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Reports of the Academy of Sciences of the USSR
- Volume 185, No. 3
UDC
GEOPHYSICS
V. M. VOLOSHCHUK
ON AN ASYMPTOTIC METHOD FOR SOLVING THE EQUATIONS OF BROWNIAN MOTION OF AEROSOL PARTICLES
(Presented by Academician E. K. Fedorov, 19 VII 1968)
Let a plane body—an obstacle with a sufficiently smooth surface \(\Gamma\)—be located in a flow of a gaseous medium. Construct an orthogonal curvilinear coordinate system \(O\xi\eta\) as follows: place the origin in the front part of \(\Gamma\), direct the axis \(O\eta\) along \(\Gamma\), and the axis \(O\xi\) perpendicular to \(\Gamma\). The equations of Brownian motion of aerosol particles, obtained in \((^1)\), upon introducing the convective velocity \(\mathbf{v}^*=\mathbf{v}+\dfrac{1}{\mathrm{Pe}}\dfrac{\partial \ln n}{\partial r}\), in the coordinate system \(O\xi\eta\) take the form:
\[ \frac{\partial n}{\partial t} +\frac{\partial n v_\xi^*}{\partial \xi} +\frac{1}{R+\xi} n v_\xi^* +\frac{R}{R+\xi}\frac{\partial n v_\eta^*}{\partial \eta} = \frac{1}{\mathrm{Pe}}\frac{R}{R+\xi} \left\{ \frac{\partial}{\partial \xi}\left(1+\frac{\xi}{R}\right)\frac{\partial}{\partial \xi} + \frac{\partial}{\partial \eta}\frac{R}{R+\xi}\frac{\partial}{\partial \eta} \right\}n; \tag{1} \]
\[ k\left\{ \frac{\partial}{\partial t} +\left(v_\xi^*-\frac{1}{\mathrm{Pe}}\frac{\partial \ln n}{\partial \xi}\right)\frac{\partial}{\partial \xi} +\frac{R}{R+\xi}\left(v_\eta^*-\frac{1}{\mathrm{Pe}}\frac{R}{R+\xi}\frac{\partial \ln n}{\partial \eta}\right)\frac{\partial}{\partial \eta} \right\} \times \left(v_\xi^*-\frac{1}{\mathrm{Pe}}\frac{\partial \ln n}{\partial \eta}\right) -\frac{k}{R+\xi} \left(v_\eta^*-\frac{1}{\mathrm{Pe}}\frac{R}{R+\xi}\frac{\partial \ln n}{\partial \eta}\right)^2 +v_\xi^* = u_\xi+F_\xi; \tag{2} \]
\[ k\left\{ \frac{\partial}{\partial t} +\left(v_\xi^*-\frac{1}{\mathrm{Pe}}\frac{\partial \ln n}{\partial \xi}\right)\frac{\partial}{\partial \xi} +\frac{R}{R+\xi}\left(v_\eta^*-\frac{1}{\mathrm{Pe}}\frac{R}{R+\xi}\frac{\partial \ln n}{\partial \eta}\right)\frac{\partial}{\partial \eta} \right\} \times \left(v_\eta^*-\frac{1}{\mathrm{Pe}}\frac{R}{R+\xi}\frac{\partial \ln n}{\partial \eta}\right) +\frac{k}{R+\xi} \left(v_\xi^*-\frac{1}{\mathrm{Pe}}\frac{\partial \ln n}{\partial \xi}\right) \times \left(v_\eta^*-\frac{1}{\mathrm{Pe}}\frac{R}{R+\xi}\frac{\partial \ln n}{\partial \eta}\right) +v_\eta^* = u_\eta+F_\eta; \tag{3} \]
\[ n\to n_\infty,\qquad \mathbf{v}_* \to \mathbf{u}+\mathbf{F}\qquad \text{as } r\to\infty. \]
For a fairly broad class of physical problems, the boundary conditions for \(n\) on \(\Gamma\) may be written in the form:
\[ n=0,\qquad \text{if } v_\xi^*(0,\eta)=0; \tag{4} \]
\[ \frac{1}{\mathrm{Pe}}\frac{\partial n}{\partial \xi} = a(n-n_0),\qquad \text{if } v_\xi^*(0,\eta)\ne 0, \tag{5} \]
where \(a\) and \(n_0\) are certain constants determined from particular physical considerations.
In this note the asymptotic behavior of \(n\) and \(\mathbf{v}^*\) is studied for small \(k\) and large \(\mathrm{Pe}\).
Let us conditionally divide the flow region into three parts: \(Q_\infty\), \(Q_\Gamma\), and \(Q_s\) (see Fig. 1). In \(Q_\infty\) the derivatives of \(n\) are small; consequently, the solution of equations (1)—(3) can be represented in the form of series
\[ n \approx n^{(1)}(\xi,\eta;k;\mathrm{Pe}) = \sum_{m\ge 0} n^{m(1)} \mathrm{Pe}^{-m}, \]
\[ \mathbf{v}^* \approx \mathbf{v}^{*(1)}(\xi,\eta;k;\mathrm{Pe}) = \sum_{m\ge 0} \mathbf{v}^{(m)(1)} \mathrm{Pe}^{-m}. \tag{6} \]
In \(Q_{\Gamma}\), by virtue of the boundary conditions (5), the concentration \(n\) in the direction toward \(\Gamma\) must change so substantially that either part of the diffusion terms in (1)—(3) becomes, as \(\mathrm{Pe}\to\infty\), of order comparable with the convective terms, or, in the part of \(Q_{\Gamma}\) adjacent to \(Q_{\infty}\), will be comparable, while in the part of \(Q_{\Gamma}\) adjacent to \(\Gamma\), will prevail over the convective terms. Therefore it is necessary first to stretch \(Q_{\Gamma}\) along the normals to \(\Gamma\) so that the rate of change of the concentration in the direction toward \(\Gamma\) in the transformed region \(Q_{\Gamma}\) has order \(O(1)\), and only after this to seek solutions of equations (1)—(3) in the form of series analogous to (6). It follows from physical considerations that in \(Q_s\) the longitudinal changes of concentration will be significant. Since, in principle, the study of the asymptotics in \(Q_s\) is analogous to its study in \(Q_{\Gamma}\), here we shall restrict ourselves to considering only the regions \(Q_{\infty}\) and \(Q_{\Gamma}\).
Fig. 1
We transform the region \(Q_{\Gamma}\) as follows:
\[ \xi=\alpha_0 \xi,\quad \eta^*=\eta;\quad \alpha_0=\alpha_0(\mathrm{Pe}),\quad \alpha_0|_{\mathrm{Pe}\to\infty}\to\infty . \tag{7} \]
Let, in the new variables (we omit the asterisk by \(\eta\)):
\[ n\approx n_1=\tilde n(\xi,\eta;k;\mathrm{Pe}), \]
\[ \mathbf v^*\approx \mathbf v_1^* =\alpha_{\xi}\tilde v_{\xi^*}(\xi,\eta;k;\mathrm{Pe})\mathbf e_{\xi} +\alpha_{\eta}\tilde v_{\eta}^*(\xi,\eta;k;\mathrm{Pe})\mathbf e_{\eta}, \tag{8} \]
\[ \alpha_{\xi}=\alpha_{\xi}(\mathrm{Pe}),\quad \alpha_{\eta}=\alpha_{\eta}(\mathrm{Pe}), \]
\[ \tilde n,\ \tilde v_{\xi^*},\ \tilde v_{\eta}^*=O(1),\quad \mathrm{Pe}\to\infty, \]
where \(\mathbf e_{\xi}\) and \(\mathbf e_{\eta}\) are the unit vectors of the coordinate system \(O\xi\eta\). Suppose that
\[ \mathbf v^{*(1)}\to O(\xi^{\beta_1})\mathbf e_{\xi} +O(\xi^{\beta_2})\mathbf e_{\eta},\quad \xi\to 0 . \tag{9} \]
Then the condition of asymptotic matching of the functions \(\mathbf v^{*(1)}\) and \(\mathbf v_s^*\) on the boundary of the region \(Q_{\Gamma}\), and the condition of comparability of the convective and diffusion terms (which determine the variation of the concentration in the direction toward the body), lead to the relations
\[ \alpha_0=\alpha_{\xi}\mathrm{Pe},\quad \alpha_{\xi}=\alpha_0^{-\beta_1},\quad \alpha_{\eta}=\alpha_0^{-\beta_2}. \tag{10} \]
Relations (10), for a broad class of physical problems, make it possible to find the principal asymptotic terms for \(n\) and \(\mathbf v^*\) in the region \(Q_{\Gamma}\):
\[ n_1\approx \tilde n(\xi,\eta;k;\infty),\quad \mathbf v^*=\alpha_{\xi}\tilde v_{\xi^*}(\xi,\eta;k;\infty)\mathbf e_{\xi} +\alpha_{\eta}\tilde v_{\eta^*}(\xi,\eta;k;\infty), \]
\[ \mathrm{Pe}\to\infty . \tag{11} \]
After this one can construct equations for the functions
\[ \tilde n(\xi,\eta;k;\mathrm{Pe})-\tilde n(\xi,\eta;k;\infty), \]
\[ \tilde v_{\xi}(\xi,\eta;k;\mathrm{Pe})-\tilde v_{\xi}(\xi,\eta;k;\infty), \tag{12} \]
\[ \tilde v_{\eta}(\xi,\eta;k;\mathrm{Pe})-\tilde v_{\eta}(\xi,\eta;k;\infty), \]
and, analyzing them in an analogous way, obtain the subsequent terms of the asymptotic expansions.
Let us write out the solutions for several concrete problems.
a) Let \(\beta_1=2\) and \(\beta_2=1\) (this case is always realized for viscous flow of a medium around an obstacle body and \(F=0\) for \(k<k_{\mathrm{cr}}\) \((^2,^3)\), where \(k_{\mathrm{cr}}\) is the value of the Stokes number at which inertial deposition begins).
particles on the body). Then, in the stationary case,
\[ \alpha_0=\mathrm{Pe}^{1/3},\qquad \alpha_\xi=\mathrm{Pe}^{-2/3},\qquad \alpha_\eta=\mathrm{Pe}^{-1/3}, \]
\[ \widetilde n(\zeta,\eta;k;\infty) = \frac{1}{n_\infty}\, n^{(1)}(0,\eta;k;\infty)\, \widetilde n(\zeta,\eta;0;\infty) \times \]
\[ \times \exp\left(-2k\int_\eta^{\eta}\frac{\widetilde u_\eta}{R}\,d\eta\right) +O(\mathrm{Pe}^{-1/3}), \]
\[ \widetilde v_\xi= \left(\widetilde u_\xi+\frac{k}{R}\widetilde u_\eta^2\right)\xi^2 +O(\mathrm{Pe}^{-1/3}), \]
\[ \widetilde v_\eta=\widetilde u_\eta\xi+O(\mathrm{Pe}^{-1/3}), \]
\[ \widetilde n(\zeta,\eta;0;\infty) = \frac{n_\infty}{\Gamma(1/3)} \gamma(1/3,1/9\,\zeta^3\varphi), \tag{13} \]
\[ \varphi = \exp\left[ -3\int^\eta \left( \frac{\widetilde u_\xi}{\widetilde u_\eta} +\frac{k}{R}\widetilde u_\eta \right)d\eta' \right] \left\{ \int_{\eta_N}^{\eta} \widetilde u_\eta^{-1} \exp\left[ -3\int^{\eta'} \left( \frac{\widetilde u_\xi}{\widetilde u_\eta} +\frac{k}{R}\widetilde u_\eta \right)d\eta'' \right]d\eta' \right\}^{-1}, \]
\[ \widetilde u_\xi= \zeta^{-2}u_\xi(\zeta\mathrm{Pe}^{-1/3},\eta)\mathrm{Pe}^{2/3}\big|_{\mathrm{Pe}\to\infty}, \]
\[ \widetilde u_\eta= \zeta^{-1}u_\eta(\zeta\mathrm{Pe}^{-1/3},\eta)\mathrm{Pe}^{1/3}\big|_{\mathrm{Pe}\to\infty}, \]
where \(\eta_N\) is the root of the equation \(\widetilde u_\eta(\eta)=0\). The values of the function \(n^{(1)}(0,\eta_N;k;\infty)\) for a sphere in Stokes flow for several \(k\) have been calculated in \((4)\). Using these data, one can conclude, on the basis of the formulas given, that for \(k\) close to \(k_{\mathrm{cr}}=1.214\), allowance for inertia substantially, by several times, changes the concentration of aerosol particles in the vicinity of the forward critical point of the flow (for \(k/k_{\mathrm{cr}}\approx 0.164\), by a factor of 1.85; for \(k/k_{\mathrm{cr}}\approx 0.328\), by a factor of 3.01). We note that solution (13) also holds for axisymmetric problems. Analogous solutions in particular form for several specific cases were obtained in \((4\text{--}7)\).
b) Let \(\beta_1=0\) and \(\beta_2=0\) (this case is realized when a body-obstacle is flowed around by the medium and when there are interaction forces between the body and the aerosol particles). After simple but cumbersome calculations we have:
\[ j_\xi=j_1+O(\mathrm{Pe}^{-1}),\qquad j_1=-\widehat n\widehat v_\xi+\partial\widehat n/\partial\xi =-\widehat n\widehat v, \]
\[ \widetilde n= \begin{cases} \widehat n - a\,\dfrac{\widehat n-n_0}{a+\widehat u}\,e^{-\widehat u\zeta} +O(k\,\mathrm{Pe}),\\[1.2em] \widehat n\left(1+\dfrac{\widehat v+\widehat u}{\widehat v^2}\, \dfrac{\zeta}{k\,\mathrm{Pe}}\right) +(n_0-\widehat n) \exp\left[-\dfrac{\widehat u}{\widehat v^2}\, \dfrac{\zeta}{k\,\mathrm{Pe}}\right] +O\left(\dfrac{1}{k\,\mathrm{Pe}}\right)+O(k), \end{cases} \tag{14} \]
\[ \widehat n=n^{(1)}(0,\eta;k;\infty),\qquad \widehat v=v_\xi^{*(1)}(0,\eta;k;\infty),\qquad \widehat u=-u_\xi(0,\eta)-F_\xi(0,\eta). \]
It follows from relations (14) that: 1) the flux of aerosol particles in the direction toward the body, \(j_\xi\), up to terms of order \(\mathrm{Pe}^{-1}\), does not depend on \(\zeta\) and is equal to the flux of aerosol particles obtained when diffusion is completely neglected; 2) the effect of inertia on the motion of aerosol particles in the region \(Q_\Gamma\) may be neglected only for \(k\ll \mathrm{Pe}^{-1}\). Comparing formulas (13) and (14), we come to the conclusion that the influence of particle inertia on their Brownian motion is strongest in the case when \(u_\xi+F_\xi\ne0\) on the surface of the body being flowed around and \(k\gg \mathrm{Pe}^{-1}\).
Institute of Experimental
Meteorology
Received
11 VII 1968
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