HYDROMECHANICS

A. A. ZAITSEV

ON THE QUESTION OF THE STABILITY OF A VISCOUS FILM ON A SOLID BODY IN A GAS FLOW

(Presented by Academician I. I. Artobolevskii, 5 XI 1959)

§ 1. Statement of the problem. On the plane (y=0) (Fig. 1) there flows a layer (0 \leqslant y \leqslant h) of a viscous heavy liquid with a linear velocity distribution

[
U=\frac{U_0}{h}y,\qquad V=0.
\tag{1}
]

In the half-space (y>h) there flows a viscous heavy gas with velocity

[
U_1=U_1(y),\qquad V_1=0.
\tag{2}
]

Fig. 1

All parameters of the flow under study do not depend on the coordinate (x). We shall denote parameters pertaining to the gas by the subscript 1, leaving without a subscript the parameters pertaining to the liquid layer. The pressure in the liquid and in the gas is distributed as follows:

[
P=P_0-\rho g(y-h),\qquad
P_1=P_0-\int_0^y \rho_1(y)g\,dy,
\tag{3}
]

where (\rho) and (\rho_1) are the densities of the liquid and gas; (g) is the acceleration of gravity. On the interface (y=h) the conditions

[
U_1(h)=U_0,\qquad
\mu_1\frac{dU_1}{dy}=\mu\frac{U_0}{h},
\tag{4}
]

are satisfied, where (\mu) and (\mu_1) are the corresponding viscosity coefficients.

We superpose small disturbances on the basic flow

[
u=U+u',\qquad u_1=U_1+u_1',\qquad v=v',\qquad v_1=v_1',\qquad p=P+p',\qquad p_1=P_1+p_1',
\tag{5}
]

whereupon the former interface (y=h) assumes the form

[
y=h+h'(x,t).
\tag{6}
]

On the interface we require the relations

[
u'=u_1',\qquad v'=v_1',\qquad
v'=\frac{\partial h'}{\partial t}+U_0\frac{\partial h'}{\partial x},\qquad
p_\tau=p_{\tau1},\qquad
p-p_{n1}=K\frac{\partial^2 h'}{\partial x^2},
\tag{7}
]

where (p_\tau) are the tangential stresses, and (p_n) the normal stresses on the interface; (K) is the coefficient of surface tension.

Introducing dimensionless parameters and the stream function (\psi),

[
\varepsilon=\mu_1/\mu,\quad \sigma=\rho_1/\rho,\quad T=K\rho h/\mu^2,
]
[
G=\rho^2 g h^3/\mu^2,\quad \operatorname{Re}=U_0 h\rho/\mu,\quad N=\mu\rho_1/\mu_1\rho=\sigma/\varepsilon
\tag{8}
]

[
u'=\partial\psi/\partial y,\quad v'=-\partial\psi/\partial x,
\tag{9}
]

we consider a particular form of the disturbances

[
h'=\delta e^{i\alpha(x-ct)},\quad \psi=\delta f(y)e^{i\alpha(x-ct)},\quad u_1'=\delta u_1(y)e^{i\alpha(x-ct)},
\tag{10}
]

[
v'=\delta U_1(y)e^{i\alpha(x-ct)},\quad p'=\delta p(y)e^{i\alpha(x-ct)},\quad p_1'=\delta p(y)e^{i\alpha(x-ct)},
]

where (x) and (y) are referred to (h); (t) to (h/U_0); (u') and (v') to (U_0); (p) to (\mu U_0/h).

Using the linearized Navier—Stokes equations and all the preceding assumptions, we reduce the boundary conditions (7) to the form

[
f'(1)=u_1(1),\quad -i\alpha f(1)=v_1(1),
\tag{11}
]

[
f(1)=c-1,
\tag{12}
]

[
f''(1)+\alpha^2(c-1)=\varepsilon\,[u_1'(1)+\alpha^2(c-1)],
]

[
f'''(1)-3\alpha^2 f'(1)+i\alpha \operatorname{Re}(c-1)[f'(1)+1-N]
+\varepsilon[2\alpha^2 f'(1)-
]

[
-u_1''(1)+i\alpha v_1'(1)]-\sigma i\alpha \operatorname{Re}(c-1)f'(1)
=i\alpha [G(1-\sigma)+\alpha^2 T]\frac{1}{\operatorname{Re}}.
\tag{13}
]

If now (\varepsilon) and (\sigma) are made to tend to zero in such a way that the limit
(\displaystyle \lim_{\varepsilon\to0,\ \sigma\to0}(\sigma/\varepsilon)=N) remains some finite quantity, then the boundary conditions (13) will not contain parameters characterizing the motion of the gas:

[
f''(1)=-\alpha^2(c-1),
]

[
f'''(1)-3\alpha^2 f'(1)+i\alpha \operatorname{Re}(c-1)[f'(1)+1-N]
=i\alpha (G+\alpha^2 T)\frac{1}{\operatorname{Re}}.
\tag{14}
]

The latter expressions may be regarded as the result of expanding the quantities entering into conditions (13) in powers of the small parameters (\varepsilon) and (\sigma), retaining only the first terms of the expansion.

At the solid wall we take the no-slip condition for the liquid

[
f(0)=f'(0)=0.
\tag{15}
]

Thus, the problem of the stability of a thin liquid film on a body in a gas stream, for sufficiently small (\varepsilon) and (\sigma), reduces to the study of a particular form of the Orr—Sommerfeld equation ((^1))

[
f^{\mathrm{IV}}(y)-2\alpha^2 f''(y)+\alpha^4 f(y)
-i\alpha \operatorname{Re}(y-c)[f''(y)-\alpha^2 f(y)]=0
\tag{16}
]

with boundary conditions (12), (14), (15).

§ 2. Characteristic equation. We shall seek the solution of equation (16) in the form of a series in powers of the parameter (\lambda=\alpha\operatorname{Re})

[
f(y)=\sum_{n=0}^{\infty}\lambda^n f_n(y).
\tag{17}
]

Substituting (17) into equation (16) and equating coefficients of equal powers of (\lambda), we obtain the system of differential equations

[
f_n^{\mathrm{IV}}-2\alpha^2 f_n''+\alpha^4 f_n=F_n,\quad n=0,1,2,\ldots,
\tag{18}
]

where

[
F_0=0,\quad F_n=i(y-c)[f_{n-1}''-\alpha^2 f_{n-1}],\quad n=1,2,3,\ldots
\tag{19}
]

This system is easily integrated in elementary functions successively, starting with (n=0). In order that the function (f(y)) satisfy the boundary conditions (12), (14), (15), it is sufficient to require the fulfillment of the following-

following relations for the solution of the system of equations (18), (19):

[
f_n(0)=f'n(0)=0,\quad n=0,1,2,\ldots,\quad
f_0(1)=c-1,\quad f''_0(1)=-\alpha^2(c-1);
]
[
f_n(1)=f'_n(1)=0,\quad n=1,2,3,\ldots;
\tag{20}
]
[
f'''_0(1)-3\alpha^2 f'_0(1)-i\alpha\,(G+\alpha^2T)\frac{1}{\operatorname{Re}}
+i\lambda(c-1)(1-N)+
]
[
+\sum\lambda^n}^{\infty
\left[f'''n(1)-3\alpha^2 f'_n(1)+i(c-1)f'(1)\right]=0.
\tag{21}
]

The general solution of the system of differential equations (18), (19) is

[
f_n(y)=\Phi_{n1}(y)\operatorname{sh}\alpha y+\Phi_{n2}(y)\operatorname{ch}\alpha y
+\Phi_{n3}(y)y\operatorname{sh}\alpha y+\Phi_{n4}(y)y\operatorname{ch}\alpha y+
]
[
+C_{n1}\operatorname{sh}\alpha y+C_{n2}\operatorname{ch}\alpha y
+C_{n3}y\operatorname{sh}\alpha y+C_{n4}y\operatorname{ch}\alpha y,
\tag{22}
]

where

[
\Phi_{n1}=-\frac{1}{2\alpha^3}\int_0^y
F_n(\eta)(\operatorname{ch}\alpha\eta-\alpha\eta\operatorname{sh}\alpha\eta)\,d\eta,
]
[
\Phi_{n2}=\frac{1}{2\alpha^3}\int_0^y
F_n(\eta)(\operatorname{sh}\alpha\eta-\alpha\eta\operatorname{ch}\alpha\eta)\,d\eta,
\tag{23}
]
[
\Phi_{n3}=-\frac{1}{2\alpha^2}\int_0^y
F_n(\eta)\operatorname{sh}\alpha\eta\,d\eta,\quad
\Phi_{n4}=\frac{1}{2\alpha^2}\int_0^y
F_n(\eta)\operatorname{ch}\alpha\eta\,d\eta.
]

Fig. 2

(C_{nk}) are constants of integration; (F_n(\eta)) is determined by formulas (19).

Using the boundary conditions (20), we find the constants of integration (C_{nk}), (n=0,1,2,\ldots;\ k=1,2,3,4). Substituting the solution thus obtained, which contains no constants of integration, into relation (21), we find the characteristic equation. Neglecting in this equation the terms containing powers of (\lambda) higher than the second, we obtain the approximate characteristic equation

[
(c-1)\left[Q_{00}+\lambda^2(Q_{22}c^2+Q_{21}c+Q_{20})\right]+
]
[
+i\left[\lambda(c-1)(Q_{11}c+Q_{10}+1-N)
-\frac{\alpha}{\operatorname{Re}}(G+\alpha^2T)\right]=0,
\tag{24}
]

where (Q_{ik}) are real quantities depending only on (\alpha).

§ 3. Neutral curves, critical Reynolds number. Taking (c) to be real, we separate in equation (24) the real and imaginary parts; then we obtain the system of equations

[
(c-1)\left[Q_{00}+\alpha^2\operatorname{Re}^2(Q_{22}c^2+Q_{21}c+Q_{20})\right]=0,
\tag{25}
]
[
(c-1)(Q_{11}c+Q_{10}+1-N)\operatorname{Re}^2-(G+\alpha^2T)=0.
\tag{26}
]

If (G\equiv T\equiv0), then (25) and (26) are satisfied for (c=1) for arbitrary (\operatorname{Re}) and (\alpha), but in this case the imaginary part of (c) is identically equal to zero, whereas loss of stability occurs when the sign of the imaginary part of (c) changes. Therefore we shall exclude this case from consideration.

If (c \ne 1), then from equation (25) we find

[
\operatorname{Re}=\sqrt{-\frac{Q_{00}}{\alpha^2\left(Q_{22}c^2+Q_{21}c+Q_{20}\right)}} .
\tag{27}
]

Substituting (27) into (26), we obtain a quadratic equation with respect to (c)

[
c^2\left[Q_{00}Q_{11}+\alpha^2Q_{22}\left(G+\alpha^2T\right)\right]
+c\left[Q_{00}\left(Q_{10}+1-N-Q_{11}\right)+\alpha^2Q_{21}\left(G+\alpha^2T\right)\right]
]
[
{}-Q_{00}\left(Q_{10}+1-N\right)-\alpha^2Q_{20}\left(G+\alpha^2T\right)=0.
\tag{28}
]

The roots of this equation are (c_1) and (c_2). Substituting (c_1) and (c_2) into (27), we find, respectively, (\operatorname{Re}_1) and (\operatorname{Re}_2). The curves of the dependence of (\operatorname{Re}) on (\alpha) are customarily called

Fig. 3

neutral. In the case (G \ge 0), the neutral curves have the form shown in Fig. 2 ((N=0,\ T=500)). The minimum value (\min \operatorname{Re}(\alpha)=\operatorname{Re}{\mathrm{cr}}) is called the critical Reynolds number. In the case (G<0) we shall not have (\operatorname{Re}) without some additional condition, since in this case, in addition to the neutral curves of the former form, there is the straight line}

[
\alpha=\sqrt{-G/T},
]

and the region of instability will lie below this straight line. In the subsequent calculations we set (G=0). This is justified by the fact that (G) enters all formulas only in the form of the combination (G+\alpha^2T); thus, the case (G>0) can be obtained from the case (G=0) by a corresponding increase in (T).

Figure 3 gives the results of calculating (\operatorname{Re}{\mathrm{cr}}) as a function of the parameters (T) and (N) ((a—\operatorname{Re}}},\ b—\operatorname{Re{2\mathrm{cr}},\ c—\operatorname{Re})).}

The Reynolds number can be written in the form (\operatorname{Re}=\rho h^2\tau/\mu^2), where (\tau=\mu U_0/h) is the shear stress on the surface of the layer. The value of (\tau) can be calculated using boundary-layer theory in the gas, or found from experiment; then, knowing the critical Reynolds number, we can determine the thickness of the stable liquid layer on the body.

The calculations were carried out on the “Strela” electronic computer.

Received
31 X 1959

REFERENCES

1. Lin Chia-chiao, Theory of Hydrodynamic Stability, IL, 1958.