Abstract
Full Text
UDC 536.2 : 517.9
MATHEMATICAL PHYSICS
V. S. RAVIN
ON A PROBLEM OF STATIONARY HEAT CONDUCTION
(Presented by Academician A. V. Shubnikov, 19 VII 1966)
In studying the kinetics of crystallization from a melt of semiconductor micromonocrystals on foreign substrates ((^{1,2})), grown by the method ((^{3,4})), it is necessary to determine the stationary temperature fields in the melt–substrate system. In the case where the crystallizing drops of melt are close in shape to a hemisphere, the following heat problem arises.
It is required to determine the temperature field (u(r,\theta)) in the hemisphere (r \le r_0), (0 \le \theta \le \pi/2), made of a material with coefficients of thermal conductivity (K) and external heat transfer (h), lying along the surface of the great circle (\theta=\pi/2) in perfect thermal contact with an infinite plane-parallel plate (-l \le z \le 0) made of a material with coefficients (K_1) and (h_1), whose temperature is (u_1(\rho,z)). Here (r,\theta) denote spherical coordinates, (\rho,z) cylindrical coordinates, with the polar angle (\theta) measured from the positive direction of the (z)-axis; (\theta=\pi/2) corresponds to (z=0).
The functions (u(r,\theta)) and (u_1(\rho,z)) satisfy the following equations and boundary conditions:
[
\Delta_{r,\theta}u=0,\qquad
(\partial u/\partial r+hu)_{r=r_0 \atop 0\le \theta\le \pi/2}=hu_0,
\tag{1}
]
[
u\bigg|{\theta=\pi/2 \atop r<r_0}
=
u_1\bigg|,
\qquad
-\frac{K}{r}\frac{\partial u}{\partial \theta}\bigg|{\theta=\pi/2 \atop r<r_0}
=
K_1\frac{\partial u_1}{\partial z}\bigg|,
\tag{2}
]
[
\Delta_{\rho,z}u_1=0,\qquad
(\partial u_1/\partial z+h_1u_1)_{z=0 \atop \rho>r_0}=h_1u_0,
]
[
-K_1\partial u_1/\partial z\bigg|{z=-l}=q_0,\qquad
\partial u_1/\partial \rho\bigg|=0,
\tag{3}
]
[
\Delta_{r,\theta}
=
\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)
+
\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}
\left(\sin\theta\frac{\partial}{\partial\theta}\right),
\qquad
\Delta_{\rho,z}
=
\frac{1}{\rho}\frac{\partial}{\partial \rho}
\left(\rho\frac{\partial}{\partial \rho}\right)
+
\frac{\partial^2}{\partial z^2},
]
[
u_0=\mathrm{const},\qquad q_0=\mathrm{const}.
]
The peculiarity of the problem is that the boundary conditions are prescribed on surfaces which are coordinate surfaces in different coordinate systems. We shall give approximate solutions for the cases (l \ll r_0) and (l \gg r_0); (h_1r_0 \ll 1).
1. For (l \ll r_0), in order to determine the function (u(r,\theta)), we introduce, proceeding from the expansion of (u_1(\rho,z)) in a Maclaurin series in (z), the approximate boundary condition
[
\frac{1}{r}\frac{\partial u}{\partial \theta}\bigg|_{\theta=\pi/2}
=
\frac{q_0}{K}.
\tag{4}
]
Then the solution of the problem (1), (4) can be expressed in terms of Legendre polynomials:
[
u(r,\theta)=u_0-\frac{q_0r_0}{K}
\left[
\frac{r}{r_0}\cos\theta
+
\nu\sum_{n=0}^{\infty}N'n
\left(\frac{r}{r_0}\right)^{2n}
P(\cos\theta)
\right],
\tag{5}
]
where
[
\nu=\frac{1+hr_0}{2},\qquad
N'n=\frac{(4n+1)P.}(0)}{(n+1)(2n-1)(2n+hr_0)
]
The result obtained can be used to determine the function (u(r,\theta)) in the following approximation. Setting
[
u_1(\rho,z)=u_0+\frac{q_0}{K_1}\frac{1-h_1z}{h_1}+v(\rho,z),
\tag{6}
]
by virtue of (2), (3), and (5), for the function (v(\rho,z)) we shall have the boundary-value problem
[
\Delta_{\rho,z}v=0,\qquad
\left(\frac{\partial v}{\partial z}+h_1v\right)_{z=0}
=-\frac{q_0}{K_1}\Psi(\rho),
]
[
\left.\frac{\partial v}{\partial z}\right|{z=-l}=0,\qquad
\left.\frac{\partial v}{\partial \rho}\right|=0,
\tag{7a}
]
where
[
\Psi(\rho)=
\begin{cases}
\alpha+\psi(\rho), & 0\le \rhor_0,
\end{cases}
\qquad
\psi(\rho)=\eta_1\nu\frac{K_1}{K}\sum_{n=1}^{\infty}N_n\left(\frac{\rho}{r_0}\right)^{2n},
\tag{7b}
]
[
\alpha=1-\nu\frac{h_1}{h}\frac{K_1}{K},\qquad
\eta_1=h_1r_0,\qquad
N_n=N'nP(0),
]
whence, instead of (4), there follows a boundary condition of the form
[
\left.\frac{1}{r}\frac{\partial u}{\partial \theta}\right|_{\theta=\pi/2}
=\frac{q_0}{k}[1+X(r)],
\tag{8}
]
[
X(\rho)=\Psi(\rho)+\frac{h_1K_1}{q_0}\varphi(\rho),\qquad
\varphi(\rho)=v(\rho,0).
\tag{9}
]
Applying the Hankel transforms to (7),
[
\bar v=\int_0^\infty \rho J_0(\rho\lambda)v\,d\rho,\qquad
v=\int_0^\infty \lambda J_0(\rho\lambda)\bar v\,d\lambda,
\tag{10}
]
whence
[
\varphi(\rho)=-\frac{q_0}{K_1}\int_0^\infty
\frac{\lambda J_0(\rho\lambda)\operatorname{ch}\lambda l}
{\lambda\operatorname{sh}\lambda l+h_1\operatorname{ch}\lambda l}
\int_0^{r_0}\rho'J_0(\rho'\lambda)\Psi(\rho')\,d\rho'\,d\lambda,
]
[
\Psi(\rho)=\int_0^\infty \lambda J_0(\rho\lambda)
\int_0^{r_0}\rho'J_0(\rho'\lambda)\Psi(\rho')\,d\rho'\,d\lambda,
]
we obtain for the function (X(\rho)) an expression through a Stieltjes integral
[
X(\rho)=\int_0^{r_0}\Psi(\rho')\frac{d}{d\rho'}[\rho'G(\rho,\rho')]\,d\rho',
\tag{11}
]
where the function
[
G(\rho,\rho')=\int_0^\infty J_0(\rho\lambda)J_1(\rho'\lambda)
\frac{\lambda\operatorname{th}\lambda l}{\lambda\operatorname{th}\lambda l+h_1}\,d\lambda
\tag{12}
]
admits a representation in the form of the series
[
G(\rho,\rho')=2h_1\sum_{m=1}^{\infty}
\frac{\beta_m I_0!\left(\frac{\rho}{l}\beta_m\right)
K_1!\left(\frac{\rho'}{l}\beta_m\right)}
{\beta_m^2+h_1l(1+h_1l)},
\qquad
\beta_m\operatorname{tg}\beta_m=h_1l.
\tag{13}
]
From the asymptotics of the cylindrical functions (I_0(x)), (K_1(x)) it follows that
(G(\rho,r_0)\to0) as (l/r_0\to0), (\rho0). On the other hand, from (11) we have
[
X(\rho)=\Psi(\xi)r_0G(\rho,r_0),\qquad
0\le \xi\le r_0,
\tag{14}
]
moreover, according to (7b), the function (\psi(\rho)) entering into (\Psi) is bounded,
[
|\psi(\rho)|\le \eta_1\nu\frac{K_1}{K}N,\qquad
N=\sum_{n=1}^{\infty}N_n<\ln 2,
\tag{15}
]
and the parameter (\eta_1) is in practice usually small. Since, moreover, the function (I_0(\rho\beta_1/l)) changes slowly as (\rho) varies from (0) to (r_0), the solution of the problem in this approximation will be well described by a formula of the form (5), with (q_0/K) replaced by (\dfrac{q_0}{K}(1+\overline X)), (\overline X = ar_0G(\xi,r_0)), (0\leq \xi\leq r_0).
2. For (r_0 \ll l), the variation of the function (u(r,\theta)) is small, which makes it possible, proceeding from the condition of balance of the heat flux through the hemisphere,
[
\iint\limits_{(S)} \frac{\partial u}{\partial r}\,dS
=
\iint\limits_{(S_0)} \frac{\partial u}{\partial z}\,dS_0
]
((S) is the surface of the hemisphere (r=r_0), (S_0) is the circle (\theta=\pi/2)), to introduce, for determining the function (u_1(\rho,z)) at (z=0), the effective boundary condition
[
\left(\frac{\partial u_1}{\partial z}+h_1'u_1\right)_{\substack{z=0\ 0\leq \rho<r_0}}
=
h'u_0,\qquad
h'=\frac{K_1}{K}\frac{S}{S_0}h.
\tag{16}
]
Then, by virtue of (3) and (16), for the function (v(\rho,z)), defined by relation (6), the boundary-value problem will hold
[
\Delta_{\rho,z}v=0,\qquad
\left(\frac{\partial v}{\partial z}+h_1v\right)_{z=0}
=
\mu\,\frac{q_0}{K_1}\Psi_1(\rho),
\tag{17a}
]
[
\partial v/\partial z\big|{z=-l}=0,\qquad
\partial v/\partial \rho\big|=0,
]
where
[
\Psi_1(\rho)=
\begin{cases}
1+\dfrac{h_1K_1}{q_0}\,\varphi(\rho), & 0\leq \rhor_0,
\end{cases}
\qquad
\varphi(\rho)=v(\rho,0),\qquad
\mu=1-\frac{h'}{h_1},
\tag{17b}
]
and from (2), (6), and (17), in order to find the function (u(r,\theta)) we obtain the boundary condition
[
\frac{1}{r}\frac{\partial u}{\partial \theta}\bigg|_{\theta=\pi/2}
=
\frac{h'}{h_1}\frac{q_0}{K}\Psi_1(r).
\tag{18}
]
Applying the transform (10) to (17), for determining (\Psi_1(\rho)) we shall have the integral equation
[
\Psi_1(\rho)
=
1+\mu h_1\int_0^{r_1}
\Psi_1(\rho')\,\frac{d}{d\rho'}\left[\rho'G_1(\rho,\rho')\right]d\rho',
\tag{19}
]
where the kernel
[
G_1(\rho,\rho')
=
\int_0^\infty
\frac{\operatorname{ch}\lambda l\, J_0(\rho\lambda)J_1(\rho'\lambda)}
{\lambda\,\operatorname{sh}\lambda l+h_1\operatorname{ch}\lambda l}\,d\lambda
\tag{20}
]
can be expressed by the series
[
G_1(\rho,\rho')
=
2\sum_{m=1}^{\infty}
\frac{\beta_mJ_1!\left(\dfrac{\rho'}{l}\beta_m\right)K_0!\left(\dfrac{\rho}{l}\beta_m\right)}
{\beta_m^2+h_1l(1+h_1l)},
\qquad
\beta_m\tg\beta_m=h_1l.
\tag{21}
]
As a result of the smallness of (r_0) ((\mu\sim 1)) in the present case, the solution of the boundary-value problem (1), (18) will practically have the form (5), with (q_0/K) replaced by
[
\frac{h'q_0}{h_1K}.
]
The author expresses sincere gratitude to G. A. Grinberg, who pointed out the possibility of an approximate solution of the problem in limiting cases, based on the introduction of effective boundary conditions with a subsequent estimate of corrections, and is also deeply grateful to B. Ya. Lyubov for discussion.
Received
30 VI 1966
CITED LITERATURE
- V. S. Ravin, Kristallografiya, 11, 295 (1966).
- V. S. Ravin, Kristallografiya, 11, 910 (1966).
- G. A. Kurov, V. D. Vasil’ev, M. G. Kosaganova, Kristallografiya, 7, 773 (1962).
- V. J. Doo, J. Electrochem. Soc., 111, 1196 (1964).
- Higher Transcendental Functions, 2, N. Y., 1953.