UDC 536.212

MATHEMATICAL PHYSICS

A. S. UMANSKII, V. M. DUBNER, Yu. A. GORSHKOV

THE NONSTATIONARY HEAT-CONDUCTION PROBLEM FOR A TWO-LAYER CYLINDER

(Presented by Academician V. A. Kirillin on 9 X 1968)

1. The nonstationary axisymmetric problem of heat propagation in two infinite coaxial cylinders with differing physical properties is solved.* The thermal contact between the cylinders is assumed to be ideal. The temperature in the system satisfies the equation

\[ Lu=\partial u/\partial \tau, \tag{1} \]

where the linear operator \(L\) is defined as \(L=a_\beta\nabla^2\); \(\beta=1,2\).

\[ u(\tau,R_2)=0; \tag{2} \]

\[ \lim_{\varepsilon\to0}\frac{\partial}{\partial r} \,[u(\tau,R_1-\varepsilon)-\xi u(\tau,R_1+\varepsilon)]=0; \tag{3} \]

\(a_\beta\) is the thermal diffusivity coefficient; \(R_\beta\) are the radii of the cylinders; \(\xi=\lambda_1/\lambda_2\) is the ratio of the thermal-conductivity coefficients; index 1 refers to the inner region \(0\le r\le R_1\), and index 2 to the outer region \(R_1\le r\le R_2\). After separation of variables the solution is written in the form

\[ u(\tau,r)=\sum_i c_i\psi_i(r)\exp(-s_i\tau), \tag{4} \]

where \(L\psi_i=s_i\psi_i\), and \(c_i\) are the coefficients in the expansion of the initial temperature field \(u(0,r)\) in \(\{\psi_i\}\). Obviously, the relaxation properties of the system are determined by the minimum eigenvalue \(s_i\). The eigenfunctions in regions 1 and 2 are linear combinations of Bessel functions of the first and second kind,

\[ \psi_i(\beta)=A_i^{(\beta)}J_0\!\left(r\sqrt{s_i/a_\beta}\right) +B_i^{(\beta)}N_0\!\left(r\sqrt{s_i/a_\beta}\right), \]

and from the boundedness of the solution it follows that \(B_i^{(1)}=0\), while from the boundary conditions (2), (3) and the continuity of the eigenfunctions it follows that the eigenvalues \(s_i\) are the roots of the transcendental equation

\[ [z\xi J_1(xyz)N_0(xy)-J_0(xyz)N_1(xy)]J_0(x) - [z\xi J_1(xyz)J_0(xy)-J_0(xyz)J_1(xy)]N_0(x)=0, \tag{5} \]

where, for convenience, the dimensionless variables have been introduced

\[ x=\sqrt{s_i/a_2}\,R_2,\qquad y=R_1/R_2,\qquad z=\sqrt{a_2/a_1}. \]

Equation (5) has an infinite number of roots, whose determination is associated with laborious computations. In the cases \(y=0\) and \(y=1\) equation (5) takes the form \(J_0(x)\) and \(J_0(xz)=0\),

\[ s_k=\alpha_k^2 a_2/R_2^2,\qquad s_k=\alpha_k^2 a_1/R_2^2, \tag{6} \]

* This problem is in many respects analogous to the problem of the propagation of electromagnetic waves in a waveguide partially filled with a dielectric.

where \(a_k\) are the zeros of the function \(J_0(a)\). It is easy to see that this is the solution of the well-known heat-conduction problem in a cylinder of radius \(R_2\). In the present paper approximate solutions and a qualitative analysis of the dependence of the eigenvalues \(x_k\) on the physical parameters of the problem are given.

1. Consider (5) in the region \(y \ll 1\), assuming \(x \gg 1\),* but \(xy \ll 1\), \(xyz \ll 1\) (the boundaries of this region are indicated below). Using the asymptotic expressions for the Bessel functions, in this region we obtain

\[ \operatorname{tg}(x-\pi/4) \simeq 2\ln(\gamma xy/2)+8/\pi\,(\xi z^2-1)x^2y^2 \tag{7} \]

or

\[ x_k=\pi(k-{}^3\!/_{4})+\operatorname{arc\,tg}\,[2\ln(\gamma xy/2)+8/\pi\,(\xi z^2-1)x^2y^2], \tag{8} \]

where \(\gamma=\exp(0.57772)=1.7811\). As \(y\to 0\),

\[ x_k=\pi({}^3\!/_{4}+k)\left[1-\frac{\pi^2(3+4k)}{2^5}(\xi z^2-1)y^2\right], \tag{8a} \]

i.e., there exists a finite region in \(y\), the size of which is determined by \(\xi z^2\), in which the presence of the inner cylinder practically does not affect the characteristic relaxation time of the system. (We note that from (5) it follows that \((dx/du)_0=0\).) We shall further assume that the ratio of the volumetric heat capacities \(\xi z^2>1\). Since in (8) the arctangent is a function with bounded variation, we investigate its argument. It has a minimum at

\[ xy=2^{3/2}\pi^{1/2}(\xi z^2-1)^{-1/2}, \tag{8b} \]

so that

\[ \pi(k-{}^3\!/_{4})+\operatorname{arc\,tg}\,[1+\ln(2\gamma^2/(\xi z^2-1))]\leq x_k\leq \pi(k+{}^1\!/_{4}), \]

and in the region under consideration the inequality \(x\gg 1\) can be violated only for \(x_1\). We shall therefore confine ourselves for the time being to the case

\[ 1\leq \xi z^2\leq 1+2e\gamma^2/\pi=6.4896, \tag{9} \]

when \(x_1\geq \pi/4\) and the violation of the inequality \(x\gg 1\) is not very substantial.

Thus, the minimum eigenvalue \(x_1\), which, as indicated, determines the relaxation properties of the system, decreases with increasing \(y\) in accordance with (8), while the product \(xy\) increases and may become greater than 1. In this connection the region of parameter values \(z\) and \(\xi\) specified by inequality (9) splits into two:

\[ \xi z^2\leq 1+8/\pi=3.5464; \tag{10a} \]

\[ 1+8/\pi\leq \xi z^2\leq 1+2e\gamma^2/\pi. \tag{10b} \]

1. In the first of these, the inequality \(xy<1\) is violated before \(x_1(y)\) reaches a minimum. This occurs at the point

\[ y_1=\{\pi/4+\operatorname{arc\,tg}[2\ln(\gamma/2)+8/\pi\,(\xi z^2-1)]\}^{-1}, \tag{11} \]

where \(4/3\pi\leq y_1\leq 0.69\).

For \(y>y_1\) the product \(xy\geq 1\), and equation (5) is approximately written in the form

\[ \xi z^2xy\sin x(1-y)\simeq 4\cos(1-y)x \tag{12} \]

or

\[ x_k=[\pi(k-1)+\operatorname{arc\,tg}(4/\xi z^2xy)]/(1-y). \tag{12a} \]

Taking (10a) into account, the arctangent here may be replaced by its argument, so that

\[ x_1(y)\simeq 2[\xi z^2y(1-y)]^{-1/2}. \tag{13} \]

Expression (13) is valid for \(y\geq y_2\), where \(y_2\) is determined from (12a) and the condition \(xy_2=1\):

\[ 0.543=(1+\operatorname{arc\,tg}(\pi/(\pi+8)))^{-1}\leq y_2\simeq 1/[1+\operatorname{arc\,tg}(4/\xi z^2)]\leq \]

\[ \leq (1+\operatorname{arc\,tg}(\pi/(\pi+2e\gamma^2)))^{-1}=0.645. \]

* We note that, with good accuracy, \(a_k\gg 1\), since

\[ a_k=\pi(k-{}^1\!/_{4})+\frac{1}{2\pi(4k+1)} -\frac{31}{6\pi^3(4k+1)^3} +\frac{3779}{15\pi^5(4k+1)^5}+\cdots \]

It is easy to verify that \(y_1 \simeq y_2\), and for values of \(y\) close to them (8) and (13) almost coincide, but, of course, (8) is a lower estimate and (13) an upper estimate describing the region near the minimum of the function \(x_1(y)\), located at \(y_m = 1/2\) and equal to \(x_m = 4(\xi z)^{-1/2}\). (13) remains valid up to the point

\[ y_3 = [1 + z \operatorname{arc\,tg} (4/z\xi)]^{-1} \simeq \xi/(4+\xi). \tag{14} \]

1. To the right of this point \(xyz \to 1\), and therefore the solution near \(y=1\) can be found in the form of a series in powers of \((1-y)\). Differentiating (5), we find

\[ x_k(y,z,\xi) = x_k(1,z)\{1-(\xi-1)(1-y)+\ldots\}. \tag{15} \]

1. Let us now consider the parameter region (10b). For \(0 \leq y \leq y_2\) the function \(x_1(y)\), according to (8), decreases, passes through a minimum, which is deeper than in item 3, and begins to increase up to the point \(y_1\), to the right of which \(x_1(y)\) is approximated by the increasing dependence (13), and further—as in item 3.

2. Let us now turn to the parameter region \(\xi z^2 > 1 + 2 e \gamma/\pi\) (for simplicity we assume that the inequality is satisfied with a large margin), when the function \(x_1(y)\) may become less than unity. Taking into account the approximate nature of the analysis, put in (8) \(x_1 = \pi/4\), and not 1. Then, solving an equation of the form \(u\ln u=-v\), which arises when the argument of the arctangent in (8) is set equal to zero, we find the point

\[ y_1 \simeq 2^{7/2}\pi^{-3/2}[(\xi z^2 - 1)\ln(\pi(\xi z^2 - 1)/2\gamma^2)]^{-1/2}, \tag{16} \]

to the left of which the solution behaves according to (8)*. For \(y>y_1\), assuming \(x \leq 1\), instead of (5) we have

\[ \xi z^2 xy \ln(\gamma xy/2) - (\xi z^2 - 1)xy \ln(\gamma x/2) + 4/xy = 0. \tag{17} \]

(17) is also reduced to the form \(u\ln u=-v\), and the solution has the form

\[ x_1 = 2^{3/2}y^{-1}\{2(\xi z^2 - 1)\ln(1/y)-\ln 2\gamma^2\}^{-1/2}. \tag{18} \]

The function \(x_1(y)\) reaches its minimum

\[ x_m = 2(\xi z^2 - 1)^{-1/2}\exp[1+\ln 2\gamma^2/2(\xi z^2 - 1)] \tag{19} \]

at the point

\[ y_m = \exp[-1-\ln(2\gamma^2)/2(\xi z^2 - 1)]. \tag{19a} \]

The approximate solution (18) is valid inside the interval \(y_2 \leq y \leq y_3\), where \(x \leq 1\). Putting \(x=1\) in (17), we find

\[ y_2 \simeq 2^{3/2}[\xi z^2 \ln(\xi z^2/8)-\ln(2/\gamma)^2]^{-1/2}, \tag{20} \]

which almost coincides with (16), and

\[ y_3 \simeq (\gamma/2)^{1/\xi z^2}\{1-(2/\gamma)^{2/\xi z^2}\cdot 8/\xi z^2\}^{1/2} \simeq 1-3\ln(2/\gamma)/\xi z^2. \tag{21} \]

To the right of the point \(y_3\) the condition \(x<1\) is violated (and almost immediately the condition \(xy<1\) as well), and therefore on the comparatively narrow interval \([y_3,1]\), for \(x_1(y)\) one should use the approximation (15).

1. The analysis carried out above refers to the case \(z<1\); however, before passing to the case \(z>1\), let us discuss the already revealed dependence of the solution on \(\xi\) and \(z\). It is easy to see that for \(y<y_3\) (where \(y_3\) is determined from (14) and (21)), \(x_1(y,\xi,z)\) depends only on the combination \(\xi z^2\), i.e., in a sufficiently broad region the relaxation properties of the system do not depend on the thermal conductivity of the inner cylinder, and its influence is manifested only

* The indicated equation also has a second root

\[ y_1' \simeq \frac{8}{\pi\gamma}\left[1-\frac{2\gamma^2}{\pi(\xi z^2-1)}\right]^{1/2}, \]

which is greater than 1 by virtue of \(\xi z^2-1 > 2e\gamma^2/\pi\).

through the ratio of the volume heat capacities of the cylinders. Moreover, taking into account the assumption made earlier, \(xyz \ll 1\), it can be shown that (10a, b) must be supplemented by the inequalities \(\xi z^2 \gg 4z^2\) and \(\xi z^2 \gg 1+2.2z^2\), as a result of which the interval \([y_3,1]\), where \(s_1\) depends not on the ratio of the volume heat capacities \(\xi z^2\), but on the ratio of the thermal conductivities \(\xi\) and the temperature diffusivity inside the cylinder, is very narrow.

Fig. 1. \(x=f(y)\), \(z\gg1\), \(\xi>4\) \((z=3,\ \xi z^2=50)\). The numbers of the formulas for \(x_1(y)\) in each region of \(y\) are indicated in parentheses.

1. Thus, let us consider (15) for \(z>1\). It is evident in advance that for \(z\sim 1\), \(x_1(y)\) is a smooth curve with a minimum, whose depth depends on \(\xi\), while \(x_1(0)\) and \(x_1(1)\) are approximately equal. It is clear that for small \(\xi z^2\), over a wide range of \(y\), the difference between \(x_1(y)\) and \(x_1(0)\) is small, and therefore it is expedient to seek a solution only for \(z\gg1\) and \(\xi z^2\gg1\).

As before, for small \(y\) the solution decreases according to (8) up to the point \(y_1\), where the condition \(x>1\) is violated. However, it is now necessary to check where the assumption \(xyz<1\) is violated. Substituting \(xy=1\) in (7), we find, with allowance for \(\xi z^2\gg1\),

\[ \tilde y_1=z^{-1}\left[\pi/4+\operatorname{arc\,tg}(2\ln(\gamma/2z))+8/\pi\xi\right]^{-1}. \tag{22} \]

We shall not solve the inequality \(y_1\le \tilde y_1\), since from (22) it is easy to obtain that under the condition

\[ z>\frac{1}{2\gamma}\exp(-1/2-4/\pi\xi)\simeq 1 \]

\(\tilde y_1\) becomes negative, so that the transition from solution (8) to (18) for \(z>1\) is analogous to that already considered in Sec. 6. Accordingly, the conclusions drawn earlier concerning the dependence of \(x_1\) on the parameters of the system remain valid. In the region \(y>y_1\), the function \(x_1(y)\) decreases, and, depending on the magnitude of \(\xi\), two cases are possible. To examine them, from (17) we find the point where \(xyz=1\),

\[ \tilde y_3\simeq \exp\left[-(8z^2+\ln 2\gamma^2)/2\xi z^2\right]. \tag{23} \]

Comparing \(\tilde y_3\) with (19), it is easy to see that for \(\xi z^2\ge 4\), \(y_1<\tilde y_3\), i.e. the minimum of the function \(x_1(y)\) is described according to (18), (19). For \(y>y_3\), instead of (5), in view of \(xyz>1\), we have

\[ \operatorname{tg}\left(xyz-\frac{\pi}{4}\right) = \frac{1}{\xi z\ln y} \left(-\frac{1}{xy}-xy\ln\frac{\gamma x}{2}\right) \tag{24} \]

or

\[ x_1= \left[ \frac{\pi}{4} +\operatorname{arctg}\left\{ \frac{1}{\xi z\ln y} \left(-\frac{1}{xy}-xy\ln\frac{\gamma x}{2}\right) \right\} \right]\bigg/yz; \tag{25} \]

since \(x\ll1\) and \(1-y_3\ll1\), (26) can be rewritten in the form

\[ x_1(y)\simeq \left[ \frac{\pi}{4} +\operatorname{arctg}\left(\frac{4}{3\pi\xi\ln(1/y)}\right) \right]\bigg/yz; \tag{25a} \]

(25) as \(y\to1\) agrees well with (6), and, moreover, from (24) we have

\[ (dx/dy)_{y=1}=x_1(1)\left[\xi-1-x\ln(\gamma x/2)\right], \tag{26} \]

which differs from the exact result (15) by a term of order \(2/\gamma e(\xi-1)\ll1\). Finally, putting \(xyz=1\) in (24), we find the left boundary of the range of applicability of (26), (25),

\[ \tilde y_4\simeq \exp\left[-(z^2-\ln z)/\xi z^2(1-\pi/4)\right], \tag{27} \]

which agrees well with \(\tilde y_3\). The general behavior of the function \(x_1(y)\) for \(z\gg1\), \(\xi>4\) is shown in Fig. 1.

If \(1\le\xi\le4\), then the dependence \(x_1(y)\) is analogous to the preceding case, with the only difference that the region of the minimum of the function is described not by expressions (17), (18), but by (24), (25).

Institute of High Temperatures
Academy of Sciences of the USSR

Received
13 IX 1968