Hydromechanics

V. L. Danilov

On the Motion of the Interface Between Viscous Fluids in a Narrow Slot

(Presented by Academician P. Ya. Kochina, 17 X 1960)

The problem of the motion of the interface between two viscous immiscible fluids in a narrow slot between parallel plates is of interest in connection with the well-known analogy of this flow with laminar filtration in a homogeneous reservoir of constant thickness. In the work of P. Ya. Polubarinova-Kochina and A. R. Shkirich (\left({}^{1}\right)), the theoretical solution of the problem of contraction of the oil-bearing contour was first compared with an experiment on a horizontal slot tray. In connection with filtration in porous media, the motion of the interface in a slot was considered in later experimental and theoretical studies by foreign authors (\left({}^{2-8}\right)). However, in these works the influence of interfacial tension—which, as is shown below, may have a substantial effect on the motion of the interface in a slot—was either not studied at all or was taken into account insufficiently.

Consider the plane motion of a system of two fluids which at the initial instant have a known interface surface (\Sigma_0), which we shall characterize by its mean cross-section, parallel to the walls of the slot—the contour (\Gamma_0), given by the equation (Fig. 1)

Fig. 1

[
F_0(x,y)=0.
\tag{1}
]

Let the dynamic viscosity of the fluid in the inner region (G_1) be denoted by (\mu_1), and that of the fluid in the outer unbounded region (G_2) by (\mu_2). The interfacial tension at their contact (\Gamma) is equal to (\sigma).

Let, at the instant (t=0), (k) sources begin to act in the region (G_1), and (l) sources (sinks) in the region (G_2), with strengths (Q_i(t)) at the points ((x_i,y_i)) ((i=1,2,\ldots,k+l)). The velocity of the fluid in each of the regions is equal to (\left({}^{9,10}\right))

[
\mathbf{v}_j=-c_j\nabla p,\qquad c_j=\frac{h^2}{12\mu_j},\qquad j=1,2
\tag{2}
]

((h) is the height of the slot), and the pressure (p) in the incompressible fluids ((\operatorname{div}\mathbf{v}_j=0)) satisfies in (G_1+G_2), everywhere outside the special points-sources, the point at infinity, and the interface (\Gamma), the two-dimensional Laplace equation:

[
\nabla^2 p=0.
\tag{3}
]

Throughout the motion, on the mean cross-section (\Gamma) of the lateral surface (\Sigma), whose equation in implicit form has the form

[
F(x,y,t)=0,
\tag{4}
]

the continuity condition for the flow, (v_n^+ = v_n^-), is satisfied, or, by virtue of (2):

[
-v_n = c_1 \frac{\partial p^+}{\partial n} = c_2 \frac{\partial p^-}{\partial n},
\tag{5}
]

and the condition for the pressure jump

[
p^+ - p^- = \sigma \left( \frac{1}{\mathfrak{R}_1} + \frac{1}{\mathfrak{R}_2} \right).
\tag{6}
]

Here (n) is the inward normal to the contour (\Gamma); the plus and minus indices denote limiting values as (\Gamma) is approached from inside and outside, respectively; (\mathfrak{R}_1) and (\mathfrak{R}_2) are the radii of curvature of the surface (\Sigma) along the section (\Gamma), respectively in the plane parallel to the walls and in the plane perpendicular to them and passing through the normal (n). The radius (\mathfrak{R}_2) may be regarded as constant along the perimeter (\Gamma). In the case of complete wetting by one of the liquids, (\mathfrak{R}_2=\pm h/2).

Introducing the known harmonic function

[
\varphi(x,y,t)
=
\frac{1}{2\pi h c_1}
\sum_{i=1}^{k} Q_i(t)\ln \frac{1}{R_i}
+
\frac{1}{2\pi h c_2}
\sum_{i=k+1}^{k+l} Q_i(t)\ln \frac{1}{R_i},
]

[
R_i=\sqrt{(x-x_i)^2+(y-y_i)^2},
\tag{7}
]

we shall seek the pressure (p) in the form

[
p(x,y,t)
=
\varphi
+
\int\limits_{(\Gamma)} \rho(x,t)\ln\frac{1}{R}\,ds
+
\int\limits_{(\Gamma)} \chi(s,t)\frac{\partial}{\partial n_1}\ln\frac{1}{R}\,ds,
]

[
R=\sqrt{(x-\xi)^2+(y-\eta)^2},
\qquad
s(\xi,\eta)\in\Gamma,
\tag{8}
]

where (\rho) and (\chi) are, respectively, the densities of logarithmic potentials of simple and double layers, continuously distributed over (\Gamma).

Using equation (4), the conditions (5) can be represented in the form

[
\left(\frac{\partial F}{\partial t}\bigg/ \frac{\partial F}{\partial n}\right)
=
c_1 \frac{\partial p^+}{\partial n}
=
c_2 \frac{\partial p^-}{\partial n},
\tag{9}
]

and, as a consequence of the known properties of potentials, we have

[
\rho(s,t)
=
-\frac{1}{2\pi}
\left(\frac{1}{c_1}-\frac{1}{c_2}\right)
\frac{\partial F}{\partial t}\bigg/ \frac{\partial F}{\partial n},
\qquad
\chi(s,t)
=
\frac{\sigma}{2\pi}
\left(\frac{1}{\mathfrak{R}_1}+\mathrm{const}\right).
\tag{10}
]

The function (8), with (10) taken into account, satisfies equation (3), condition (6), and has the prescribed singularities. Satisfying the conditions (9), after transformations we obtain the functional equation of motion of the contour (\Gamma)

[
\left(\frac{\partial F}{\partial t}\bigg/ \frac{\partial F}{\partial n}\right)
-
\frac{\lambda}{\pi}
\int\limits_{(\Gamma)}
\left(\frac{\partial F}{\partial t}\bigg/ \frac{\partial F}{\partial n}\right)
\frac{\partial}{\partial n}
\ln\frac{1}{R}\,ds
-
]

[

\frac{c_1c_2}{c_1+c_2}\,
\frac{\sigma}{\pi}
\int\limits_{(\Gamma)}
\frac{\partial}{\partial s}\left(\frac{1}{\mathfrak{R}_1}\right)
\frac{\partial}{\partial \tau}
\ln\frac{1}{R}\,ds
=
\frac{2c_1c_2}{c_1+c_2}
\frac{\partial\varphi}{\partial n},
\tag{11}
]

[
s(\xi,\eta),\ \tau(x,y)\in\Gamma,
\qquad
\lambda=\frac{c_1-c_2}{c_1+c_2}
]

((\tau) is the arc abscissa of the point of the contour ((x,y))).

Thus, determination of the law of motion of the boundary (\Gamma) is reduced to the Cauchy problem for equation (11) with the initial condition (see (1))

[
F(x,y,0)=F_0(x,y)=0.
\tag{12}
]

It is assumed that the contour (\Gamma) possesses differentiable curvature (11).

In the case of a star-shaped interface, by introducing polar coordinates ((r,\theta)) and the equation of the contour (\Gamma) in the form

[
F(r,\theta,t)=r-f(\theta,t)=0
\tag{13}
]

the equation of motion (12) is transformed into the form

[
\begin{aligned}
& f(\theta,t) f_t(\theta,t)
-\frac{\lambda}{\pi}\int_0^{2\pi} f(\nu,t) f_t(\nu,t) K(\theta,\nu,t)\,d\nu
\
&\quad
+\frac{c_1c_2}{c_1+c_2}\frac{\sigma}{\pi}
\int_0^{2\pi}
\frac{\partial}{\partial \nu}
\left(\frac{1}{\mathfrak{R}1}\right)
L(\theta,\nu,t)\,d\nu
\
&=
\frac{1-\lambda}{2\pi h}\sum Q_i(t)K_i(\theta,t)}^{k
+
\frac{1+\lambda}{2\pi h}\sum_{i=k+1}^{k+l} Q_i(t)K_j(\theta,t),
\end{aligned}
\tag{14}
]

where (f_t=\partial f/\partial t); (K) and (L) are functions of (\theta,\nu,f(\theta,t), f(\nu,t)), and (f_\theta(\theta,t)); (K_i) are functions of (\theta, f(\theta,t)), and (f_\theta(\theta,t)) ((f_\theta=\partial f/\partial \theta)). The sought function (f(\theta,t)) satisfies the initial condition

[
f(\theta,0)=f_0(\theta).
\tag{15}
]

The Cauchy problem (14)—(15) can be solved by the finite-difference method—a step-by-step process in time (t), similar to that described in ((^{11,12})). In doing so, one must take into account the singularity of the second integral (the function (L(\theta,\nu,t)) has a pole of first order when (\nu=\theta)).

Fig. 2

According to the proposed algorithm, the problem was programmed and solved by A. S. Zhardikova on the Strela electronic computer of the Computing Center of the Academy of Sciences of the USSR for the particular case of a single sink with constant flow rate (Q), located eccentrically with respect to the circular initial contour (\Gamma_0) (Fig. 1). The calculation was performed with initial data taken from the experiment: (R=7.04) cm, (a=2.29) cm, (h=0.24) cm, (Q=-0.512) cm(^3)/sec, (\mu_1=2.7) poise, (\mu_2=0), (\sigma=31) dyn/cm. Figure 2 shows the initial and two intermediate positions of the boundary, taken from photographs (solid lines) and from the calculation (points). Figure 3 shows plots of the displacement of the contour (\Gamma) in the principal direction ((\theta=0)). The agreement of the experimental data with the results of the numerical solution may be considered quite satisfactory. The curves of boundary motion without distinction of viscosities and without interfacial tension ((\lambda=0,\ \sigma=0)), and with only the distinction of viscosities ((\lambda\ne0,\ \sigma=0)), show a strong discrepancy with experiment.

Fig. 3. (a) — experimental points;
(b) — calculated points

To estimate the influence of interfacial tension, the dimensionless criterion* is introduced

[
B=\frac{\sigma h^3}{12(\mu_1+\mu_2)|Q|R_0},
]

* Saffman and Taylor ((^4)), in the particular case (\mu_2=0), used the quantity (\mu_1 V/\sigma) ((V) is velocity), which is essentially the reciprocal of (B).

where (R_0) is the shortest distance from the sink to the initial interface (Fig. 1). It follows from the equation of motion (14).

As a measure of the degree of influence of interfacial tension on the motion of the boundary, it is convenient to take (T_{\mathrm{d}}/T_{\mathrm{e}})—the ratio of the time (T_{\mathrm{d}}) for the boundary to approach the sink, calculated for (\sigma = 0) and with all the other parameters from experiment ((^{12})) retained, to the value of this time (T_{\mathrm{e}}) in the experiment.

Fig. 4

Yu. A. Teplov, at the Physico-Technical Institute of the Kazan Branch of the Academy of Sciences of the USSR, carried out numerous experiments on the contraction of the boundary (\Gamma) for various values of (B) (the values of (\mu_1, \mu_2, R_0, h, Q), and (\sigma) were varied). The results of their processing are given in Fig. 4, from which it follows that already at (B < 0.0025) the discrepancy between (T_{\mathrm{e}}) and (T_{\mathrm{d}}) does not exceed 10%. Therefore, by choosing (B) sufficiently small, one can eliminate the undesirable influence of interfacial tension when modeling the motion of the boundary on a Hele-Shaw cell, which confirms the conclusion drawn in ((^1)).

Received
11 X 1960

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