f(𝜂) of the original equation (10) (at p =1). The homotopy-Maclaurin series of 𝑓̅(𝜂; p) expansion
𝑓̅(𝜂; 𝑝) = 𝑓0(𝜂) + ∑±∞𝑚=1𝑓𝑚(𝜂)𝑝𝑚 , (17) where
𝑓𝑚(𝜂) = 1
𝑚!
𝜕𝑚𝑓(𝜂;𝑝)
𝜕𝑝𝑚 ⎹𝑝=0, (18) If the initial guessf0(𝜂), the auxiliary operator ℒ, and the convergence-control parameter c0 are properly chosen so that the above homotopy-Maclaurin series absolutely converges at p = 1, we have the homotopy series solution
𝑓(𝜂) = 𝑓0(𝜂) + ∑±∞𝑚=1𝑓𝑚(𝜂), (19) wherefm(𝜂) is governed by the mth-oder deformation equation
ℒ[𝑓𝑚(𝜂) − 𝜒𝑚𝑓𝑚−1(𝜂)] = 𝑐0𝛿𝑚−1(𝜂), (20) where
𝜒𝑚= {0, 𝑚 ≤ 1, 1, 𝑚 > 1, and
𝛿𝑚(𝜂) = 3𝑓𝑚′′′(𝜂) + ∑ 𝑓𝑚−𝑖′ (𝜂)
𝑚
𝑖=0
𝑓𝑖′(𝜂) + ∑ 𝑓𝑚−𝑖′ (𝜂)
𝑚
𝑖=0
𝑓𝑖′′(𝜂)
For details please refer to Liao 2009a. The exponentially decaying solution of (10) can be expressed in the form 𝑓(𝜂) = ∑±∞𝑚=0𝐴𝑚(𝜂)exp(−𝑚𝜂) (21)
25 whereAm(η) is a polynomial to be determined. Considering one unknown parameter 𝜎 =f0(±∞) which provide us additionally one degree of freedom and satisfying with the boundary condition (11), we choose the following initial guess
𝑓0(𝜂) = 𝜎 +43𝜎 exp(−𝜂) −1
3𝜎 exp(−2𝜂), (22) and the auxiliary linear operator
ℒ[𝑓(𝜂)] = 𝑓′′′(𝜂) − 𝑓′(𝜂), (23) The auxiliary linear operator (23) has the property
ℒ[𝐴0+ 𝐴1exp(−𝜂) + 𝐴2exp(+𝜂)] = 0, (24) A0-A2 are the constants, let for any given function 𝑓∗(𝜂), the special solution can be denoted as
𝑓𝑚∗(𝜂) = 𝜒𝑚𝑓𝑚−1(𝜂) + 𝑐0ℒ−1[𝛿𝑚−1(𝜂)], Therefore, the general solution as
𝑓𝑚(𝜂) = 𝜒𝑚𝑓𝑚−1(𝜂) + 𝑐0ℒ−1[𝛿𝑚−1(𝜂)] + 𝐴0+ 𝐴1exp(−𝜂) + 𝐴2exp (+𝜂). (25)
4. Results and Discussion 4.1 1st Solution 1st form
The homotopy-approximation is dependent upon two unknown parameters: 𝜎 is the initial guess and c0 is the convergence-control parameter, respectively. The minimum squared residual is ensured by trailing with the different values of σ and the range for the admissible c0-values is −1 ≤ 𝑐0< 0. After that, the curves of average squared residual Em versus c0 are shown in Figure 1 with the optimal value of σ = 3/2.
Figure 1: Squared residual vs convergence-control parameter c0.
Table 1: The average squared residual at mth-order homotopy approximation when c0 = -1/2 and σ =3/2 Order of approximation, m f(0) Squared residual, Em
2 0 6.0 × 10-3
4 -3.6 × 10-14 7.4 × 10-4
6 -2.4 × 10-14 2.2 × 10-4
8 -2.4 × 10-14 9.6 × 10-5
10 -2.4 × 10-14 4.9 × 10-5
16 -2.4 × 10-14 9.6 × 10-6
20 -2.4 × 10-14 3.6 × 10-6
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Squared residual, Em
c0
7th-order Homotopy approximation Similarity Law (closed form)
26 Figure 2: Comparison of homotopy approximation results of f(𝜂) with existing closed-form solution.
The summarized results in Table 1 clearly indicate that the accurate homotopy-series solution whose 𝑓′′(0) converges to - 2.4 × 10-14. Thus, 𝑓′′(0) quickly converges to the closed-form solution as shown in Figure 2, and the 7th-order homotopy-approximation of f(𝜂) agrees well.
It indicates that 3rd-order homotopy-approximation, the optimal convergence-control parameter c0 is about -1/2 with the optimal value of σ = 3/2. Therefore, corresponding to the optimal convergence-control parameter c0 =-1/2 and initial guess σ = 3/2, the averaged squared residual of (10) over the interval 𝜂 ∈ [0,3] decreases from 6.3 × 10-3 to 9.6
× 10-5 regarding the order of homotopy-approximation from 2 to 8 as shown in Table1.
Similarly, the 7th-order homotopy-approximation confirms well with the closed-form solution over the interval 𝜂 ∈ [0,12] of 𝑓′(𝜂) as shown in Figure 3.
Figure 3: Comparison of homotopy approximation results of 𝑓′(𝜂) with existing closed-form solution 4.2 1st Solution 2nd form
We consider λ > 0 denote a kind spatial scale-parameter and by means of transformation 𝑓(𝜂) = 𝜆−1𝑢(𝜉), 𝜉 = 𝜆𝜂,
The original equation (10) becomes
3𝑢′′′(𝜉) + 1
𝜆2[(𝑢′ (𝜉))2+ 𝑢(𝜉)𝑢′′(𝜉)] = 0, (26) boundary conditions become
𝑢′(±∞) = 0, 𝑢(0) = 𝑢′′(0) = 0, (27) where the prime denotes the derivative with respect to ξ.
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
f(ƞ)
ƞ
7th-order Homotopy approximation Similarity Law (closed form)
0.0 0.1 0.2 0.3 0.4 0.5
0 2 4 6 8 10 12
𝑓′(ƞ)
ƞ
7th-order Homotopy approximation Similarity Law (closed form)
27 We obtained the homotopy-series converges corresponding to the optimal convergence-control parameter c0
= -1/2 and initial guess σ = 3/2 with the value λ = 2, initial guess (22) and the linear auxiliary operator (23), respectively. The averaged squared residual of equation (26) with the boundary conditions over the interval 𝜂 ∈ [0,2]
decreases from 1.3 × 10-3 to 4.9 × 10-5 regarding the order of homotopy-approximation from 4 to 14 as shown in Table 2.
Table 2: The average squared residual at mth-order homotopy approximation when c0 = -1/2, σ =3/2 and λ=2 for 1st solution of 2nd form
Order of approximation, m f(0) Squared residual, Em
4 0 1.3 × 10-3
6 7.5 × 10-11 4.4 × 10-4
8 1.4 × 10-11 2.1 × 10-4
10 -8.0 × 10-11 1.2 × 10-4
12 -8.0 × 10-11 7.4 × 10-5
14 -8.0 × 10-11 4.9 × 10-5
16 -8.0 × 10-11 3.4 × 10-5
20 -8.0 × 10-11 1.9 × 10-5
The results in Table 2 indicate that the accurate homotopy-series solution whose 𝑓′′(0) converges to - 8.0 × 10-11. Moreover, the 13th-order homotopy-approximation of f(𝜂) agrees well the existing closed-form solution with each other attributing a negligible error as shown in Table 3.
Also, it is interesting that the results of 1st solution of 1st form gain from 1st solution 2nd form just setting λ = 1 with the optimal convergence-control parameter c0 = -1/2 and initial guess σ = 3/2. The results of both 1st and 2nd form of 1st solution is presented in Table 4 corresponding to the order of homotopy-approximation. The results in Table 4 reveal that the 5th to 20th homotopy-approximation tend to the same value 𝑓′′(0) = - 2.4 × 10-14 for 1st form and the 13th to 20th homotopy-approximation tend to the same value 𝑓′′(0) = - 8.0 × 10-11 in case of 2nd form of 1st solution.
Table 3: Comparison of results of 1st solution of 2nd form with existing closed form solution 𝜂 Closed form solution 2nd form with 13th-order
approximation
Error
0 0.0000 0.0000 0.0000
0.2 0.090764 0.089397 0.001368
0.4 0.180982 0.178368 0.002614
0.6 0.270118 0.266467 0.003651
0.8 0.357666 0.353221 0.004445
1.0 0.443152 0.438144 0.005008
1.2 0.526152 0.520751 0.005401
1.4 0.606294 0.600578 0.005716
1.6 0.683264 0.677204 0.006060
1.8 0.756809 0.750265 0.006544
2.0 0.826738 0.819467 0.007271
Although, the homotopy-series converges with the different values of 𝑓′′(0) = - 2.4 × 10-14 and 𝑓′′(0) = - 8.0 × 10-
11 for 1st and 2nd form of 1st solution respectively, both these two forms of solution agree well with the closed-form solution. Thus, the results in Table 4 indicate the multiform of same solution of two-dimensional laminar jet.
28 Table 4: 1st solution of two forms at mth-order homotopy approximation by means of initial guess equation (22) and auxiliary linear operator equation (23)
Order of approximation, m 1st form f(0) 2nd form f(0)
1 0 0
5 -2.4 × 10-14 4.9 × 10-11
7 -2.4 × 10-14 -3.6 × 10-11
13 -2.4 × 10-14 -8.0 × 10-11
17 -2.4 × 10-14 -8.0 × 10-11
20 -2.4 × 10-14 -8.0 × 10-11
4.3 2nd Solution
A novel technique so-called Gram-Schimdt orthogonalization method is employed to obtain the initial guess for 2nd solution. According to Gram-Schimdt process two functions 𝑢𝑛(𝑥) and 𝜓𝑛(𝑥) are orthogonal over the interval −1 ≤ 𝑥 ≤ 1 with weighting function 𝑤(𝑥) if
〈𝑢𝑛(𝑥)|𝜓𝑛(𝑥)〉 = ∫ 𝑢−11 𝑛(𝑥)𝜓𝑛(𝑥)𝑤(𝑥)𝑑𝑥 = 0, (28) if in addition,
∫ [𝑢−11 𝑛(𝑥)]2𝑤(𝑥)𝑑𝑥 = 1, and ∫ [𝜓−11 𝑛(𝑥)]2𝑤(𝑥)𝑑𝑥 = 1. (29)
Figure 4: Initial guess for 1st and 2nd solutions, 2nd initial guess obtained by means orthogonalization of 1st initial guess
Let us consider, an original set of linearly independent functions[𝑢𝑛]𝑛=0∞ , let [𝜓𝑛]𝑛=0∞ denote the orthogonalized (but not normalized) functions,[𝜙𝑛]𝑛=0∞ , denote the orthonormalized functions, and define
𝜓0(𝑥) ≡ 𝑢0(𝑥) ≡ 𝜎 +4
3𝜎 exp(−𝜂) −1
3𝜎 exp(−2𝜂), (30) 𝜙0(𝑥) ≡ 𝜓0(𝑥)
√∫ [𝜓−11 0(𝑥)]2𝑤(𝑥)𝑑𝑥
, (31) then taking
𝜓1(𝑥) = 𝑢1(𝑥) + 𝑎10𝜙0(𝑥), (32) where we require
∫ 𝜓−11 1(𝑥)𝜙0(𝑥)𝑤(𝑥)𝑑𝑥 = 0, (33)
∫ 𝑢−11 1(𝑥)𝜙0(𝑥)𝑤(𝑥)𝑑𝑥 + 𝑎10∫ [𝜙−11 0(𝑥)]2𝑤(𝑥)𝑑𝑥 = 0, (34) 0.0
0.2 0.3 0.5 0.6 0.8 0.9 1.1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
f(ƞ)
ƞ
Iniatial guess for 2nd solution Iniatial guess for 1st solution
29 by definition,
∫ [𝜙−11 0(𝑥)]2𝑤(𝑥)𝑑𝑥 = 1, so, 𝑎10= − ∫ 𝑢−11 1(𝑥)𝜙0(𝑥)𝑤(𝑥)𝑑𝑥. (35) The first orthogonalized function is therefore
𝜓1(𝑥) = 𝑢1(𝑥) − [∫ 𝑢−11 1(𝑥)𝜙0(𝑥)𝑤(𝑥)𝑑𝑥] 𝜙0(𝑥). (36) The obtained initial guess for 2nd solution considering 𝑢1(𝑥) = 1 and σ = 1 by using equations (28) to (36) is shown in Figure 4.
The same auxiliary linear operator (23), and the similar procedure of 1st solution is followed for the 2nd solution. The obtained result of two different positive solutions of nonlinear boundary-layer equation (10) of two-dimensional laminar jet as listed in Table 5. The numerical comparison between the 1st and 2nd form of solutions are presented in Table 5. The obtained results in Table 5, show that the flow begins from the entrance part and they finish at a certain level for 1st solution branch. On the other hand, 2nd branch of solution rest part of the flow level of the two dimensional laminar jet. The significance of 2nd solution is described in the section 4.4.
Table 5: Two positive solutions of homotopy approximation when c0 = -1/2, σ =3/2 with auxiliary linear operator equation (23) and the initial guess Figure 4.
𝜂 1st solution 2nd solution
0 0 1.000000
0.4 0.182846 0.979887
0.8 0.360977 0.960293
1.2 0.529828 0.941719
1.4 0.609498 0.932955
1.8 0.757342 0.916692
2.2 0.888216 0.902296
Figure 5: Streamlines a) 1st solution, (b) 2nd solution obtained from equation (7) with M = 1000 kg/sec2, ρ = 1000 kg/m3, ν = 0.01 m2/sec.
4.4. Stream Function and Velocity Profile
The steam function obtained from the equation (7) is plotted to describe the streamlines for the flow generated by the jet by means of homotopy-approximation both for 1st and 2nd solution as shown in Fig 5. Fig 5a is almost similar to be Mei (2002) whereas Fig 5b has less significant meanings for two-dimensional laminar jet. The nature of stream
a b
30 lines show the maximum velocity is at centre at the exit part of the jet for 1st form of solutions. On the other hand, the nature of the stream lines show the flow level at the edge and opposite part of the jet for 2nd form of solution. In fact, it is essential the maximum velocity is at centre for flowing jet, which is satisfy in the case of 1st form of solutions.
Therefore, the first form of solution is more significant than the 2nd form of solutions. Also, the velocity components of u and v are graphically shown in Figs 6-7 which obtained from equations (8) and (9) by means of homotopy- approximation both for 1st and 2nd solution with the jet properties M = 1000 kg/sec2, ρ = 1000 kg/m3, ν = 0.01 m2/sec.
These two figures clearly show physical implication of two-dimensional laminar jet as described Mei (2002) for 1st branch of solution i.e., (i) the jet width 𝛿 ∝ 𝑥23, (ii) the velocity at centerline 𝑈 = 𝑢𝑚𝑎𝑥∝ 𝑥−13, (iii) there is entrainment of jet at edges and (iv) Reynolds’s number 𝑅 = 𝑢𝑚𝑎𝑥𝛿𝑣∝ 𝑥13. Also, the velocity profiles illustrate in figures 6 to 8 the influence of jet at the exit level.
Fig 6: Velocity profile u-component (a) 1st solution, (b) 2nd solution obtained from equation (8) with M = 1000 kg/sec2, ρ = 1000 kg/m3, ν = 0.01 m2/sec
Fig 7: Velocity profile v-component (a) 1st solution, (b) 2nd solution obtained from equation (9) with M = 1000 kg/sec2, ρ = 1000 kg/m3, ν = 0.01 m2/sec.
a b
a b
31 Fig 8: Velocity profile resultant of u and v-components (a) 1st solution, (b) 2nd solution
5. Conclusions
In this study, homotopy analysis (exponentially decay solution) method is successfully applied to solve the nonlinear ODE in an infinite interval for two-dimensional laminar jet. The convergence of series is properly examined by the minimum of the squared residual of the governing equation (10) corresponding to the optimal convergence-control parameter c0 and initial guess σ, and the analytical solutions are presented in this paper. From this study, the following conclusion can be drawn: (1)the homotopy-series converges with the multiform of same solution and the comparison of these forms agree well with the existing closed-form solution, (2) the new approach of 2nd solution shows significant meanings for the velocity profile although it indicates less significant to illustrate the stream function, and (3) the streamlines and velocity obtained from homotopy-approximation are also satisfied the existing closed-form solution and these graphs agree well with the physical implication of two-dimensional laminar jet.
Acknowledgement
The research reported in this paper was fully supported by a grant from the City University of Hong Kong [SFA ID 000477] Postgraduate Studentship (by UGC-allocated funds).
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