Full Text
UDC 517.945.7 : 532.526.2
HYDROMECHANICS
O. A. OLEINIK
ON A SYSTEM OF BOUNDARY-LAYER EQUATIONS FOR AXISYMMETRIC FLOWS
(Presented by Academician I. G. Petrovskii on May 7, 1967)
We shall consider the system of boundary-layer equations for axial stationary and nonstationary flow past bodies of revolution. For these systems, solutions will be constructed below by means of the method of lines; theorems on uniqueness and stability of solutions will be obtained; and the question of the behavior of solutions as time increases without bound will be investigated. In paper \((^1)\), by the method of lines, solutions were constructed for the system of boundary-layer equations for nonstationary axisymmetric and plane-parallel symmetric flows over some finite time interval. Problems concerning continuation of the boundary layer for stationary and nonstationary flows were studied in papers \((^{1-3})\). We note that the proof of the existence of a solution of the Prandtl system that will be indicated below contains an approximate method for constructing the solution and a proof of its convergence. The method of A. A. Dorodnitsyn for the approximate solution of the system of boundary-layer equations is set forth in \((^4,^5)\).
1. Stationary axisymmetric boundary layer
The system of boundary-layer equations for an axisymmetric three-dimensional stationary flow of an incompressible fluid in the corresponding coordinates has the form
\[ uu_x+vu_y=UU_x+\nu u_{yy}, \qquad (ru)_x+(rv)_y=0 \tag{1} \]
in the domain \(D_\theta\{0\le x\le \theta,\ 0\le y<\infty\}\), under the conditions
\[ u|_{x=0}=0,\qquad u|_{y=0}=0,\qquad v|_{y=0}=v_0(x),\qquad u\to U(x)\quad \text{as } y\to\infty . \tag{2} \]
The function \(r(x)\) determines the surface of the body being flowed around; \(r(0)=0;\ r_x(0)\ne 0;\ U\) is the velocity of the external flow; \(\nu\) is the coefficient of viscosity (see \((^6)\), p. 174).
If one introduces new independent variables
\[ \xi=x,\qquad \eta=u(x,y)/U(x), \tag{3} \]
then for the function \(w=u_y/U\) we obtain the equation
\[ \nu w^2 w_{\eta\eta}-\eta U w_\xi + A w_\eta + B w = 0 \tag{4} \]
in the domain \(G_\theta\{0\le \xi\le \theta,\ 0\le \eta\le 1\}\), with conditions
\[ w|_{\eta=1}=0,\qquad (\nu w w_\eta-v_0 w+C)|_{\eta=0}=0, \tag{5} \]
where \(A=(\eta^2-1)U_x,\ B=-\eta U_x+\eta r_x U/r,\ C=U_x\).
Using the method of lines, we shall prove, under certain natural assumptions, the existence and uniqueness of a solution of problem \((4),(5)\), which, through the transformation \((3)\), leads to a solution of problem \((1),(2)\). Let \(f^k(\eta)\) denote \(f(kh,\eta)\); \(h=\mathrm{const}>0;\ k=0,1,2,\ldots\). We replace equation \((4)\) with conditions \((5)\) by the system of ordinary differential equations
\[ \nu (w^k)^2 w^k_{\eta\eta} -\eta U^k (w^k-w^{k-1})/h +A^k w^k_\eta +B^k w^k=0, \tag{6} \]
\(0 \leqslant \eta \leqslant 1;\ k=0,1,\ldots,[\theta/h]\), with the conditions
\[ w^k(1)=0,\qquad \left(\nu w^k w_\eta^k-v_0^k w^k+C^k\right)\bigm|_{\eta=0}=0. \tag{7} \]
We shall assume that \(U\) and \(r\) are twice continuously differentiable functions; \(U(x)>0\), \(r(x)>0\) for \(x>0\); \(U=ax+b(x)\), where \(a=\mathrm{const}>0\); \(|b|\leqslant K_1x^2\); \(v_0\), \(v_{0x}\), \(B_\xi\) are bounded, \(U_x>0\). It is obvious that \(|B|\leqslant K_2\xi\); \(K_1,K_2=\mathrm{const}\).
Lemma 1. The system (6), (7) has a solution \(w^k(\eta)\), continuous for \(0\leqslant\eta\leqslant1\) and possessing all derivatives for \(\eta<1\).
Lemma 2. For the solution of the problem (6), (7) the estimate
\[ M(1-\eta)\bigl[|\ln(1-\eta)\beta_1|^{1/2}-M_0\bigr]e^{-\gamma_1kh} \leqslant w^k(\eta) \leqslant M(1-\eta)|\ln(1-\eta)\beta_2|^{1/2}e^{\gamma_2kh}, \tag{8} \]
holds, where \(M^2=4a\); \(\gamma_1,\gamma_2,M_0,\beta_1,\beta_2\) are certain positive constants independent of \(h\), and \(|\ln(1-\eta)\beta_1|^{1/2}-M_0>0\).
Lemma 3. For \(kh\leqslant x_0\), where \(x_0>0\) depends on \(U,r,v_0\), the following estimates are valid for the solution \(w^k(\eta)\) of the system (6), (7):
\[ -M_1|\ln(1-\eta)\beta_2|^{1/2} \leqslant w_\eta^k(\eta) \leqslant -M_2|\ln(1-\eta)\beta_1|^{1/2}, \tag{9} \]
\[ \left|(w^k-w^{k-1})/h\right| \leqslant M_3(1-\eta)|\ln(1-\eta)\beta_2|^{1/2}, \qquad |w^k w_{\eta\eta}^k|\leqslant M_4, \tag{10} \]
where the constants \(M_i\) are positive and do not depend on \(h\).
Estimates for \(w^k=z^k\) and \((w^k-w^{k-1})/h=r^k\) can be obtained by considering successively the equations
\[ \nu(w^k)^2 z_{\eta}^k+A^k z^k=\eta U^k r^k-B^k w^k, \tag{11} \]
\[ \nu(w^{k+1})^2 r_{\eta\eta}^{k+1} -\eta U^k(r^{k+1}-r^k)/h +A^{k+1}r_\eta^{k+1} +r^{k+1}\left[B^{k+1}-\eta(U^{k+1}-U^k)/h+\right. \]
\[ \left. +(w^{k+1}+w^k)(\eta U^k r^k-A^kz^k-B^k w^k)/(w^k)^2\right] +w_\eta^k(A^{k+1}-A^k)/h +w^k(B^{k+1}-B^k)/h=0 \tag{12} \]
for \(k=0\), then for \(k=1,2,\ldots,[\theta/h]\), and taking into account the boundary conditions:
\(r^{k+1}(1)=0\);
\(\nu r_\eta^{k+1}-C^{k+1}r^{k+1}/w^k w^{k+1}+(C^{k+1}-C^k)/w^k h+(v_0^{k+1}-v_0^k)/h=0\) at \(\eta=0\);
\(\nu z^k-v_0+C^k/w^k=0\) at \(\eta=0\), as well as the inequalities (8), from which estimates of the form (9) follow for \(z^k\) for some sequence of points \(\eta_n\) tending to \(\eta=1\) as \(n\to\infty\). Estimate (10) for \(w^k w_{\eta\eta}^k\) follows directly from equations (6) and the preceding inequalities. By Lemmas 1 and 2, the functions \(w^k\) form a compact family in the sense of uniform convergence. The limiting function \(w\) as \(h\to0\) is continuous in \(G_{x_0}\) and has a bounded derivative \(w_\xi\); the derivatives \(w_\eta\), \(w_{\eta\eta}\) are such that \(ww_\eta\), \(ww_{\eta\eta}\) are bounded. Since at the interior points of \(G_{x_0}\) equation (4) is parabolic, the derivatives \(w_\xi\), \(w_\eta\), \(w_{\eta\eta}\) satisfy the Hölder condition in any interior subdomain \(G_{x_0}\), and within \(G_{x_0}\) equation (4) is valid for \(w\). By virtue of the boundedness of \(w_{\eta\eta}^{\,k}\) for \(\eta<1-\delta\) and any \(\delta>0\), \(w_\eta\) satisfies a Lipschitz condition in \(\eta\) for \(\eta<1-\delta\), and condition (5) is fulfilled for \(\eta=0\). Hence we obtain the following assertion.
Theorem 1. If \(\theta\leqslant x_0\), then in \(D_\theta\) there exists a solution of the problem (1), (2) possessing the following properties: \(u,u_y\) are continuous and bounded in \(D_\theta\); \(u_{yy},u_x,v_y\) are bounded and continuous in \(y\) in \(D_\theta\) and continuous in \(x\) at the interior points of \(D_\theta\); \(v\) is continuous in \(y\) in \(D_\theta\), in \(x\) at the interior points of \(D_\theta\), and bounded for bounded \(y\); \(u_y\to0\) as \(y\to\infty\), \(u_y>0\) for \(0\leqslant y<\infty\); \(u_{yyy}\) is bounded in \(D_\theta\); \(u_{yx}\) is bounded in \(D_\theta\) for bounded \(y\); \(u_{yx},u_{yyy}\) are continuous at the interior points of \(D_\theta\). The solution of the problem (1), (2) with the indicated properties is unique.
2. Nonstationary axisymmetric boundary layer. The Prandtl system for an axisymmetric three-dimensional un-
of the steady flow (see (6))
\[ u_t+uu_x+vu_y=UU_x+U_t+\nu u_{yy},\qquad (ru)_x+(rv)_y=0 \tag{13} \]
in the domain \(Q_\theta\{0\leq t<\infty,\ 0\leq x\leq \theta,\ 0\leq y<\infty\}\), with the conditions
\[ u\big|_{t=0}=u_0(x,y),\qquad u\big|_{x=0}=0,\qquad u\big|_{y=0}=0,\qquad v\big|_{y=0}=v_0(t,x), \]
\[ u\to U(t,x)\quad \text{as } y\to\infty, \tag{14} \]
like problem (1), (2), by means of the transformation of the independent variables
\[ \tau=t,\qquad \xi=x,\qquad \eta=u(t,x,y)/U(t,x) \tag{15} \]
and the introduction of the new function \(w=u_y/U\), is reduced to the equation
\[ \nu w^2w_{\eta\eta}-w_\tau-\eta U w_\xi+\widetilde A w_\eta+\widetilde B w=0 \tag{16} \]
in the domain \(\Omega_\theta\{0\leq \tau<\infty,\ 0\leq \xi\leq \theta,\ 0\leq \eta\leq 1\}\), with the conditions
\[ w\big|_{\tau=0}=u_{0y}/U(0,x)\equiv w_0,\quad w\big|_{\eta=1}=0,\quad (\nu w w_\eta-v_0w+\widetilde C)\big|_{\eta=1}=0, \tag{17} \]
where
\[ \widetilde A=(\eta^2-1)U_x+(\eta-1)U_t/U,\qquad \widetilde B=\eta r_xU/r-U_t/U-\eta U_x, \]
\[ \widetilde C=U_x+U_t/U. \]
We shall assume that \(r(t,x)\) and \(U(t,x)\) are functions twice continuously differentiable in \(Q_\theta\); \(U>0\) and \(r>0\) for \(x>0\); \(U=ax+b(t,x)\), where \(a=\mathrm{const}>0\), \(|b|\leq K_3x^2\), \(K_3=\mathrm{const}\); \(r(t,0)=0\), \(r_x(t,0)\ne0\); \(v_0(t,0)=\mathrm{const}\); the functions \(U_x\), \(U_t/U\), \(r_xU/r\), \(v_0\) and their derivatives with respect to \(t\) and \(x\) are bounded in \(Q_\theta\), and the derivatives of these functions with respect to \(t\), as well as \(U_t/U\), do not exceed in absolute value \(K_4x\), where \(K_4=\mathrm{const}\).
Let \(f^{mk}(\eta)\equiv f(mh,kh,\eta)\). We replace equation (16) in the domain \(\Omega_\theta\), with conditions (17), by a system of ordinary differential equations
\[ \nu (w^{mk})^2 w_{\eta\eta}^{mk} -\frac{w^{mk}-w^{m-1\,k}}{h} -\eta U^{mk}\frac{w^{mk}-w^{mk-1}}{h} +\widetilde A^{mk}w_\eta^{mk} +\widetilde B^{mk}w^{mk}=0, \tag{18} \]
\[ 0\leq \eta\leq 1;\qquad m=1,2,\ldots;\qquad k=0,1,\ldots,[\theta/h];\qquad h=\mathrm{const}>0, \]
with the conditions
\[ w^{mk}(1)=0,\qquad (\nu w^{mk}w_\eta^{mk}-v_0^{mk}w^{mk}+\widetilde C^{mk})\big|_{\eta=0}=0,\qquad w^{0k}=w_0(kh,\eta). \tag{19} \]
With respect to the function \(w_0(\xi,\eta)\) we shall assume the following: \(w_0\leq M(1-\eta)|\ln(1-\eta)\alpha_1|^{1/2}e^{\gamma\xi}\), where \(M^2=4a\); \(w_0\geq K_4(1-\eta)|\ln(1-\eta)\alpha_2|^{1/2}e^{-\gamma\xi}\), moreover \(K_4^2\geq 2a\), \(|w_{0\xi}|\leq K_5(1-\eta)|\ln(1-\eta)\alpha_2|^{1/2}\), if \(|\widetilde B_\xi|\leq K_6\), and \(K_4\) is an arbitrary sufficiently small constant, while \(|w_{0\xi}|\leq K_7(1-\eta)\xi|\ln(1-\eta)\alpha_2|^{1/2}\), if \(|\widetilde A_\xi|+|\widetilde B_\xi|+|\widetilde C_\xi|\leq K_8\xi\); let, in addition, \(w_{0\eta}\leq0\), \(w_{0\eta}\geq -K_9|\ln(1-\eta)\alpha_1|^{1/2}\),
\[ \left|\nu w_0^2w_{0\eta\eta}-\eta U(0,\xi)w_{0\xi} +\widetilde A(0,\xi,\eta)w_{0\eta} +\widetilde B(0,\xi,\eta)w_0\right| \leq K_{10}\xi(1-\eta)|\ln(1-\eta)\alpha_2|^{1/2} \]
and the compatibility condition is fulfilled
\[ \nu w_0(\xi,0)w_{0\eta}(\xi,0)-v_0(0,\xi)w_0(\xi,0)+\widetilde C(0,\xi) \equiv \nu u_{0yy}(x,0)+(UU_x+U_t)\big|_{t=0}-v_0(0,x)u_{0y}(x,0)=0. \]
Here \(K_i\), \(\alpha_i\), \(\gamma\) are certain positive constants. Obviously, all these conditions on \(w_0\) will be satisfied if, for example, \(u_0(x,y)\) is a solution of problem (1), (2), or coincides with it in a neighborhood of the straight line \(x=0\), and also for large \(y\), with \(U\), \(r\), \(v_0\) taken at \(t=0\). This means that \(w_0\) is a certain perturbation of the velocity \(u\) in the steady boundary layer.
Lemma 4. The system (18) with conditions (19) has a solution \(w^{mk}\), continuous for \(0\leq\eta\leq1\) and possessing all continuous derivatives for \(\eta<1\). For \(kh\leq \tilde x_0\) and \(0\leq mk<\infty\), the following estimates hold for \(w^{mk}\):
\[ M_5(1-\eta)|\ln(1-\eta)\alpha_4|^{1/2} \leq w^{mk}\leq M(1-\eta)|\ln(1-\eta)\alpha_3|^{1/2}e^{\gamma kh}, \]
\[ 0\geq w_\eta^{mk}\geq -M_6|\ln(1-\eta)\alpha_3|^{1/2},\qquad \left|\frac{w^{mk}-w^{m-1\,k}}{h}\right| \leq \]
\[ \leq M_7kh(1-\eta)|\ln(1-\eta)\alpha_4|^{1/2}, \tag{20} \]
\[ \left|\frac{w^{mk}-w^{mk-1}}{h}\right| \leq M_8(1-\eta)|\ln(1-\eta)\alpha_4|^{1/2}. \]
Here \(M_i\) and \(\alpha_i\) are certain positive constants independent of \(h\).
Theorem 2. Let \(U, r, v_0, u_0\) satisfy the conditions indicated above in Sec. 2. Then in the domain \(Q_\theta\), for \(\theta \leqslant \tilde{x}_0\), there exists a solution of problem (13), (14) having the following properties: \(u, u_y\) are continuous and bounded in \(Q_\theta\); \(u_{yy}, u_x, u_t, v_y\) are bounded and continuous with respect to \(y\) in \(Q_\theta\); \(v\) is continuous with respect to \(y\) and bounded for bounded \(y\); \(u_y \to 0\) as \(y \to \infty\), \(u_y > 0\) for \(y \geqslant 0\); \(u_{yyy}\) is bounded in \(Q_\theta\), \(u_{yx}, u_{yt}\) are bounded in \(Q_\theta\) for bounded \(y\). The solution of problem (13), (14) possessing these properties is unique.
3. Stability. Denote by \(\tilde{u}(t,x,y)\) the solution of problem (13), (14) for
\[
U=\tilde{U}(t,x),\quad r=\tilde{r}(t,x),\quad v_0=\tilde{v}_0(t,x)
\]
and \(u_0(x,y)\), and by \(u(x,y)\) the solution of problem (1), (2) corresponding to the functions \(U(x), r(x), v_0(x)\). Suppose that these functions satisfy the conditions of Theorems 2 and 1, respectively.
Theorem 3. Let the functions
\[
\tilde{U}-U,\quad \tilde{U}_x-U,\quad \tilde{U}_t/\tilde{U},\quad \tilde{v}_0-v_0,\quad \tilde{r}_x\tilde{U}/\tilde{r}-r_xU/r
\tag{21}
\]
tend to zero as \(t\to\infty\) uniformly in \(D_\theta\). Then for any \(\delta>0\) there exists a constant \(C_1(\delta)\) such that in \(D_\theta\)
\[
\left|\tilde{u}(t,x,y)/\tilde{U}(t,x)-u(x,y)/U(x)\right|\leqslant \delta+C_1(\delta)e^{-\sigma t},
\]
where \(\sigma=\mathrm{const}>0\) depends on the data of problem (1), (2), \(\theta=\min(x_0,\tilde{x}_0)\).
Theorem 4. If the functions (21) are equal to zero for \(t\geqslant t_0>0\), where \(t_0=\mathrm{const}<\infty\), then in \(D_\theta\), for \(y\leqslant y_0\),
\[
|\tilde{u}(t,x,y)-u(x,y)|\leqslant C_2U(x)e^{-\sigma t},
\]
where the constant \(C_2\) depends on \(y_0\), \(\theta=\min(x_0,\tilde{x}_0)\).
Theorem 5. Let the functions (21) not exceed \(\varepsilon>0\) in absolute value, and let
\[
|\tilde{w}(0,\xi,\eta)-w(\xi,\eta)|\leqslant \varepsilon,
\]
where \(w\) is the solution of problem (4), (5), and \(\tilde{w}\) is the solution of problem (16), (17), corresponding to the functions \(u\) and \(\tilde{u}\). Then
\[
\left|\tilde{u}(t,x,y)/\tilde{U}(t,x)-u(x,y)/U(x)\right|\leqslant C_3\varepsilon
\]
for all \(t\) in \(D_\theta\), where the constant \(C_3\) depends on the data of problem (1), (2) and on the constants entering into (20).
Let us note that, by an analogous method, one can investigate the boundary-layer equations for plane-parallel symmetric flows of an incompressible fluid. The question of the stability of solutions of the problem of the continuation of the boundary layer was considered in work (7).
Moscow State University
named after M. V. Lomonosov
Institute of Mechanics Problems
Academy of Sciences of the USSR
Received
6 V 1967
References Cited
- O. A. Oleinik, DAN, 176, No. 6 (1967).
- O. A. Oleinik, Zhurn. vychisl. mat. i matem. fiz., 3, No. 3, 489 (1963).
- O. A. Oleinik, PMM, 30, No. 5, 801 (1966).
- A. A. Dorodnitsyn, Zhurn. prikl. mekh. i tekhn. fiz., 1, No. 3, 111 (1960).
- O. M. Belotserkovskii, P. I. Chushkin, Zhurnal vychislit. matem. i matem. fiz., 2, No. 5, 731 (1962).
- G. Schlichting, Boundary-Layer Theory, IL, 1956.
- O. A. Oleinik, PMM, 30, No. 3, 417 (1966).