An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching

International Journal of Non-Linear Mechanics 43 (2008) 1000 -- 1006

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An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching Tiegang Fang ∗ , Ji Zhang Mechanical and Aerospace Engineering Department, North Carolina State University, 3182 Broughton Hall, Campus Box 7910, 2601 Stinson Drive, Raleigh, NC 27695, USA

A R T I C L E

I N F O

Article history: Received 8 April 2008 Accepted 15 May 2008

Keywords: Similarity solution Falkner-Skan equation Stretching surface Mass transfer Analytical solution

A B S T R A C T

In this paper, an exact analytical solution of the famous Falkner-Skan equation is obtained. The solution involves the boundary layer flow over a moving wall with mass transfer in presence of a free stream with a power-law velocity distribution. Multiple solution branches are observed. The effects of mass transfer and wall stretching are analyzed. Interesting velocity profiles including velocity overshoot and reversal flows are observed in the presence of both mass transfer and wall stretching. These solutions greatly enrich the analytical solution for the celebrated Falkner-Skan equation and the understanding of this important and interesting equation. © 2008 Elsevier Ltd. All rights reserved.

The famous Falkner-Skan equation was first obtained for the boundary layer flow with stream-wise pressure gradient [1]  f  + ff  + (1 − f 2 ) = 0

(1)

with the boundary conditions (BCs) f (0) = ,

f  (0) = , and f  (∞) = 1,

(2a−2c)

where  is the wall mass transfer parameter showing the strength of the mass transfer at the wall,  is the wall movement parameter indicating the strength of the wall stretching, and  is a parameter of the stream-wise pressure gradient. The original Falkner-Skan involved  = 0 and  = 0 for a fixed and impermeable wedge flow. Later the numerical solution was calculated by Hartree [2] for this case. This celebrated equation played an important role in the development of boundary layer theory in fluid mechanics. Thereafter, most of the work were concentrated on such kind of BCs with f (0) = f  (0) = 0. The mathematically rigorous analysis for this equation was initiated by Weyl [3] for  = 0, namely the Blasius equation. Early works on this equation about fluid flow can be found in a book edited by Rosehead [4] and the mathematical treatment can be seen in a text by Hartman [5]. Some interesting characteristics of the Falkner-Skan equation were observed by these early researchers. Stewartson [6] showed that multiple solutions existed for the Falkner-Skan equation when −0.19884    0. Libby and Liu [7] observed a family of

∗

Corresponding author. Tel.: +1 919 515 5230; fax: +1 919 515 7968. E-mail address: [email protected] (T. Fang).

0020-7462/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2008.05.006

separated velocity profiles for  < − 0.19884. For this parameter domain, Stewartson proved that there was velocity over shooting near the wall. As a matter of fact, there are infinite number of solutions for the Falkner-Skan equation in this region of . A new solution branch of the Falkner-Skan equation was found by Zaturska and Banks [8] for a positive  greater than one. The solution for the Falkner-Skan equation with mass transfer at the wall, namely f (0)0 and f  (0) = 0 were first discussed by Schlichting and Bussmann [9] for  = 0. A general treatment of mass transfer at the wall was conducted by Nickel [10] for arbitrary values of . Analytical solution for the Falkner-Skan equation with  = −1 was presented by Yang and Chien [11] for f (0) =  and f  (0) = 0. The flow with boundary stretching or movement was first investigated by Sakiadis [12,13] for f (0) = 0, f  (0) = 1, and f  (∞) = 0 with  = 0. This is a pioneer work for a new area of stretching boundary flow. The moving wall problem of a plate in a free stream was first studied by Klemp and Acrivos [14] for f (0) = 0, f  (0) = , and f  (∞) = 1 with  = 0. Then the problem with mass injection at the wall for drag reduction was investigated by Vajrvelu and Mohapatra [15]. Recently the general cases with f (0) = , f  (0) = , and f  (∞) = 1 with  = 0 were numerically studied by Fang [16] and Weidman et al. [17] independently. For a non-zero value of , the solution was given by Riley and Weidman [18] for a moving impermeable wall and an exact solution for the Falkner-Skan equation with f (0) = 0, f  (0) = , and f  (∞) = 1 was presented for  = −1. Due to the strong non-linearity of the Falkner-Skan equation, analytical solutions were very rarely available. For a general value of , an analytical solution was published by Liao [19] using the homotopy analysis method for a fixed impermeable wedge flow. Most of the analytical solution was given for  = −1. The solution of Yang and Chien [11]

T. Fang, J. Zhang / International Journal of Non-Linear Mechanics 43 (2008) 1000 -- 1006

was an example of the analytical solution to the mass transfer case, while the solution by Riley and Weidman [18] was for a non-fixed wall. Most recently, an analytical solution for the Falkner-Skan equation was presented for the non-fixed wall problem in a new form [20]. The objective of the current short communication is to present an analytical solution in a more general condition for a moving and permeable wall. The solution for Yang and Chien [11] and Riley and Weidman [18] are only two special cases of the current results. New solution branch for the permeable wall was observed with different velocity profiles different from the results of Yang and Chien [11]. We will study a special case of the Falkner-Skan equation with  = −1,  f  + ff  + f 2 − 1 = 0

(3)

f  (0) = , and f  (∞) = 1.

(4a−4c)

This equation can be integrated twice as   2 2 f2  = + ( + ) + + , f + 2 2 2

(5)

= f  (0). In order to solve Eq. (5), by defining f () =

where  2(u ()/u()), we obtain    2 1 2  + ( + ) + + u=0 u − 2 2 2

f () →  + A

and

f  () → 1.

(11a,11b)

Substitute them to Eq. (5) gives 1+

( + A)2 2 = + ( + ) + 2 2

Then we obtain

 +  = A and

A2 + 2



2 2



2 2

 + .

(12)

 +  − 1 = 0.

(13a,13b)

Eqs. (13a) and (13b) imply

    2 2   +(+)+  + −1 2 2 u (∞) u (0)  = and = . √ u(0) 2 u(∞) 2 2

   2 1 1 1 2 + = = . 1 − ( + ) + 4 2 2 2 Then

   √  1, 32 , z + 6C z 32 , 32 , z + 43 z 2, 52 , z w (z)

  = 2 √ w(z) C z 12 , 12 , z + z 1, 32 , z =2+ √

(7a−7b)

Defining a new variable z = ( +  + ) /2 with z0 = ( + ) /2 and u() = e−z/2 w(z) and substituting them into Eq. (6) yield,   

1 1 1 2 − z − w 1 − ( + )2 + + w z + w = 0. 2 4 2 2 (8) 

√

1 √

erf( z) + 2C

2e−z √ . z

and



√ √ 3 5 1 4  1, , z + z 2, , z =  1, , z = zez erf z + 1, 2 3 2 2 the stream function can be simplified as √ 2 2e−(++) /2 . f () =  +  +  + √ √ ( /2) erf[( +  + )/ 2] + C The velocity solution becomes √

√ 2 2e(++) /2 ( +  + )(( /2) erf[( +  + )/

(16)

2] + C)

√ √ [( /2) erf[( +  + )/ 2] + C]2

.

(17)

The constant C can be determined by

There is a general solution for w(z) as   √ 1 1 3 w(z) = C1  , , z + C2 z  + , , z , 2 2 2

√ (9)

where  is defined as    2 1 1 + , = 1 − ( + )2 + 4 2 2

2 2e−(+) /2

 =  +  + √ √ ( /2) erf[( + )/ 2] + C with √ 2 √ √ 2e−(+) /2 (18) − ( /2) erf[( + )/ 2],  − ( + )  where +=± 2 + 2 − 2. The condition for a real valued solution is

C=

and [a, b, z] is the confluent hypergeometric function of the first kind [21]. Then it is obtained that √  w (z) −1 , (10) f () = F(z) = 2z 2 w(z) where

   √ 1 3 3 2 3 5 w (z)  + 2 , 2 , z +12C z +1, 2 , z + 3 (1+2)z + 2 , 2 , z

  2 = √ w(z) C z , 1 , z +z + 1 , 3 , z 2

(15)

By using the properties of the confluent hypergeometric functions of the first kind   3 z √ e erf z [a, a, z] = ez ,  1, , z = 2 4z

2

2 e−(++) −1 −

(14)

Thus

(6)

with BCs as

f  () = 1 +

with C = C1 /C2 . Based on the BCs for  → ∞,

2 + 2 − 2 = ( + )2 .

with the BCs f (0) = ,

1001

2 2

2 + 2 − 2  0.

(19)

Based on the above equation it is seen that both the wall stretching parameter and the wall mass transfer parameter can take either a positive value or a negative value. The “+” and “−” signs indicate that there are two solution branches for a given pair of  and .

T. Fang, J. Zhang / International Journal of Non-Linear Mechanics 43 (2008) 1000 -- 1006

2

6

4 -1 2 -1 0 -1 -8

-6

-4

18 14 --1 8--102-2-26 16 -1 42408-2 0-1 68 0 3

-2

4

-1 6

4

20

24 22

8

5

12 10

1002

0

-12 -4-1681-4-102 -2226-424 -2

2

15

II

I 10

0-1 8

-4

0

-2 -4 -2-24 0

V

VI

-2 0

0

8 -1

-1

2 2

6 --180-41 1-21 -42-6--42 0 2 -1 8-

6

-4

-3

10

-2

-1

0

20

2

3

-20

24

4

8 6

16 14

0 -2 8 -1 -1 6

-2

2

2 -1

II

5

4

-2--801-2422--611 2-0641 8

2

8 -2

VI

0

-2

-4

-2--2 601-2-482 0

0

62 4-8-11 -1-

-1

-2

-1

0

1

6

3

4

-15 -20

10 12

-25

14 16

4

2

6

-2

-3

2

-1 -1 6 1 4 8

-4

8

-6

-5

2

III

2 -1

-2 0 -2

-4

4

0

-3-20 -2 8

4

IV

-4

-3

-10

2

0 --602-14 -2-83

-2

5

-1 0

-6

-5

V

42 0 --8 -612 -8-11-62-2 -1-01

15

-1 4

10

I

-1-20-2463 0

1 0

-2 6

-8

-21-2-02-46 3-02 8

20

-2 4

-4

8-4-81 -1 2-4220--6112 -061 48

3

-1 0

-6

18

0

4

γ

1

-15

22

λ

5

-5

-10

18

16

14 12

8

2

4

-5

8

6

III -8

-4

-5 10 12 14

0

-2

16 -1-4 -2-42 2

-1 0

-2 0

-1 2

-3

0

4

IV 4

-2

5

0

2 2-6 -6181-4-012 -1-

γ

1

-5

20

5

-30

λ Fig. 1. The solution domain for f  (0) at different values of  and  for the two solution branches. (a) for the ”+” sign in the top plot and (b) for the ”−” sign in the bottom plot.

The solution in this paper is an extension to the previous studies. For an impermeable stretching wall, it requires   1 [18], and for a fixed permeable wall, we need 2 −2  0 [11]. Due to this generalization, the velocity variation behavior in the boundary layers will be greatly enriched. The variation of , namely the wall stress, with the variations of  and  are shown in Fig. 1(a) and (b) for the upper solution (“+” sign of the square root) and the lower solution (“−” sign of the square root) branches, respectively. Interesting solution domains are found in the contour plots. In both figures, the white areas indicate the no-solution domain for 2 + 2 − 2 < 0. The contour curves

with  = 0 show the shear free solutions. These shear free contour curves and the solution domain boundary divide the solution existence domain into five regions as shown in Fig. 1(a) and (b). There are two saddle points for the two solution branches at  = 1,  = 1 for the upper solution branch and  = 1,  = −1 for the lower solution branch. For the saddle point, C → ∞, therefore the solution will be f () =  + 1 for the upper solution branch and f () =  − 1 for the lower solution branch. As a matter of fact, a similar solution exists for  > 0 at  = 1 with a shear-free solution as f () =  +  for the upper solution branch and f () =  −  for  < 0 for the lower solution.

T. Fang, J. Zhang / International Journal of Non-Linear Mechanics 43 (2008) 1000 -- 1006

4

5 λ = 1.0

γ = 1.5 γ= 2 γ= 3 γ = -1.45 γ = -1.5

3.5

4

3 2.5

γ = -2.0, -1.5, -1.0, -0.5 for the upper solution branch

2

f' ()

3

f' ()

1003

2

1.5 1

1

0.5 0

0

γ = 2.0, 1.5, 1.0, 0.5 for the lower solution branch

-1

-1 0

1

2

3 

4

5

6

Fig. 2. Some examples of the velocity profiles in the boundary layers at  = 1 with  < 0 for the upper solution branch and with  > 0 for the lower solution branch.

8 γ = 1.5 γ=2 γ=3 γ = -1.5 γ = -2 γ = -3

7 6

f' ()

5 4

λ=0 Upper Solution Branch

3 2 1 0

0

0.5

1

1.5

2

2.5 η

3

3.5

4

4.5

λ=0 Lower Solution Branch

-0.5

5

Fig. 3. The velocity profiles of the upper solution branch for a fixed wall at different mass transfer parameters.

With mass suction or injection, the solution for  = 1 are quite different from the impermeable wall solution. Some velocity profiles for  = 1 with  < 0 for the upper solution branch and with  > 0 for the lower solution branch are shown in Fig. 2. For the upper solution branch, mass injection at the wall makes velocity over shoot in the boundary. But for the lower solution branch, mass suction results in negative velocity in the boundary layer, indicating reversal flows in the fluid near the wall. These solutions are very different from the trivial solution for an impermeable wall as f () = . In the previous results for a fixed plate with mass transfer, only the upper solution branch was discussed in the paper [11] with a positive . However, for a fixed permeable wall, a solution also ex√ ists for mass injection, namely   − 2. Some velocity profiles for  = 0 are shown in Fig. 3 for the upper solution branch. Yang and Chien [11] only discussed the mass suction velocity in their paper. Mass injection at the wall makes the velocity profiles greatly distinct

0

1

2

3 η

4

5

6

Fig. 4. The velocity profiles of the lower solution branch for a fixed wall at different mass transfer parameters.

from the mass suction results. Velocity overshoot in the boundary layers is seen in the mass injection cases. A stronger mass injection leads to a higher value of overshoot velocity in the boundary layers. One thing somehow counter-intuitive is that the boundary layer thickness becomes thinner for a higher mass injection rate, which is not consistent with most of the observations in the boundary layer theory. The lower solution branch for a fixed wall also shows interesting results as seen in Fig. 4. For the mass suction cases, a reversal flow region is observed in the boundary layer. Also the reversal flow becomes stronger with increasing mass suction rate. But for mass injection cases, both reversal flow and velocity overshoot can be seen in the velocity profiles. For a large mass injection parameter, the velocity in the boundary layer can be infinite, which will be discussed in detail in a later section. As seen in the solution domain as a function of  and  in Fig. 1, different types of solution can be found for the whole solution domain with their unique features. Typical solutions of Regions I, II, and IV for the upper solution branches are shown in Fig. 5. For the upper solution branch, in Regions I and II, the velocity profiles are monotonous functions. The maximum or minimum values occurs at the boundaries at  = 0 or ∞. For Region I, the solution is a monotonously decreasing function, while for Region II, the solution is monotonously increasing. But for Region IV, the velocity first increases to a peak value higher than both boundary velocities, then decreases to the ambient velocity in the free stream. The lower solution branch, however, shows quite different variation behaviors as seen in Fig. 6. For Region I, the velocity first decreases to a minimum velocity and then increases to the free stream velocity. The minimum velocity can be negative with reversal flow in the boundary layer. For Region II, the velocity is still a monotonously increasing function, however, the increasing rate is different from the upper solution branch. The shear stress function is not a monotonous function for the lower solution branch. But it is monotonous for the upper solution branch. The velocity for Region IV first increases to a maximum value higher than the velocities at the wall and in the free stream, then it decreases to a negative velocity with reversal flow, and finally it increases again to the free stream velocity. Solution behavior in the small solution region, namely Region V, is also distinct from other regions. Two typical velocity profiles are shown in Fig. 7 for the two solution branches. For the upper solution branch, the velocity first drops and then increases to the free stream velocity,

1004

T. Fang, J. Zhang / International Journal of Non-Linear Mechanics 43 (2008) 1000 -- 1006

f' ()

2.5

Region I λ =2 γ=3

2 1.5 1

0

0.5

1

1.5

2

2.5

3

3.5

4

2.5

3

3.5

4

2.5

3

3.5

4

1

f' ()

0

Region II λ = -2 γ=3

-1 -2 -3

0

0.5

1

1.5

2

f' ()

10

Region IV λ=2 γ = -3

5

0

0

0.5

1

1.5

η

2

Fig. 5. Typical velocity profiles of the upper solution branch in solution Regions I, II, and IV.

f' ()

2 Region I λ =2 γ=3

1 0 -1

0

1

2

3

4

5

6

7

4

5

6

7

4

5

6

7

2

f' ()

1 0

Region II λ = -2 γ=3

-1 -2

0

1

2

f' ()

4

3

Region IV λ= 2 γ = -3

2 0 0

1

2

3



Fig. 6. Typical velocity profiles of the lower solution branch in solution Regions I, II, and IV.

while for the lower solution branch, the velocity first increases then it decreases to the free stream velocity. However, there is no reversal flow in the boundary layers for both solution branches.

The solutions in Region III for the two solution branches are quite different and the velocity profiles are not finite for most of the solution domain in this region. A velocity profile is shown in Fig. 8. It is

T. Fang, J. Zhang / International Journal of Non-Linear Mechanics 43 (2008) 1000 -- 1006

1005

0.95 0.9

Region V λ = 0.7 γ = 0.9 Upper solution branch

f' ()

0.85 0.8 0.75 0.7 0.65 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

2.5 η

3

3.5

4

4.5

5

1.3 1.2

f' ()

1.1 1

Region V λ = 0.7 γ = -0.9 Lower solution branch

0.9 0.8 0.7 0

0.5

1

1.5

2

Fig. 7. Typical velocity profiles of the two solution branches in solution Region V.

2

In summary, an exact solution for the Falkner-Skan equation of

 = −1 was presented in this paper for general BCs with both mass

1.8

transfer and wall movement. Compared with the previous published results, the current solution provides some new velocity variation behaviors. Different solution behaviors were identified in different solution regions. These solutions greatly enriched the analytical solution for the celebrated Falkner-Skan equation and the understanding of this important and interesting equation.

1.6

f' ()

1.4 1.2

References

1 λ = 0.5 γ = -1.5 Lower solution branch

0.8 0.6 0.4 0.2

0

1

2

3

η

4

5

6

7

Fig. 8. An example of the velocity profile of the low solution branch in Region III with a finite fluid velocity.

seen that the velocity first decreases, then increases to a peak value greater than the two boundary velocities and finally decreases again to the free stream velocity. However, for most of the solution domain, the velocity is not finite. √ This infiniteness is because of the fact that √ ( /2) erf[( +  + )/ 2] + C = 0 at a certain value of . In order to √ further √ analyze this point, we define a function h()=( /2) erf[( +  + )/ 2]+C. Then it is known that h() is√a monotonously increas√ ing function of . Therefore, hmin = h(0) = ( /2) erf[( + )/ 2] + C √ and hmax = h(∞) = /2 + C. Thus the conditions for infinite velocthere ity profiles are hmin  0 and hmax  0. Under these conditions, √ √ exists a certain value of  with ( /2) erf[( +  + )/ 2] + C = 0.

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[19] S.J. Liao, A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate, J. Fluid Mech. 385 (1999) 101–128. [20] P.L. Sachdev, R.B. Kudenatti, N.M. Bujurke, Exact analytic solution of a boundary value problem for the Falkner-Skan equation, Stud. Appl. Math. 120 (2008) 1–16. [21] S. Wolfram, Mathematica—A System for Doing Mathematics by Computer, second ed., Addison-Wesley Publishing Company, Inc., New York, 1993.