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UDC 517.946
MATHEMATICS
T. D. DZHURAEV
ON A SYSTEM OF BOUNDARY-LAYER EQUATIONS FOR AN AXISYMMETRIC STATIONARY FLOW OF A COMPRESSIBLE FLUID
(Presented by Academician I. G. Petrovskii on 14 II 1969)
The question of the existence, uniqueness, and stability of solutions of the system of boundary-layer equations for stationary and nonstationary flows of an incompressible fluid has been studied in the works of O. A. Oleinik (see (¹)). In (²) the problem of continuation of the boundary layer for a stationary flow of a compressible fluid is considered under the assumption that the Prandtl number \(\sigma\) is equal to one.
In the present note we give results concerning the existence and uniqueness of the solution of the system of boundary-layer equations for a stationary axisymmetric flow of a compressible fluid for \(\sigma=1\).
For \(\sigma=1\) there exists a quadratic relation between the enthalpy and the longitudinal component of velocity (³). Therefore we consider the dynamical system of equations
\[
\rho u \frac{\partial u}{\partial x}+\rho \vartheta \frac{\partial u}{\partial y}
=
-\frac{dp}{dx}
+
\frac{\partial}{\partial y}\left(\mu \frac{\partial u}{\partial y}\right),
\]
\[
\frac{\partial(\rho r_0 u)}{\partial x}
+
\frac{\partial(\rho r_0 \vartheta)}{\partial y}
=0
\tag{1}
\]
in the domain \(D_x\{0\le x\le X,\ 0\le y<\infty\}\) under the conditions
\[ u\big|_{x=0}=0,\qquad u\big|_{y=0}=0,\qquad \vartheta\big|_{y=0}=\vartheta_0(x),\qquad \lim_{y\to\infty}u(x,y)=u_l(x). \tag{2} \]
The function \(r_0(x)\) determines the surface of the body being flowed around, \(r_0(0)=0\), \(r_0'(0)\ne0\), \(r_0(x)>0\) for \(x>0\); \(u_l(x)\) is the longitudinal component of the velocity at the outer boundary of the boundary layer, \(u_l(0)=0\), \(u_l(x)>0\) for \(x>0\), \(u_l'(x)>0\) for \(x\ge0\). In the case under consideration we have
\[
\mu=\mu_0 f\left(1-\frac{u^2}{2h_0}\right),\qquad
\rho=\rho_0\left(1-\frac{u_l^2}{2h_0}\right)^{k/(k-1)}
\left(1-\frac{u^2}{2h_0}\right)^{-1},
\tag{3}
\]
\[
p=p_0\left(1-\frac{u_l^2}{2h_0}\right)^{k/(k-1)},
\]
where \(f\) is a certain prescribed positive function; \(k,h_0,\mu_0,\rho_0,p_0\) are certain positive constants (see (³)).
We shall study problem (1), (2) by the methods developed in (¹). If we introduce new independent variables
\[ \xi=x,\qquad \eta=u(x,y)/u_l(x), \tag{4} \]
then for the function \(W=\mu u_y/u_l\) we obtain the equation
\[ W^2 W_{\eta\eta}-\eta A W_\xi+B W_\eta+CW=0 \tag{5} \]
in the domain \(G_x\{0\le x\le X,\ 0\le \eta<1\}\) with the conditions
\[ W\big|_{\eta=1}=0,\qquad \left(WW_\eta-DW+E\right)\big|_{\eta=0}=0, \tag{6} \]
where \(A(\xi,\eta)=a(\xi,\eta)u_l\),
\[ a(\xi,\eta)\equiv \mu\rho =\mu_0\rho_0(1-\alpha_l^2)^{k/(k-1)} \frac{f(1-\alpha_l^2\eta^2)}{1-\alpha_l^2\eta^2}, \]
\[ B(\xi,\eta)=a\frac{\eta^2-1}{1-\alpha_l^2}\,u_l', \qquad C(\xi,\eta)=\eta A\frac{r_0'}{r_0} +\eta\frac{\partial A}{\partial \xi} -\frac{\partial B}{\partial \eta}, \qquad \alpha_l=\frac{u_l}{\sqrt{2h_0}}, \]
\[ D=\rho_0(1-\alpha_l^2)^{k/(k-1)}v_0(\xi), \qquad E=\rho_0\mu_0 f(1)(1-\alpha_l^2)^{1/(k-1)}u_l'. \]
The boundary condition (6) for \(\eta=0\) means that the first equation of the system (1) is satisfied for \(y=0\), while the condition for \(\eta=1\) ensures fulfillment of the condition \(u\to u_l\) as \(y\to\infty\).
Equation (5), under the conditions of the problem on continuation of the boundary layer, was considered in (2). It is proved there that equation (5) has a solution in the class of functions \(W\) possessing bounded first derivatives in \(G_X\). We note that problem (5), (6) has no solution in the indicated class for arbitrarily smooth data. Therefore, below we shall establish existence and uniqueness theorems for the solution of problem (5), (6) in another class of functions, namely in the class of functions possessing certain singularities at \(\eta=1\) (see (1)).
Let \(\varphi^k(\eta)\equiv \varphi(kh,\eta)\). Following O. A. Oleinik, we replace equation (5) with conditions (6) by the system of ordinary differential equations
\[ (W^k)^2W_{\eta\eta}^k -\bigl(\eta A^k+\mu_k h\bigr)(W^k-W^{k-1})/h +B^kW_\eta^k+C^kW^k=0, \tag{7} \]
\[ 0\leq \eta<1,\qquad k=0,1,2,\ldots,[X/h] \]
with the conditions
\[ W^{(k)}(1)=0,\qquad \left.(W^kW_\eta^k-D^kW^k+E^k)\right|_{\eta=0}=0, \tag{8} \]
where \(\mu_k\) is a certain positive constant for \(k\geq 1\), determined by the data of the problem, and \(\mu_0=0\).
We shall assume that the functions \(f,u_l,r_0\) are twice continuously differentiable, and that \(v_0(x)\) has a bounded derivative; moreover,
\(|u_l''|\leq N_1x,\ v_0(x)\leq N_2x,\ N_i=\mathrm{const}>0\).
Lemma 1. The problem (7), (8) has a solution \(W^k(\eta)\), continuous for \(0\leq\eta\leq1\) and possessing, for \(0\leq\eta<1\), derivatives up to and including the third order. For this solution \(W^k(\eta)\) the estimate holds
\[ M_1(1-\eta)\sigma\leq W^k(\eta)\leq M_2(1-\eta)\sigma, \qquad \sigma=\sqrt{-\ln\mu(1-\eta)} \tag{9} \]
for \(kh\leq X\), where \(M_1,M_2,1>\mu\) are positive constants independent of \(h\).
Lemma 2. For \(kh\leq X_1\), where \(X_1>0\) depends on \(f,u_l,r_0,v_0\), the following estimates are valid for the solution \(W^k(\eta)\) of problem (7), (8):
\[ -M_3\sigma\leq W_\eta^k\leq M_4\sigma, \]
\[ \frac{1}{h}\left|W^k-W^{k-1}\right| \leq M_5(1-\eta)\sigma, \tag{10} \]
\[ W^kW_{\eta\eta}^k\leq -M_6, \qquad \left|W^kW_{\eta\eta\eta}^k\right|\leq M_7, \]
where \(\sigma\) is the same as in Lemma 1, and \(M_i\) are positive constants independent of \(h\).
Theorem 1. Suppose that the assumptions stated above concerning the functions \(f,u_l,r_0,v_0\) are fulfilled. Then in the domain \(G_{X_1}\{0\leq\xi\leq X_1,\ 0\leq\eta<1\}\) there exists a unique solution of problem (5), (6) having the following properties: \(W(\xi,\eta)\) is continuous in \(\overline{G}_{X_1}\) and \(\hat M_1(1-\eta)\sigma\leq W\leq \hat M_2(1-\eta)\sigma\); \(W_\eta\) is continuous in \(\eta\) for \(\eta<1\), \(-M_3\sigma\leq W_\eta\leq -M_4\sigma\) in \(G_{X_1}\); \(|W_\xi|\leq M_5(1-\eta)\sigma\); \(WW_{\eta\eta}<0\) and \(WW_{\eta\eta\eta}\) is bounded in \(\widehat G_{X_1}\). In any closed domain lying strictly inside \(G_{X_1}\), \(W\) and its derivatives, in-
entering equation (5) satisfy the Hölder condition. The function \(W\) satisfies equation (5) at all interior points of \(G_{X_1}\), and for \(0 \le \xi \le X_1\) the conditions (6) are fulfilled.
The uniqueness of the solution \(W\) of problem (5), (6) follows from the maximum principle applied to the equation for the difference of two solutions. The functions \(W^h(\eta)\), which are solutions of equations (7) with conditions (8), form, by virtue of Lemmas 1 and 2, a compact family in the sense of uniform convergence. By virtue of the estimates (9), (10), the limiting (as \(h \to 0\)) function \(W\) is continuous in \(G_{X_1}\) and has the properties indicated in Theorem 1.
Since at the interior points of \(G_{X_1}\) equation (5) is parabolic, the derivatives \(W_\xi, W_\eta, W_{\eta\eta}\) satisfy the Hölder condition in any subdomain \(G_{X_1}\), and inside \(G_{X_1}\), \(W\) satisfies equation (5). By virtue of the boundedness of \(W_{\eta\eta}^h\) for \(\eta < 1\), the derivative \(W_\eta\) satisfies the Lipschitz condition with respect to \(\eta\) for \(\eta < 1\), and condition (6) is fulfilled for \(\eta = 0\). As a consequence of Theorem 1 we obtain the following assertion.
Theorem 2. For \(x \le X_1\) in the domain \(D_{X_1}\) there exists a unique solution of problem (1), (2) possessing the following properties: \(u/u_1\), \(u_y/u_1\) are bounded and continuous in \(D_{X_1}\), \(u>0\) for \(y>0\) and \(x>0\), \(u(x,y) \to u_1(x)\) as \(y \to \infty\); \(u_y/u_1>0\) for \(y \ge 0\), \(u_y/u_1 \to 0\) as \(y \to \infty\); \(u_x, v_y, u_{yy}\) are bounded and continuous with respect to \(y\) in \(D_{X_1}\) and continuous with respect to \(x\) and \(y\) at the interior points of \(D_{X_1}\); \(v(x,y)\) is continuous with respect to \(y\) in \(D_{X_1}\), with respect to \(x\) and \(y\) at the interior points of \(D_{X_1}\), and is bounded for bounded \(y\); the derivative \(u_{yyy}\) is bounded in \(D_{X_1}\), \(u_{xy}\) is bounded for bounded \(y\); \(u_{xy}\) and \(u_{yyy}\) are continuous at the interior points of \(D_{X_1}\); \(u_{yy}/u_y\) is continuous with respect to \(y\) in \(D_{X_1}\). The functions \(u, v\) satisfy inside \(D_{X_1}\) equations (1) and \(u|_{x=0}=u|_{y=0}=0\); \(v|_{y=0}=v_0(x)\).
Received
7 II 1969
CITED LITERATURE
- O. A. Oleinik, Uspekhi Mat. Nauk, 23, no. 3, 3 (1968).
- T. D. Dzhuraev, Izv. AN UzSSR, Ser. Phys.-Math. Sci., No. 6 (1968).
- L. G. Loitsyanskii, Laminar Boundary Layer. Moscow, 1962.