- Email: [email protected]
An explicit, totally analytic solution of laminar viscous flow over a semi-infinite flat plate
LIAO:
An Explicit,
Totally
Analytic
Solution
...
53
An Explicit, Totally Analytic Solution of Laminar Viscous Flow over a Semi-Infinite Flat Plate 1 Shi-Jun LIAO (College of Naval Architecture Shanghai 200030, China) E-mail: [email protected]
& Ocean Engineering, Shanghai Jiao Tong University,
Abstract: In this paper, a new kind of analytic technique for nonlinear problems, namely the Homotopy Analysis Method, is applied to give an explicit, totally analytic solution of the Blasius’ flow, i.e. the two dimensional (2D) laminar viscous flow over a semi-infinite flat plate. This analytic solution is valid in the whole region having physical meanings. To our knowledge, it is the first time in history that such a kind of explicit, totally analytic solution is given. This fact well verifies the great potential and validity of the Homotopy Analysis Method as a kind of powerful analytic tool for nonlinear problems in science and engineering. Keywords: Blasius’ viscous flow, explicit analytic solution, new analytic technique, the Homotopy Analysis Method
Introduction It is well-known that perturbation techniques are too strongly dependent. upon the socalled “small parameters”. Thus, it is worthwhile developing some new analytic techniques independent upon small parameters. Some attempts have been made in this directions. Based on the homotopy in topology, Liao ~9~1proposed such a kind of analytic technique, namely the Homotopy Analysis Method (HAM). The validity of the HAM is independent upon whether or not nonlinear problems under consideration contain “small parameters”. Thus, the HAM is valid for more of nonlinear problems, especially those with strong nonlinearity. Furthermore, the HAM provides us with great freedom to select related initial approximations, governing equations of auxiliary sub-problems and also some auxiliary parameters. It is just this kind of freedom which provides us with a larger possibility to ensure the corresponding approximation sequence of the HAM convergent. Besides, based on the HAM, Liao [3$41further developed some new numerical techniques such as the “general boundary element method” 141,and even a non-iterative numerical technique for strongly nonlinear problems L31.All of these verify the validity of the HAM. For example, Liao L21applied the HAM to solve the Blasius’ flow governed by
with boundary conditions f(0) = f’(0)
= 0,
f’(+oo)
= 1
(2)
where the prime denotes derivatives w.r.t.7. L&i21 gave a family of power series in parameter fL, which contains the well-known power series solution given by Blasius L5] in 1908. We know that Blasius’ power series solution is valid in a rather restricted region (77) < PO, where po M 6.69. However, Liao’s power series solution L21can be valid in the whole region 7 E [O,+~Q) as A tends to zero (-2 < ti < O)! For details, please refer to Ref. [2]. This ‘The paper was received on April
10, 1998
54
Communications in Nonlinear Science & Numerical Simulation
Vol.3, No.2 (Jun. 1998)
verifies the validity of the HAM as a new kind of analytic tool. However, Blasius’ I51 and even Liao’s 121power series solutions are only semi-analytic and semi-numerical ones, because both of them need the value f”(0) which must be given by numerical techniques. To our knowledge, up to now, no one has given an explicit, totally analytic solution of Blasius’ flow and even an analytic value of f”(0). In [2], we used a very simple auxiliary linear operator. However, the HAM provides us with great freedom to select some other operators as our auxiliary ones. In this paper, we will illustrate that by means of using an auxiliary linear operator better than that used in [2], we can give an explicit, analytic solution of the Blasius’ flow governed by (1) and (2).
1. The Analytic
Solution
of Blasius’ Flow
Let
denote an auxiliary linear operator. We construct a family of differential equations
rl E LO,+cQ>, h # 0, P > 0, P E LO,11
(4
with boundary conditions F(O, h P,P) =z Jw,
fi, AP) = 0, q+m
h B,P) = 1, P E [O, 11, fi # 0, P > 0
(5)
where is the initial approximation and p = 1, we have
f0(rl) = rl - P - exd-PdllP (f-5) which satisfies the boundary conditions (2). Clearly, when p = 0
F(v, h A 0) = fo(77L
11E PA +=J>, fL # 0, B > 0
(7)
F(rl, hP, 1) = f(77),
77E [O, +m>, fi # 0, D > 0
(8)
respectively. Therefore, the process of p varying from 0 to 1 is just the continuous variation of the real function F(q, fL, p, p) from the known initial approxirnation fo(q) to the unknown solution f(q) of Eqs.(l) and (2). Assume that both of /3 and zl are properly selected so that the variation F(r], A, /3,p) is smooth enough, then, the so-called Icth-order (Jz 2 1) deformation derivative
(9) exists. Thus, according to (7) and the Taylor formula, we have
Clearly, the convergence region of the above infinite sequence is dependent upon both li and p. Assume that both of them are properly selected so that the above infinite sequence is convergent at p = 1. Then, due to (8) and (lo), we get the relationship
f!%gA, , D) +O” f(v)=fo(d+c+O” =
k=l
~(Pkhfi,b? k=O
(11)
55
LIAO: An Explicit, Totally Analytic Solution . . .
between the initial approximation
fs(n) and the solution f(n), where we define
(12) The governing equation of the unknown function (~~(7, A, /3) (m 2 1) is obtained by first differentiating Eqs.(4) and (5) m times w. r. t. p and then setting p = 0 and finally dividing it by m!, i.e. cp; (7, h B) + P&h
A, 8) = Gn (7, h 8)
(13)
with the related boundary conditions %n(O, A, PI = cp:,(O, h P> = cp:,(+m
fi, P) = 0
(14)
where n E [0, +oo), p > 0, fi. # 0, m 2 1, the prime denotes the partial derivatives w. r. t. n and
+ h
Cm22) (16) I
Notice that the above linear differential Eq.(13) with the linear boundary condition (14) can be solved one after another in order. By means of the symbolic computation software MATHEMATICA., we find, in a little surprise, that (~~(q, Ii, ,8) has such a structure m+l vh(rl,
h P) = c
*‘m,k(% h PbP(-kh)~
(17)
m 1 0
k=O
where the real function qm,k(7], ti, a) is defined by 2(m+l-k) %,k(7],
h,@
=
c
b;,kA;,k$,
m
1
0,
0 5
k < m +
1
08)
i=o
Here, A& is defined by
0, i=j=O, X~j
= I
0, i>O, j=O, 0, j>i+l 0, k>2(i+l-j) 1, otherwise
k>2 k>l
and b&,k are coefficients dependent upon both ,8 and FL. Thus, substituting we get an explicit analytic solution
=
t+
lim M++fXJ
(19)
(17) into (ll),
)I(20)
56
Communications in Nonlinear Science & Numerical Simulation
Vo1.3, No.2 (Jun. 1998)
where for m 2 ho 5 n L m + 1,O I k I 2(m - n f l), we rigorously deduce by logic the following recurrence formulas 2m-1 =
bL,o
XmbL,o
-8-l
c
Tn,l&
q=o
2(m-n+l)
mtl -
(n - Vf+4,0
+
n=2 4
C g=l
r%,,
cn,n
hl,o
(v4,o
- 14,~ - B-%i,J
2m-1 bZ,l
=
xmbL,1+
P-’
c
1
(21)
r;,,p;l,,
q=o
2(m-n+l)
m+l +c
nro,,t#:,o
+
n=2 [
c q=l
- P&)
2m-1 C,l
=
bk,,
=
Xmbk-l,l
+
1
(22)
l
C rk,~P~,k7 q=k-1
(23)
2m-1 c
%w-- 1~ k -< 2m
rf,&,k,
(24)
q=k-1
2(m-n+l)
bk,n
=
xmbLl,n
-
c q=k
r$&.i:k9 ’ ’
0
-
-< m
(25)
2(m-n+l)
bh,, = -
C
r&,,p,
q=k
2(m-n)+l
bO,,m+l
(26) (27)
-rOm,m+l&+l,o
Here, Xm
0, 1,
=
when m = 1 otherwise
(28)
(29)
d ‘E
1 (n -
l)q-k+lPq-k+3
’
Oik_
(30)
The coefficient IYL,, is defined by cl,1
cn,n
O
=
fi bL,l
=
h5;y
(32)
=
fL(5k,m+l
(33)
=
+ %,I)
7
fik%-l,n +@n,n> > Oiq_<2(m--n),2
WL ’n 0
(31)
2(m - n) + 1 5 4 5 2(m - n + l), 2 < 72< m (34) otherwise
57
LIAO: An Explicit, Totally Analytic Solution . . .
The related coefficient Sk,, is defined by ;z Sk,
min{q,2(k-j+l)) c
-iy+11
Ch,jb~l-k,n-jX~,-,,n-j
= k=O j=max{l,n+k-m}
(35)
i=max{O,q-2(m-k-n+j)}
where CL k
=
(i + l)(i
din:k
=
(i +
=
+..)c2C;-“+l)
dz;-k+l)
+ 2)b$;X$2,
l)c$:
-
Owing to the initial approximation,
- 2(kp)(i
+ l)bz,:X$;
+ (k/?)2bk,kXi,k
0 5 i 5 2(” - Ic) + 1
(k/+$$
(36)
(37) (38)
we get the first three coefficients
t$jo = -p-l,
b;,. = 1,
b& = /?-l
(39)
Then, using above recurrence formulas and these first three coefficients, we can calculate all coefficients of I&,. Thus, formula (20) gives us an ezpcplicit,totally analytic solution. Our calculations indicate that the mth-order approximation of f”(O), say, 0, = $m,W) k=O
= fyc&
(40)
k=O n=l
contains the term p(l + ti)“. Thus, the solution (20) is certainly divergent when fi. > 0 or fi < -2. Our calculations indicate that the solution (20) is convergent in the whole region q E [0, +co) to the solution of Eqs.(l) and (2), when -2 < Ii < 0,
p > PO
(41)
where @eM 2.5. For example, when h = -g/10, p = 3, our 35th-order approximation agrees very well with Howarth’s numerical result and gives such an analytic value f”(O) = 0.33206.
2. Conclusions In this paper, by means of the Homotopy Analysis Method (HAM), we obtain an explicit, totally analytic solution of Blasius’ flow governed by the nonlinear differential Eqs.(l) and (2). We emphasize that, to our knowledge, it is the first time in history that such a kind of explicit, totally analytic solution has been given. This well verifies the validity and great potential of the HAM. So, we firmly believe that, combined with high performance computers, the HAM might become a powerful analytic tool for nonlinear problems in science and engineering. References [I] Liao, S. J., An approximate solution technique not depending on small parameters: a special example. Int. J. Non-Linear Mechanics, 1995, 30(3): 371-380. [2] Liao, S. J., An approximate solution technique not depending on small parameters (Part 2): an application in fluid mechanics. Int. J. Non-Linear Mechanics, 1997, 32(5): 815-822. [3] Liao, S. J., Numerically solving non-linear problems by the homotopy analysis method, Computational Mechanics, 1997, 20: 530-540. [4] Liao, S. J., On the general boundary element method, Engineering Analysis with Boundary Elements, 1998, 21(1):39-51. [5] Blasius, H., Z. Math. Phys., 1908, 56: l-37.
An Explicit,
Totally
Analytic
Solution
...
53
An Explicit, Totally Analytic Solution of Laminar Viscous Flow over a Semi-Infinite Flat Plate 1 Shi-Jun LIAO (College of Naval Architecture Shanghai 200030, China) E-mail: [email protected]
& Ocean Engineering, Shanghai Jiao Tong University,
Abstract: In this paper, a new kind of analytic technique for nonlinear problems, namely the Homotopy Analysis Method, is applied to give an explicit, totally analytic solution of the Blasius’ flow, i.e. the two dimensional (2D) laminar viscous flow over a semi-infinite flat plate. This analytic solution is valid in the whole region having physical meanings. To our knowledge, it is the first time in history that such a kind of explicit, totally analytic solution is given. This fact well verifies the great potential and validity of the Homotopy Analysis Method as a kind of powerful analytic tool for nonlinear problems in science and engineering. Keywords: Blasius’ viscous flow, explicit analytic solution, new analytic technique, the Homotopy Analysis Method
Introduction It is well-known that perturbation techniques are too strongly dependent. upon the socalled “small parameters”. Thus, it is worthwhile developing some new analytic techniques independent upon small parameters. Some attempts have been made in this directions. Based on the homotopy in topology, Liao ~9~1proposed such a kind of analytic technique, namely the Homotopy Analysis Method (HAM). The validity of the HAM is independent upon whether or not nonlinear problems under consideration contain “small parameters”. Thus, the HAM is valid for more of nonlinear problems, especially those with strong nonlinearity. Furthermore, the HAM provides us with great freedom to select related initial approximations, governing equations of auxiliary sub-problems and also some auxiliary parameters. It is just this kind of freedom which provides us with a larger possibility to ensure the corresponding approximation sequence of the HAM convergent. Besides, based on the HAM, Liao [3$41further developed some new numerical techniques such as the “general boundary element method” 141,and even a non-iterative numerical technique for strongly nonlinear problems L31.All of these verify the validity of the HAM. For example, Liao L21applied the HAM to solve the Blasius’ flow governed by
with boundary conditions f(0) = f’(0)
= 0,
f’(+oo)
= 1
(2)
where the prime denotes derivatives w.r.t.7. L&i21 gave a family of power series in parameter fL, which contains the well-known power series solution given by Blasius L5] in 1908. We know that Blasius’ power series solution is valid in a rather restricted region (77) < PO, where po M 6.69. However, Liao’s power series solution L21can be valid in the whole region 7 E [O,+~Q) as A tends to zero (-2 < ti < O)! For details, please refer to Ref. [2]. This ‘The paper was received on April
10, 1998
54
Communications in Nonlinear Science & Numerical Simulation
Vol.3, No.2 (Jun. 1998)
verifies the validity of the HAM as a new kind of analytic tool. However, Blasius’ I51 and even Liao’s 121power series solutions are only semi-analytic and semi-numerical ones, because both of them need the value f”(0) which must be given by numerical techniques. To our knowledge, up to now, no one has given an explicit, totally analytic solution of Blasius’ flow and even an analytic value of f”(0). In [2], we used a very simple auxiliary linear operator. However, the HAM provides us with great freedom to select some other operators as our auxiliary ones. In this paper, we will illustrate that by means of using an auxiliary linear operator better than that used in [2], we can give an explicit, analytic solution of the Blasius’ flow governed by (1) and (2).
1. The Analytic
Solution
of Blasius’ Flow
Let
denote an auxiliary linear operator. We construct a family of differential equations
rl E LO,+cQ>, h # 0, P > 0, P E LO,11
(4
with boundary conditions F(O, h P,P) =z Jw,
fi, AP) = 0, q+m
h B,P) = 1, P E [O, 11, fi # 0, P > 0
(5)
where is the initial approximation and p = 1, we have
f0(rl) = rl - P - exd-PdllP (f-5) which satisfies the boundary conditions (2). Clearly, when p = 0
F(v, h A 0) = fo(77L
11E PA +=J>, fL # 0, B > 0
(7)
F(rl, hP, 1) = f(77),
77E [O, +m>, fi # 0, D > 0
(8)
respectively. Therefore, the process of p varying from 0 to 1 is just the continuous variation of the real function F(q, fL, p, p) from the known initial approxirnation fo(q) to the unknown solution f(q) of Eqs.(l) and (2). Assume that both of /3 and zl are properly selected so that the variation F(r], A, /3,p) is smooth enough, then, the so-called Icth-order (Jz 2 1) deformation derivative
(9) exists. Thus, according to (7) and the Taylor formula, we have
Clearly, the convergence region of the above infinite sequence is dependent upon both li and p. Assume that both of them are properly selected so that the above infinite sequence is convergent at p = 1. Then, due to (8) and (lo), we get the relationship
f!%gA, , D) +O” f(v)=fo(d+c+O” =
k=l
~(Pkhfi,b? k=O
(11)
55
LIAO: An Explicit, Totally Analytic Solution . . .
between the initial approximation
fs(n) and the solution f(n), where we define
(12) The governing equation of the unknown function (~~(7, A, /3) (m 2 1) is obtained by first differentiating Eqs.(4) and (5) m times w. r. t. p and then setting p = 0 and finally dividing it by m!, i.e. cp; (7, h B) + P&h
A, 8) = Gn (7, h 8)
(13)
with the related boundary conditions %n(O, A, PI = cp:,(O, h P> = cp:,(+m
fi, P) = 0
(14)
where n E [0, +oo), p > 0, fi. # 0, m 2 1, the prime denotes the partial derivatives w. r. t. n and
+ h
Cm22) (16) I
Notice that the above linear differential Eq.(13) with the linear boundary condition (14) can be solved one after another in order. By means of the symbolic computation software MATHEMATICA., we find, in a little surprise, that (~~(q, Ii, ,8) has such a structure m+l vh(rl,
h P) = c
*‘m,k(% h PbP(-kh)~
(17)
m 1 0
k=O
where the real function qm,k(7], ti, a) is defined by 2(m+l-k) %,k(7],
h,@
=
c
b;,kA;,k$,
m
1
0,
0 5
k < m +
1
08)
i=o
Here, A& is defined by
0, i=j=O, X~j
= I
0, i>O, j=O, 0, j>i+l 0, k>2(i+l-j) 1, otherwise
k>2 k>l
and b&,k are coefficients dependent upon both ,8 and FL. Thus, substituting we get an explicit analytic solution
=
t+
lim M++fXJ
(19)
(17) into (ll),
)I(20)
56
Communications in Nonlinear Science & Numerical Simulation
Vo1.3, No.2 (Jun. 1998)
where for m 2 ho 5 n L m + 1,O I k I 2(m - n f l), we rigorously deduce by logic the following recurrence formulas 2m-1 =
bL,o
XmbL,o
-8-l
c
Tn,l&
q=o
2(m-n+l)
mtl -
(n - Vf+4,0
+
n=2 4
C g=l
r%,,
cn,n
hl,o
(v4,o
- 14,~ - B-%i,J
2m-1 bZ,l
=
xmbL,1+
P-’
c
1
(21)
r;,,p;l,,
q=o
2(m-n+l)
m+l +c
nro,,t#:,o
+
n=2 [
c q=l
- P&)
2m-1 C,l
=
bk,,
=
Xmbk-l,l
+
1
(22)
l
C rk,~P~,k7 q=k-1
(23)
2m-1 c
%w-- 1~ k -< 2m
rf,&,k,
(24)
q=k-1
2(m-n+l)
bk,n
=
xmbLl,n
-
c q=k
r$&.i:k9 ’ ’
0
-
-< m
(25)
2(m-n+l)
bh,, = -
C
r&,,p,
q=k
2(m-n)+l
bO,,m+l
(26) (27)
-rOm,m+l&+l,o
Here, Xm
0, 1,
=
when m = 1 otherwise
(28)
(29)
d ‘E
1 (n -
l)q-k+lPq-k+3
’
Oik_
(30)
The coefficient IYL,, is defined by cl,1
cn,n
O
=
fi bL,l
=
h5;y
(32)
=
fL(5k,m+l
(33)
=
+ %,I)
7
fik%-l,n +@n,n> > Oiq_<2(m--n),2
WL ’n 0
(31)
2(m - n) + 1 5 4 5 2(m - n + l), 2 < 72< m (34) otherwise
57
LIAO: An Explicit, Totally Analytic Solution . . .
The related coefficient Sk,, is defined by ;z Sk,
min{q,2(k-j+l)) c
-iy+11
Ch,jb~l-k,n-jX~,-,,n-j
= k=O j=max{l,n+k-m}
(35)
i=max{O,q-2(m-k-n+j)}
where CL k
=
(i + l)(i
din:k
=
(i +
=
+..)c2C;-“+l)
dz;-k+l)
+ 2)b$;X$2,
l)c$:
-
Owing to the initial approximation,
- 2(kp)(i
+ l)bz,:X$;
+ (k/?)2bk,kXi,k
0 5 i 5 2(” - Ic) + 1
(k/+$$
(36)
(37) (38)
we get the first three coefficients
t$jo = -p-l,
b;,. = 1,
b& = /?-l
(39)
Then, using above recurrence formulas and these first three coefficients, we can calculate all coefficients of I&,. Thus, formula (20) gives us an ezpcplicit,totally analytic solution. Our calculations indicate that the mth-order approximation of f”(O), say, 0, = $m,W) k=O
= fyc&
(40)
k=O n=l
contains the term p(l + ti)“. Thus, the solution (20) is certainly divergent when fi. > 0 or fi < -2. Our calculations indicate that the solution (20) is convergent in the whole region q E [0, +co) to the solution of Eqs.(l) and (2), when -2 < Ii < 0,
p > PO
(41)
where @eM 2.5. For example, when h = -g/10, p = 3, our 35th-order approximation agrees very well with Howarth’s numerical result and gives such an analytic value f”(O) = 0.33206.
2. Conclusions In this paper, by means of the Homotopy Analysis Method (HAM), we obtain an explicit, totally analytic solution of Blasius’ flow governed by the nonlinear differential Eqs.(l) and (2). We emphasize that, to our knowledge, it is the first time in history that such a kind of explicit, totally analytic solution has been given. This well verifies the validity and great potential of the HAM. So, we firmly believe that, combined with high performance computers, the HAM might become a powerful analytic tool for nonlinear problems in science and engineering. References [I] Liao, S. J., An approximate solution technique not depending on small parameters: a special example. Int. J. Non-Linear Mechanics, 1995, 30(3): 371-380. [2] Liao, S. J., An approximate solution technique not depending on small parameters (Part 2): an application in fluid mechanics. Int. J. Non-Linear Mechanics, 1997, 32(5): 815-822. [3] Liao, S. J., Numerically solving non-linear problems by the homotopy analysis method, Computational Mechanics, 1997, 20: 530-540. [4] Liao, S. J., On the general boundary element method, Engineering Analysis with Boundary Elements, 1998, 21(1):39-51. [5] Blasius, H., Z. Math. Phys., 1908, 56: l-37.