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On One Problem of Heat Propagation in a Rotating Cylinder
R. S. Minasyan
1. Formulation of the problem and construction of the solution
The question of determining the thermal field in a rod and a cylinder moving in the direction of their axis was considered by Carslaw and Jaeger [1]. A number of problems on the distribution of temperature in a homogeneous plate, cylinder, and sphere with moving heat sources were solved in studies [2–7]. The problem of heat propagation in a rotating composite sphere was solved in [8].
In the present paper we give the solution of the problem of planar heat propagation in an infinite circular homogeneous cylinder rotating with constant angular velocity \(\omega\), when heat exchange with two different media takes place on the surface of the cylinder (see Fig.). We assume that the temperature of the surrounding media, varying arbitrarily along the circumference at the surface of the cylinder, does not depend on time. In this case the thermal field of the cylinder with respect to the external medium will be quasi-stationary [3], and if we denote by \(U\) the temperature inside the cylinder, then \(U\) will satisfy the following differential equation in a fixed coordinate system [3, 9]:
\[ -\omega \frac{\partial U}{\partial \varphi} = a\left( \frac{\partial^{2}U}{\partial r^{2}} + \frac{1}{r}\frac{\partial U}{\partial r} + \frac{1}{r^{2}}\frac{\partial^{2}U}{\partial \varphi^{2}} \right) \tag{1.1} \]
and the conditions on the surface of the cylinder \(r=R\)
\[ \left. \frac{\partial U(r,\varphi)}{\partial r} \right|_{r=R} = h_{1}\,[S_{1}(\varphi)-U(R,\varphi)] \quad \text{for } \alpha<\varphi<2\pi-\alpha; \tag{1.2} \]
\[ \left. \frac{\partial U(r,\varphi)}{\partial r} \right|_{r=R} = h_{2}\,[S_{2}(\varphi)-U(r,\varphi)] \tag{1.3} \]
for \(0 \leq \varphi<\alpha\) and \(2\pi-\alpha<\varphi\leq 2\pi\).
Here \(a\) is the thermal diffusivity coefficient of the cylinder; \(h_1, S_1(\varphi)\) are, respectively, the heat-transfer coefficient and the temperature of the surrounding medium on the surface segment \(CE'C'\) \((\alpha < \varphi < 2\pi-\alpha)\), and \(h_2, S_2(\varphi)\) are the corresponding quantities on the segment \(C'EC\).
With regard to \(S_1(\varphi)\) and \(S_2(\varphi)\) we assume that they have bounded variation.
To find the solution of the problem, we first expand the function \(U(r,\varphi)\) in a series in the functions \(e^{ik\varphi}\):
\[ U(r,\varphi)=\sum_{k=-\infty}^{\infty} F_k(r)e^{ik\varphi}, \tag{1.4} \]
where
\[ F_k(r)=\frac{1}{2\pi}\int_0^{2\pi} U(r,\varphi)e^{-ik\varphi}\,d\varphi . \]
Multiplying equation (1.1) by \(\frac{1}{2\pi}e^{-ik\varphi}\,d\varphi\) and integrating from \(0\) to \(2\pi\), to determine \(F_k(r)\) we obtain the following equation:
\[ F_k''(r)+\frac{1}{r}F_k'(r)+\left(\frac{ik\omega}{a}-\frac{k^2}{r^2}\right)F_k(r)=0. \tag{1.5} \]
Solving equation (1.5) and taking into account the condition that the solution be bounded at \(r=0\), we have
\[ F_k(r)=A_kJ_k(\sqrt{i}\,\gamma_k r);\qquad F_0(r)=A_0, \tag{1.6} \]
where \(J_k\) is the Bessel function of the first kind of order \(k\); \(\gamma_k=\sqrt{\frac{\omega k}{a}}\).
Then, grouping in (1.4) the mutually conjugate terms, we shall have
\[ U(r,\varphi)=A_0+\sum_{k=1}^{\infty}\bigl[f_k(r)\sin k\varphi+g_k(r)\cos k\varphi\bigr], \tag{1.7} \]
where
\[ \begin{aligned} f_k(r)&=c_ku_k(\gamma_k r)+d_kv_k(\gamma_k r),\\ g_k(r)&=d_ku_k(\gamma_k r)-c_kv_k(\gamma_k r). \end{aligned} \tag{1.8} \]
Here \(u_k(x)=\operatorname{ber}_k x\) and \(v(x)=\operatorname{bei}_k x\) are Thomson functions of the first kind of order \(k\), representing the real and imaginary parts of the Bessel function of a complex argument: \(J_k(i\sqrt{i}\,x)=u_k(x)+iv_k(x)\).
Before proceeding to determine the unknowns \(c_k\) and \(d_k\) entering into expressions (1.8), we represent the boundary conditions (1.2) and (1.3) in the following form:
\[ \left.\frac{\partial U(r,\varphi)}{\partial r}\right|_{r=R} +\left[\frac{h_1+h_2}{2}+\psi(\varphi)\right]U(R,\varphi)=S(\varphi), \tag{1.9} \]
where the function
\[ \psi(\varphi)=(h_1-h_2)\left[\frac{1}{2}-\frac{\alpha}{\pi}-\frac{2}{\pi}\sum_{k=1}^{\infty}\frac{\sin k\alpha\cos k\varphi}{k}\right] \]
is equal to
\(\dfrac{h_1-h_2}{2}\) for \(\alpha<\varphi<2\pi-\alpha\) and \(-\dfrac{h_1-h_2}{2}\) in the intervals \((0,\alpha)\) and \((2\pi-\alpha,2\pi)\), while \(S(\varphi)\) is equal to \(h_1S_1(\varphi)\) in \((\alpha,2\pi-\alpha)\) and to \(h_2S_2(\varphi)\) in \((0,\alpha)\) and \((2\pi-\alpha,2\pi)\).
Multiplying condition (1.9) successively by \(\dfrac{1}{\pi}\sin k\varphi\,d\varphi\) and \(\dfrac{1}{\pi}\cos k\varphi\,d\varphi\), and integrating from \(0\) to \(2\pi\), we shall have
\[ f'_k(R)+\left[H+\frac{(h_1-h_2)\sin 2k\alpha}{2k\pi}\right]f_k(R) = \]
\[ =\frac{2(h_1-h_2)}{\pi} \sum_{j=1}^{\infty}{}' \frac{k\cos k\alpha \sin j\alpha-j\cos j\alpha \sin k\alpha}{j^2-k^2}\, f_j(r) +\frac{1}{\pi}\int_{0}^{2\pi} S(\varphi)\sin k\varphi\,d\varphi; \]
\[ g'_k(R)+\left[H-\frac{(h_1-h_2)\sin 2k\alpha}{2k\pi}\right]g_k(R) = \frac{2(h_1-h_2)}{\pi} \left[ \frac{\sin k\alpha}{k}A_0+ \right. \]
\[ \left. +\sum_{j=1}^{\infty}{}' \frac{j\sin j\alpha \cos k\alpha-k\sin k\alpha \cos j\alpha}{j^2-k^2}\, g_j(R) \right] +\frac{1}{\pi}\int_{0}^{2\pi} S(\varphi)\cos k\varphi\,d\varphi; \tag{1.10} \]
\[ HA_0= \frac{h_1-h_2}{\pi} \sum_{j=1}^{\infty} \frac{\sin j\alpha}{j}\,g_j(R) + \frac{1}{2\pi}\int_{0}^{2\pi} S(\varphi)\,d\varphi, \]
where
\[ H=\left(1-\frac{\alpha}{\pi}\right)h_1+\frac{\alpha}{\pi}h_2, \]
and the prime on the summation sign denotes that, in summation, the index \(j=k\) is omitted.
Thus, for determining the constants \(c_k\) and \(d_k\) entering into the expressions for \(f_k(r)\) and \(g_k(r)\), we have obtained the collection of two infinite systems of linear algebraic equations (1.10).
2. Investigation of the collection of infinite systems (1.10). Special cases
We shall investigate the solvability of the obtained infinite systems (1.10) and the rate of decrease of their solutions \(c_k\) and \(d_k\). First we transform them, denoting
\[ c_k= \frac{m_k u_k(\gamma_k R)-n_k v_k(\gamma_k R)} {k^{3/2}\left[u_k^2(\gamma_k R)+v_k^2(\gamma_k R)\right]}; \tag{2.1} \]
\[ d_k= \frac{m_k v_k(\gamma_k R)+n_k u_k(\gamma_k R)} {k^{3/2}\left[u_k^2(\gamma_k R)+v_k^2(\gamma_k R)\right]}; \qquad A_0=\frac{n_0}{2}. \]
Substituting the values of \(c_k\) and \(d_k\) from (2.1) into (1.8), from (1.10), for determining the new unknowns \(m_k\) and \(n_k\), after some transformations we obtain the following collection of two infinite systems:
\[ m_k= \frac{2(h_1-h_2)k^{3/2}}{G_k\pi} \left\{ M_k \sum_{j=1}^{\infty}{}' \frac{k\cos k\alpha \sin j\alpha-j\cos j\alpha \sin k\alpha} {j^{3/2}(j^2-k^2)} \,m_j - \right. \]
\[ - N_k \left[ \frac{\sin k\alpha}{2k} n_0 + \sum_{j=1}^{\infty}{}' \frac{j \sin j\alpha \cos k\alpha - k \sin k\alpha \cos j\alpha} {j^{\frac32}(j^2-k^2)} n_j \right] \Bigg\} + q_k; \tag{2.2} \]
\[
n_k =
\frac{2(h_1-h_2)k^{\frac32}}{G_k\pi}
\Bigg\{
L_k \left[
\frac{\sin k\alpha}{2k} n_0 +
\sum_{j=1}^{\infty}{}'
\frac{j \sin j\alpha \cos k\alpha - k \sin k\alpha \cos j\alpha}
{j^{\frac32}(j^2-k^2)} n_j
\right]
+
\]
\[
+ N_k \sum_{j=1}^{\infty}{}'
\frac{k \cos k\alpha \sin j\alpha - j \cos j\alpha \sin k\alpha}
{j^{\frac32}(j^2-k^2)} m_j
\Bigg\} + t_k;
\]
\[ n_0 = \frac{2(h_1-h_2)}{\pi H} \sum_{j=1}^{\infty} \frac{\sin j\alpha}{j^{\frac32}} n_j + t_0, \]
where, for brevity, the following notation has been introduced:
\[
G_k =
\gamma_k^2 \left[ u_k'^2(\gamma_k R) + v_k'^2(\gamma_k R) \right]
+ 2\gamma_k H \left[ u_k(\gamma_k R)u_k'(\gamma_k R) +
\right.
\]
\[
\left.
+ v_k(\gamma_k R)v_k'(\gamma_k R) \right]
+
\left[ H^2 - (h_1-h_2)^2 \frac{\sin^2 2k\alpha}{4k^2\pi^2} \right]
\left[ u_k^2(\gamma_k R) + v_k^2(\gamma_k R) \right];
\]
\[
M_k =
\gamma_k \left[ u_k(\gamma_k R)u_k'(\gamma_k R) +
v_k(\gamma_k R)v_k'(\gamma_k R) \right] +
\]
\[
+
\left[ H - (h_1-h_2)\frac{\sin 2k\alpha}{2k\pi} \right]
\left[ u_k^2(\gamma_k R) + v_k^2(\gamma_k R) \right];
\]
\[ L_k = M_k + (h_1-h_2)\frac{\sin 2k\alpha}{k\pi} \left[ u_k^2(\gamma_k R)+v_k^2(\gamma_k R) \right]; \tag{2.3} \]
\[ N_k = \gamma_k \left[ u_k(\gamma_k R)V_k'(\gamma_k R) - v_k(\gamma_k R)u_k'(\gamma_k R) \right]; \]
\[ q_k = \frac{k^{\frac32}}{G_k\pi} \int_0^{2\pi} S(\varphi)(M_k \sin k\varphi - N_k \cos k\varphi)\,d\varphi; \]
\[ t_k = \frac{k^{\frac32}}{G_k\pi} \int_0^{2\pi} S(\varphi)(L_k \cos k\varphi + N_k \sin k\varphi)\,d\varphi; \qquad t_0 = \frac{1}{H\pi}\int_0^{2\pi} S(\varphi)\,d\varphi. \]
Let us estimate the sum of the moduli of the coefficients of the unknowns \(m_j\) and \(n_j\) in each of equations (2.2). If we denote by \(\sigma_k\) the sum of the moduli of the coefficients of the \(k\)-th equation of the first system (2.2), we shall have
\[
\sigma_k =
\frac{2|h_1-h_2|k^{\frac32}}{G_k\pi}
\Bigg\{
M_k \sum_{j=1}^{\infty}{}'
\frac{\left| k\cos k\alpha \sin j\alpha - j\cos j\alpha \sin k\alpha \right|}
{j^{\frac32}|j^2-k^2|}
+
\]
\[
+
|N_k|
\left[
\frac{|\sin k\alpha|}{2k}
+
\sum_{j=1}^{\infty}{}'
\frac{\left| j\sin j\alpha \cos k\alpha - k\sin k\alpha \cos j\alpha \right|}
{j^{\frac32}|j^2-k^2|}
\right]
\Bigg\}.
\tag{2.4}
\]
Let us first estimate the ratios \(\dfrac{M_k}{G_k}\) and \(\dfrac{|N_k|}{G_k}\). For this we use the representation of the sum of squares \(u_\nu^2(x)+v_\nu^2(x)\) for an arbitrary index \(\nu>0\) in the form of the series [10]
\[ u_\nu^2(x)+v_\nu^2(x)= \sum_{m=0}^{\infty} \frac{\left(\dfrac{1}{2}x\right)^{2\nu+4m}} {m!\Gamma(\nu+m+1)\Gamma(\nu+2m+1)}, \tag{2.5} \]
and also the easily obtained differential equations satisfied by the functions \(u_\nu(x)=\operatorname{ber}_\nu x\) and \(v_\nu(x)=\operatorname{bei}_\nu x\):
\[ u_\nu''(x)+\frac{1}{x}u_\nu'(x)-\frac{\nu^2}{x^2}u_\nu(x)=-v_\nu(x); \]
\[ v_\nu''(x)+\frac{1}{x}v_\nu'(x)-\frac{\nu^2}{x^2}v_\nu(x)=u_\nu(x). \tag{2.6} \]
Differentiating relation (2.5) twice with respect to \(x\) and taking (2.6) into account, we have
\[
u_\nu'^2(x)+v_\nu'^2(x)
-\frac{1}{x}\left[u_\nu(x)u_\nu'(x)+v_\nu(x)v_\nu'(x)\right]
+\frac{\nu^2}{x^2}\left[u_\nu^2(x)+v_\nu^2(x)\right]
\]
\[
=
\frac{1}{2x^2}\sum_{m=0}^{\infty}
\frac{(2\nu+4m)(2\nu+4m-1)\left(\dfrac{1}{2}x\right)^{2\nu+4m}}
{m!\Gamma(\nu+m+1)\Gamma(\nu+2m+1)},
\tag{2.7}
\]
whence
\[
u_\nu'^2(x)+v_\nu'^2(x)=
\frac{1}{x^2}\sum_{m=0}^{\infty}
\frac{(\nu^2+8\nu m+8m^2)\left(\dfrac{1}{2}x\right)^{2\nu+4m}}
{m!\Gamma(\nu+m+1)\Gamma(\nu+2m+1)}
>
\]
\[
>
\frac{\nu^2}{x^2}\left[u_\nu^2(x)+v_\nu^2(x)\right].
\tag{2.8}
\]
Further, from (2.5) we have, for \(x>0\),
\[
u_\nu(x)u_\nu'(x)+v_\nu(x)v_\nu'(x)=
\frac{1}{x}\sum_{m=0}^{\infty}
\frac{(\nu+2m)\left(\dfrac{1}{2}x\right)^{2\nu+4m}}
{m!\Gamma(\nu+m+1)\Gamma(\nu+2m+1)}
>
\]
\[
>
\frac{\nu}{x}\left[u_\nu^2(x)+v_\nu^2(x)\right].
\tag{2.9}
\]
From inequalities (2.8) and (2.9) we easily obtain
\[ \frac{M_k}{G_k}<\frac{R}{k}; \]
\[ \frac{|N_k|}{G_k} \le \frac{\gamma_k}{G_k} \sqrt{\left[u_k^2(\gamma_k R)+v_k^2(\gamma_k R)\right] \left[u_k'^2(\gamma_k R)+v_k'^2(\gamma_k R)\right]} < \frac{R}{k}. \tag{2.10} \]
Then, estimating the sums of the series in (2.4), we have
\[ \sum_{j=1}^{\infty} {}' \frac{|k\cos k\alpha\sin j\alpha-j\cos j\alpha\sin k\alpha|} {j^{3/2}|j^2-k^2|} < \frac{2}{k}+\frac{\ln k+6}{k^{3/2}}. \tag{2.11} \]
We obtain the same estimate also for the second sum. Taking (2.10) and (2.11) into account, from (2.4), after simplifications, we shall have
\[ \sigma_k < \frac{4|h_1-h_2|R}{\pi}\left(\frac{2}{k^{1/2}}+\frac{\ln k+6}{k}\right). \tag{2.12} \]
The expression on the right-hand side of inequality (2.12), beginning with the value
\[ k \ge \frac{1}{2}P^2\left(1+\sqrt{1+\frac{6}{P}}\right)^2, \]
where
\[ P=\frac{8|h_1-h_2|R}{\pi}, \]
becomes less than 1; moreover, as \(k\) increases it decreases with rapidity \(\dfrac{P}{k^{1/2}}\). Analogous estimates are obtained also for the second of systems (1.10).
Thus, the systems (1.10) are quasiregular, and the sums of the moduli of the coefficients tend to zero. On the other hand, taking into account the condition of bounded variation of the functions \(S_1(\varphi)\) and \(S_2(\varphi)\), from (2.3) we obtain that the free terms \(q_k\) and \(t_k\) of systems (2.2) tend to zero with rapidity \(O(k^{-1/2})\).
From the theory of infinite systems [11] it follows that the infinite systems of linear equations (2.2) have a unique bounded solution, which can be found by the method of successive approximations. Hence it follows that the constants \(c_k\) and \(d_k\), according to (2.1), tend to zero with rapidity
\[ O\left\{k^{-3/2}\left[u_k^2(\gamma_kR)+v_k^2(\gamma_kR)\right]^{-1/2}\right\}. \]
Substituting the values \(c_k\) and \(d_k\) from (2.1) into (1.7), for the function \(U(r,\varphi)\) we finally obtain
\[ \begin{aligned} U(r,\varphi)=\frac{n_0}{2} &+\sum_{k=1}^{\infty} \frac{1}{k^{3/2}\left[u_k^2(\gamma_kR)+v_k^2(\gamma_kR)\right]} \Big\{[m_ku_k(\gamma_kR) \\ &\quad -n_kv_k(\gamma_kR)] \left[u_k(\gamma_kr)\sin k\varphi-v_k(\gamma_kr)\cos k\varphi\right] \\ &\quad +[m_kv_k(\gamma_kR)+n_ku_k(\gamma_kR)] \left[v_k(\gamma_kr)\sin k\varphi+u_k(\gamma_kr)\cos k\varphi\right]\Big\}. \end{aligned} \tag{2.13} \]
The series entering into expression (2.13) for \(U(r,\varphi)\) converges very rapidly in the domain under consideration, including the boundary. Assigning values to the quantities \(h_1\), \(h_2\), the functions \(S_1(\varphi)\) and \(S_2(\varphi)\), and also to the ratio \(\dfrac{\omega}{a}\) and the angle \(a\), from (2.2) we obtain upper and lower estimates for \(m_k\) and \(n_k\), after which, by the method set forth in [12], we find from (2.13) the values of \(U(r,\varphi)\) with excess and deficiency.
We note that if one passes to the moving coordinate system \((r,\varphi_1)\), rotating together with the cylinder, then in expression (2.13) \(\varphi\) must be replaced by \(\varphi_1+\omega t\).
In conclusion let us consider some special cases:
a) if \(h_1=h_2\), i.e. the surrounding medium is homogeneous, then the series entering into (2.2) vanish, the systems of infinite algebraic equations (2.2) reduce to equalities, and the solution coincides with the solution obtained in the usual way;
b) if the angular velocity \(\omega=0\), we obtain the stationary regime. In this case, taking into account that
\[ \lim_{\omega\to 0}\omega^{-k/2}J_k\left(e^{3\pi i/4}\gamma_kr\right) = \frac{1}{2^k\Gamma(k+1)}e^{3k\pi i/4}\left(\frac{k}{a}\right)^{k/2}r^k, \]
we have
\[ U(r,\varphi)=\frac{n_0}{2} +\sum_{k=1}^{\infty}\frac{1}{k^{3/2}} \left(\frac{r}{R}\right)^k (m_k\sin k\varphi+n_k\cos k\varphi). \tag{2.14} \]
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Received by the editors
November 9, 1964
Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR