Abstract
Full Text
HYDROMECHANICS
G. V. GIL’ and A. D. MYSHKIS
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A NONLINEAR BOUNDARY-VALUE PROBLEM IN BOUNDARY-LAYER THEORY
(Presented by Academician L. I. Sedov on 9 XI 1956)
In boundary-layer theory the following boundary-value problem is well known:
[
y'''(t)+2y(t)y''(t)+2\beta\bigl(k^2-[y'(t)]^2\bigr)=0
\quad (0\le t<\infty),
]
[
y(0)=y'(0)=0,\qquad y'(\infty)=k
\quad (\beta\ge 0,\ k>0).
\tag{1}
]
This problem has been considered by a number of authors (see ((^1)), references on pp. 80, 83, 113, 118, 129; see also the later works ((^{2-5}))).
The purpose of this note is to investigate the asymptotic behavior, as (t\to\infty), of the solution of the boundary-value problem (1).
It is of interest to obtain a method for investigating the asymptotic behavior of solutions of a broader class of boundary-value problems, including (1) as a special case.
Let (y(t)) ((0\le t<\infty)) be a solution of the boundary-value problem (1).
Lemma 1.
[
y'(t)<k \qquad (0\le t<\infty).
]
Proof. Suppose that the assertion is false and that (\tau) is the first root of the equation (y'(t)=k). Since (y'(t)<k) ((0\le t<\tau)), we have (y''(\tau)\ge 0). If (y''(\tau)=0), then, by the uniqueness theorem for the solution of a differential equation with prescribed initial conditions,
[
y(t)\equiv y(\tau)+(t-\tau)k,
]
i.e. (y'(0)=k), which is impossible. Hence (y''(\tau)>0), i.e. (y'(t)) increases at the point (\tau). But (y'(\infty)=k), and therefore (y'(t)) attains, at some point (t_1>\tau), its greatest value on the interval ([\tau,\infty)). At the point (t_1) we have (y''=0), (y'''\le 0); but (y'(t_1)>k), whence
[
y'''(t_1)=-2\beta\bigl(k^2-[y'(t_1)]^2\bigr)>0
]
if (\beta>0). If, however, (\beta=0), then, as above,
[
y(t)\equiv y(t_1)+(t-t_1)y'(t_1),
]
i.e. (y'(0)=y'(t_1)>0).
The contradiction obtained in all cases proves Lemma 1.
We transform the boundary-value problem (1) to a form more convenient for further consideration. Let (t_0\ge 0) be the last root of the equation (y'(t)=0). Then for (t\ge t_0) the function (y(t)) increases, and therefore (t), and hence also (y'(t)), is a function of (y). Making the substitution of the unknown function
[
v=k^2-[y'(t)]^2=v(y)\qquad (y_0=y(t_0)\le y<\infty),
]
we obtain the boundary-value problem
[
\sqrt{k^2-v(y)}\,v''(y)+2y v'(y)-4\beta v(y)=0
\quad (y_0<y<\infty),
]
[
v(y_0)=k^2,\qquad v(\infty)=0,\qquad 0<v(y)<k^2
\quad (y_0<y<\infty).
\tag{2}
]
Lemma 2.
[
v'(y)<0 \qquad (y_0<y<\infty).
]
Proof. At the roots of the equation (v'(y)=0)
[
v''(y)=\frac{4\beta v}{\sqrt{k^2-v}}>0
\qquad (\text{for } \beta>0),
]
i.e., all stationary points of (v) must be minima of (v(y)), which is impossible; in the case (\beta=0), with (v'(y_1)=0) ((y_1>y_0)), we would have (v(y)\equiv v(y_1)), which is also impossible. Lemma 2 is proved.
Lemma 3. The function (v''(y)) has, for (y>y_0), at most one zero.
Proof. From relations (2) it is clear that for (y>y_0) the derivative (v'''(y)) exists and
[
\sqrt{k^2-v}\,v'''-\frac{1}{2\sqrt{k^2-v}}\,v''v' + 2y v''+2(1-2\beta)v'=0.
\tag{3}
]
Hence, if (\beta\ne \tfrac12), then at the points where (v''(y)=0),
[
(2\beta-1)v'''=\frac{2(2\beta-1)^2}{\sqrt{k^2-v}}\,v'<0
]
(by Lemma 2), i.e. (v''') has one and the same sign. Our assertion follows from this.
Let now (\beta=\tfrac12). Then, if (v''(y_1)=0) ((y_1>y_0)), by the uniqueness theorem for a solution applied to equation (3),
[
v(y)\equiv v(y_1)+(y-y_1)v'(y_1)\qquad (y_0<y<\infty).
]
Since (v(\infty)=0), we obtain (v'(y_1)=0), which contradicts Lemma 2. Lemma 3 is proved.
Corollary.
[
v'(\infty)=0.
]
This relation, in view of the convergence of the integral (\displaystyle \int^\infty v'(y)\,dy), follows at once from Lemma 3.
Lemma 4.
[
v''(\infty)=0.
]
Proof. Since (v'(y)<0) ((y_0<y<\infty)), (v'(\infty)=0), and for some (y^>y_0) the function (v''(y)) ((y\ge y^)) has no zeros, it follows that (v''(y)>0) ((y\ge y^*)). It remains to prove that the set of all zeros of the function (v'''(y)) is bounded above. Suppose this is not so; then, at points where (v'''(y)=0), as a result of differentiation we obtain
[
\sqrt{k^2-v}\,v^{\mathrm{IV}}
=
v''\left[
\frac{1}{4\sqrt{(k^2-v)^3}}\,v'^2
+
\frac{1}{2\sqrt{k^2-v}}\,v''
+
4(\beta-1)
\right].
\tag{4}
]
If (\beta\ge 1), then at these points, for (y\ge y^*), we have (v^{\mathrm{IV}}(y)>0), whence a contradiction easily follows.
If (0\le\beta<1) and (v''(y)\not\to 0) (as (y) tends to (\infty)), choose a number
(\varepsilon\in\bigl(0,\min{8k(1-\beta),\overline{\lim}_{y\to\infty}|v''(y)|}\bigr)), and denote by (F) the closed set of points on the ray ([y^*,\infty)) at which (v''(y)\ge\varepsilon), and by (E) its complement. Since (\displaystyle \int^\infty v''(y)\,dy<\infty), the set (E) (as also (F)) is unbounded. Take any of the intervals of which (E) consists. At its endpoints (v''(y)=\varepsilon), and inside it (v''(y)<\varepsilon). Hence, on this interval, at some point (\eta), a minimum of (v''(y)) is attained. Take a sequence of points (\eta_i) for which (\lim \eta_i=\infty); for sufficiently large (i), at the points (\eta_i),
[
\frac{1}{2\sqrt{k^2-v}}\,v''
+
\frac{1}{4\sqrt{(k^2-v)^3}}\,v'^2
<
\frac{\varepsilon}{2k}+o(1)<4(1-\beta),
]
and from (4) we obtain (v^{\mathrm{IV}}(\eta_i)<0). But this contradicts the fact that at the points (\eta_i) a minimum of (v''(y)) is attained. Lemma 4 is proved.
Corollary.
[
\lim_{y\to\infty} yv'(y)=0.
]
This relation follows immediately from equation (2) and the boundary conditions.
Lemma 5. If (\beta>0), then
[
\lim_{y\to\infty}\frac{yv'(y)}{v(y)}=-\infty .
]
Proof. Let (yv'v^{-1}=z). Then, with the aid of (2), we obtain
[
z=\frac{4\beta y}{\sqrt{k^2-v}}+z\left(\frac1y-\frac{2y}{\sqrt{k^2-v}}-\frac{z}{y}\right).
\tag{5}
]
If for some (y>y_0) we have (z'(y)=0), then for this (y)
[
z''=4\beta\frac{2(k^2-v)+yv'}{2(k^2-v)^{3/2}}
+z\left(-\frac1{y^2}-\frac{2(k^2-v)+yv'}{(k^2-v)^{3/2}}+\frac{z}{y^2}\right).
]
In view of the corollary to Lemma 4 and the fact that (z<0) ((y\ge y_0)), for sufficiently large (y), at the stationary points of (z(y)) we shall have (z''(y)>0). Hence it follows that (z(y)), beginning with some (y), is a monotone function.
Suppose that (z(\infty)=c>-\infty) ((c\le 0)). From relation (5) we obtain (z'(\infty)=\infty), which is impossible. Thus (z(\infty)=-\infty), as was required to prove.
Theorem. The solution of the boundary-value problem (1) and its first and second derivatives have, as (t\to\infty), the following asymptotic representation:
[
y(t)=kt-C+t^{-2-2\beta+o(1)}e^{-kt^2+2Ct},
]
[
y'(t)=k-t^{-1-2\beta+o(1)}e^{-kt^2+2Ct},
]
[
y''(t)=t^{-2\beta+o(1)}e^{-kt^2+2Ct},
]
where (C>0) is a certain constant (equal to (\displaystyle \lim_{t\to\infty}(kt-y(t)))).
Proof. From equation (2) we obtain
[
\frac{v''(y)}{v'(y)}
=
-\frac{2y}{\sqrt{k^2-v}}
+\frac{4\beta v}{v'\sqrt{k^2-v}}
=
-\frac{2y}{k}+4yu(y),
]
where, by Lemma 5, (u(y)\to0) as (y\to\infty). Hence, integrating over the interval from some (y_1>y_0), we obtain
[
v'(y)=-C_1 e^{-\frac{y^2}{k}+4\int_{y_1}^{y}su(s)\,ds}
\equiv -C_1\chi(y)
\qquad (C_1>0;\ y>y_0)
\tag{6}
]
and, consequently,
[
\lim_{y\to\infty}\frac{yv}{v'}
=
\lim_{y\to\infty}
\left(
-\frac{\displaystyle\int_y^\infty \chi(s)\,ds}{y^{-1}\chi(y)}
\right)
=
-\frac{k}{2}
\tag{7}
]
(the limit is easily found by l’Hospital’s rule),
[
v=\frac{C_2+o(1)}{y}\chi(y)
\qquad
\left(y_0<y<\infty;\ C_2=\frac{k}{2}C_1\right).
\tag{8}
]
Next, by l’Hospital’s rule we have
[
\lim_{y\to\infty}y^2\left(k-\sqrt{k^2-v}\right)
=
\lim_{y\to\infty}
\frac{v'/2\sqrt{k^2-v}}{-2y^{-3}}
=
\lim_{y\to\infty}
\frac{-C_1y^3\chi(y)}{-4\sqrt{k^2-v}}
=0,
]
whence, in view of (7), (\displaystyle \lim_{y\to\infty}y^2u(y)=-\frac{\beta}{2}). Therefore, by l’Hospital’s rule,
[
\lim_{y\to\infty}
\frac{\displaystyle\int_{y_1}^{y}su(s)\,ds}{\ln y}
=
\lim_{y\to\infty}y^2u(y)
=
-\frac{\beta}{2}.
\tag{9}
]
Therefore, by virtue of (8),
[
\sqrt{k^2-v}=k-y^{-1-2\beta+o(1)} e^{-\frac{y^2}{k}} .
\tag{10}
]
Thus, first of all, since (\sqrt{k^2-v}=y'(t)), we have (y(t)\sim kt), and on the basis of (10)
[
y'(t)=k-e^{-kt^2+o(t^2)}\qquad (t\ge t_0).
]
Integrating, we obtain
[
y=kt-C+\int_t^\infty e^{-k\tau^2+o(\tau^2)}\,d\tau
\qquad
\left(
C=kt_0-y_0+\int_{t_0}^\infty e^{-k\tau^2+o(\tau^2)}\,d\tau
\right).
]
Put, for (t\ge t_0),
[
\int_t^\infty e^{-k\tau^2+o(\tau^2)}\,d\tau
=
e^{-kt^2+\gamma(t)t^2}.
]
Then it is easy to show that (\gamma(\infty)=0). Indeed, by l’Hospital’s rule,
[
\frac{\displaystyle \int_t^\infty e^{-\alpha \tau^2}\,d\tau}
{e^{-\alpha t^2}t^{-1}}
\underset{t\to\infty}{\longrightarrow}
\frac{1}{2\alpha}
\qquad (\alpha>0);
]
but for any (\varepsilon>0) ((\varepsilon<k)), for sufficiently large (t),
[
e^{-(k+\varepsilon)t^2}
<
e^{-kt^2+o(t^2)}
<
e^{-(k-\varepsilon)t^2},
]
and therefore
[
t^{-1}e^{-(k+\varepsilon)t^2}
\left(\frac{1}{2(k+\varepsilon)}+o(1)\right)
<
\int_t^\infty e^{-k\tau^2+o(\tau^2)}\,d\tau
<
t^{-1}e^{-(k-\varepsilon)t^2}
\left(\frac{1}{2(k-\varepsilon)}+o(1)\right),
]
whence our assertion follows at once. Thus,
[
y=kt-C+e^{-kt^2+o(t^2)} .
\tag{11}
]
Hence (C=\lim_{t\to\infty}(kt-y)); from Lemma 1 it follows that the difference (kt-y(t)), for (0\le t<\infty), increases, and, since (y(0)=0), (C>0).
Substituting (11) into (10), we obtain the expression for (y'(t)) given in the formulation of the theorem. Further, for (t>t_0), according to (6),
[
y''(t)=\frac{-v'(y)}{2}
=
\frac{C_1}{2}
e^{-\frac{y^2}{k}+4\int_{y_1}^{y} s u(s)\,ds}
=
y^{-2\beta+o(1)} e^{-\frac{y^2}{k}}
]
(see (9)). Using (11), we obtain the required expression for (y''(t)).
Finally, integrating the expression for (y'(t)) and taking into account that
[
\int_t^\infty \tau^{-1-2\beta+o(1)} e^{-k\tau^2+C\tau}\,d\tau
=
t^{-2-2\beta+o(1)} e^{-kt^2+2Ct}
]
(the proof is analogous to the proof that (\gamma(\infty)=0)), and then comparing the result obtained with (11), we obtain the required representation of (y(t)).
Let us note in conclusion that the value of the constant (C) is unknown. It would be important to obtain an estimate, or a method for approximate computation, or an expansion in a series of some kind, etc., for this constant.
Received
18 VI 1956
CITED LITERATURE
- G. Shlikhting, Boundary-Layer Theory, IL, 1956.
- R. Iglish, Zs. angew. Math. u. Mech., 34, No. 12, 441 (1954).
- S. Furiya, Comment. Math. Univ. St. Paul, 1, 81 (1953).
- R. Iglish, Zs. angew. Math. u. Mech., 33, No. 4, 143 (1953).
- K. Stewartson, Proc. Cambr. Phil. Soc., 50, No. 3, 454 (1954).