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Spatial Heat-Conduction Problems for Certain Bodies Consisting of Inhomogeneous Material
V. I. Makhovikov
Using the method set forth in [1], an exact solution is obtained for spatial problems of steady-state heat conduction for a cylinder and a hollow sphere made of inhomogeneous material. The heat-conduction equation for an inhomogeneous medium, in the absence of heat sources in the medium, has the form [2, 3]
\[ N(T)\equiv \frac{\partial}{\partial x}\left(\lambda \frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(\lambda \frac{\partial T}{\partial y}\right) + \frac{\partial}{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right)=0, \tag{1} \]
where \(\lambda\) is a variable coefficient of thermal conductivity.
1. Problems for a cylinder. Let the generators of the cylinder be parallel to the \(z\)-axis, and let its cross section be arbitrary, bounded by a contour \(K\). The height of the cylinder is \(2h\), and \(-h \leq z \leq h\). Let
\[ \lambda=\mu_0(x,y)\mu_1(z), \tag{2} \]
where \(\mu_0(x,y)\) and \(\mu_1(z)\) are differentiable functions of the variables \(x,y\) and \(z\), respectively. Taking (2) into account, we transform equation (1) to the form
\[ N(T)\equiv \lambda\left[ \Delta_1+\frac{\partial^2}{\partial z^2} +\omega(z)\frac{\partial}{\partial z} \right]T=0, \tag{3} \]
where
\[ \omega(z)=\mu_1'(z):\mu_1(z), \tag{4} \]
\[ \Delta_1= \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{1}{\mu_0(x,y)} \times \]
\[ \times \left[ \frac{\partial}{\partial x}\mu_0(x,y)\frac{\partial}{\partial x} + \frac{\partial}{\partial y}\mu_0(x,y)\frac{\partial}{\partial y} \right]. \tag{5} \]
We assign to the temperature the form
\[ T=\operatorname{Re}\left[ \sum_{j=0}^{\infty}\lambda_j(z)\Phi_j + \sum_{k=0}^{n}\varphi_k(z)\Lambda^k q \right]. \tag{6} \]
Here the notation is as follows: \(\operatorname{Re} F\) is the real part of the function \(F\); \(\Phi_j, q\) are functions of \(x,y\) satisfying the equations
\[ \Lambda_j\Phi_j=0,\qquad \Lambda^{\,n+1}q=0; \tag{7} \]
\(\Lambda_j, \Lambda\) are differential operators
V. I. MAKНOVIKOV
\[ \Lambda_j=\Delta_1-\alpha_j^2,\qquad \Lambda=\Delta_1+\alpha^2, \tag{8} \]
\(\Lambda^k=\Lambda\Lambda^{k-1},\quad k=1,2,\ldots,\quad \Lambda^0=1;\ \alpha,\alpha_j\) are complex constants; \(\lambda_j(z), \varphi_k(z)\) are functions of \(z\) satisfying the equations
\[ \lambda_j''(z)+\omega(z)\lambda_j'(z)+\alpha_j^2\lambda_j(z)=0,\qquad j=0,1,2,\ldots; \tag{9} \]
\[ \varphi_k''(z)+\omega(z)\varphi_k'(z)-\alpha^2\varphi_k(z)+\varphi_{k-1}(z)=0,\qquad k=0,1,\ldots, \tag{10} \]
\(\varphi_{-1}(z)=0\). It is not difficult to verify that the temperature (6), under conditions (7)—(10), satisfies equation (3). Let us note that equation (9), by the substitution \(\lambda_j(z)=\rho_j(z)e^{-\frac12\int \omega(z)\,dz}\), is transformed into the equation
\[ \rho_j''(z)+\left[\alpha_j^2-\frac12\omega'(z)-\frac14\omega^2(z)\right]\rho_j(z)=0, \]
and the determination of \(\rho_j(z)\) and \(\alpha_j\) can be carried out by the methods for solving the Sturm—Liouville problem.
In what follows we restrict ourselves to the case
\[ \mu_1(-z)=\mu_1(z),\qquad \omega(-z)=-\omega(z). \tag{11} \]
Consider the solution of problems A and B under the following boundary conditions on the end faces \(z=\pm h\) of the cylinder:
\[ \text{problem A}\quad T=\operatorname{Re}q;\qquad \text{problem B}\quad \frac{\partial T}{\partial z}=\pm \operatorname{Re}q. \tag{12} \]
On the lateral surface \(\Pi\) of the cylinder let the temperature be given by
\[ T=\operatorname{Re}Q(z,s). \tag{13} \]
Here \(Q(z,s)\) is a function of the variable \(z\) and of the arc \(s\) of the contour \(K\), and
\[ Q(-z,s)=Q(z,s), \tag{14} \]
which for \(-h<z<h\) is assumed to be expandable in the series
\[ Q(z,s)=\sum_{j=0}^{\infty} a_j(s)\lambda_j(z). \tag{15} \]
For the special case
\[ \mu_1(z)=1,\qquad \omega(z)=0 \tag{16} \]
the form of the functions \(a_j(s)\) is given below.
To satisfy the boundary conditions (12), we take account of (6) and require
\[ \text{in problem A}\quad \lambda_j(\pm h)=0;\qquad \text{in problem B}\quad \lambda_j'(\pm h)=0; \tag{17} \]
\[ \begin{aligned} &\text{in problem A}\quad \varphi_0(\pm h)=1;\quad \varphi_\gamma(\pm h)=0;\\ &\text{in problem B}\quad \varphi_0'(\pm h)=\pm 1,\quad \varphi_\gamma'(\pm h)=0, \end{aligned} \tag{18} \]
where \(\gamma=1,2,\ldots,n\). Obviously, by virtue of (11), (9), (10) and (14), (17), (18), the functions \(\lambda_j(z)\), \(\varphi_k(z)\) must be even in \(z\), and the conditions (17), (18) are quite realistically satisfiable for these functions. The boundary conditions (12) may thus be regarded as satisfied.
In the special case (16), the solution of equations (9), (10) for problems A, B takes the form
\[ \lambda_j(z)=\cos \alpha_j z,\quad j=1,2,\ldots,\quad \varphi_k(z)=c_k\operatorname{ch}\alpha z+v_k(z), \tag{19} \]
where \(\lambda_0(z)=1\), \(c_k\) are constants, \(v_0(z)=0\), and \(v_k(z)\) \((k=1,2,\ldots)\) is a particular solution of the equation \(v_k''(z)=\alpha^2 v_k(z)-v_{k-1}(z)-c_{k-1}\operatorname{ch}\alpha z\). The parameter \(\alpha_j\) \((j=1,2,\ldots)\) has the form
\[ \text{for problem A}\quad \alpha_j=\frac{\pi}{2h}(2j-1);\qquad \text{for problem B}\quad \alpha_j=\frac{\pi}{h}j, \tag{20} \]
and \(\Phi_0=0\) for problem A. It is easy to see, taking into account (19), (20), that conditions (17) are satisfied. Equations (18) will be satisfied with the aid of the constants \(c_k\), which are obtained from (18) in the form:
\[ \text{for problem A}\quad c_0=\operatorname{sch}\alpha h,\quad c_\gamma=-v_\gamma(h)\operatorname{sch}\alpha h; \]
\[ \text{for problem B}\quad c_0=\frac{1}{\alpha}\operatorname{csch}\alpha h,\quad c_\gamma=-\frac{1}{\alpha}v_\gamma'(h)\operatorname{csch}\alpha h. \]
Let us now consider the satisfaction of the boundary condition (13) on the surface \(\Pi\) of the cylinder. For the function \(\varphi_k(z)\) (10), for \(-h<z<h\), evidently the expansion
\[ \varphi_k(z)=\sum_{j=0}^{\infty}a_{j,k}\lambda_j(z), \tag{21} \]
is valid, where \(a_{j,k}\) are constants. In the particular case (16) we have (19), (20) and, according to the properties of Fourier series [4], for the constants \(a_{j,k}\) we shall have the formula
\[ a_{j,k}=\frac{2}{h}\int_0^h \varphi_k(z)\lambda_j(z)\,dz,\quad j=1,2,\ldots, \tag{22} \]
where for problem A \(a_{0,k}=0\); for problem B
\[ a_{0,k}=\frac{2}{h}\int_0^h \varphi_k(z)\,dz,\quad \lambda_0(z)=1. \]
For the coefficients \(a_j(s)\) of the expansion (15), taking into account (19), (20) and [4], we shall have the formula
\[ a_j(s)=\frac{2}{h}\int_0^h Q(z,s)\lambda_j(z)\,dz,\quad j=1,2,\ldots, \tag{23} \]
where for problem A \(a_0(s)=0\); for problem B
\[ a_0(s)=\frac{2}{h}\int_0^h Q(z,s)\,dz. \]
Taking into account (6), (21), we obtain, for \(-h<z<h\), for the temperature (6) the representation
\[ T=\operatorname{Re}\sum_{j=0}^{\infty}\lambda_j(z)(\Phi_j+q_j), \tag{24} \]
where
\[ q_j=\sum_{k=0}^{n}a_{j,k}\Lambda^k q. \]
Taking into account (15), (24), we satisfy the boundary condition (13) on the surface \(\Pi\) by solving the following plane problems on the contour \(K\) with respect to the functions \(\Phi_j\) (7):
\[ \Phi_j=a_j(s)-q_j,\quad j=0,1,2,\ldots. \tag{25} \]
For solving the plane problems (25), one can use known methods. Heat-conduction problems with a temperature distribution antisymmetric with respect to the plane \(z=0\) are solved analogously. In this case the functions \(\lambda_j(z)\), \(\varphi_k(z)\) must be odd in \(z\).
Let us note that, since no restrictions have been imposed on the parameter \(\alpha\), the function \(q\) satisfying equations (7), (8) may be assigned to a comparatively broad class of functions.
Consider the solution of the problem for a cylinder when, in formula (2),
\[ \mu_1(z)=e^{2az}, \tag{26} \]
where \(a\) is a real constant. From (4) we have \(\omega(z)=2a\), and, giving the functions \(\lambda_j(z)\), \(\varphi_k(z)\) the form
\[ \lambda_j(z)=e^{-az} f_j(z),\qquad f_0(z)=0,\qquad \varphi_k(z)=e^{-az}\psi_k(z), \tag{27} \]
we obtain from (9), (10) the conditions
\[ f_j''(z)+\beta_j^2 f_j(z)=0,\qquad \beta_j^2=\alpha_j^2-a^2,\qquad j=1,2,\ldots, \tag{28} \]
\[ \psi_k''(z)=\beta^2\psi_k(z)-\psi_{k-1}(z),\qquad \beta^2=\alpha^2+a^2,\qquad k=0,1,2,\ldots \tag{29} \]
It is not difficult to verify that the functions (27), under conditions (26), (28), (29), satisfy equations (9), (10).
On the ends \(z=\pm h\) and on the lateral surface \(\Pi\) of the cylinder, let the temperature be prescribed as
\[ T=e^{\mp ah}\operatorname{Re} q \quad \text{for } z=\pm h, \tag{30} \]
\[ T=e^{-az}\operatorname{Re} Q_0(z,s) \quad \text{on } \Pi, \tag{31} \]
where \(q\) satisfies equation (7), and let \(Q_0(z,s)\) be subject to the condition
\[ Q_0(-z,s)=Q_0(z,s) \tag{32} \]
and be expandable for \(-h<z<h\) in the series
\[ Q_0(z,s)=\sum_{j=1}^{\infty} b_j(s)f_j(z), \tag{33} \]
where \(b_j(s)\) are determined below. Equations (28), (29) will be satisfied if
\[ f_j(z)=\cos \beta_j z,\qquad \beta_j=\frac{\pi}{2h}(2j-1),\qquad j=1,2,\ldots, \tag{34} \]
\[ \psi_k(z)=e_k\operatorname{ch}\beta z+u_k(z),\qquad k=0,1,\ldots, \tag{35} \]
where \(e_k\) are constants, \(u_0(z)=0\), and \(u_k(z)\) \((k=1,2,\ldots)\) is a particular solution of the equation
\[ u_k''(z)=\beta^2 u_k(z)-u_{k-1}(z)-e_{k-1}\operatorname{ch}\beta z. \]
It is easy to see, taking into account (6), (27), (34), that if the equalities
\[ \psi_0(\pm h)=1,\qquad \psi_\gamma(\pm h)=0,\qquad \gamma=1,2,\ldots,n, \tag{36} \]
hold, then the boundary condition (30) is satisfied. The equalities (36) will be fulfilled by means of the constants \(e_k\), and, taking (35) into account, they are obtained from (36) in the form
\(e_0=\operatorname{sch}\beta h,\ e_\gamma=-u_\gamma(h)\operatorname{sch}\beta h\).
To satisfy the boundary condition (31) on the surface \(\Pi\) of the cylinder, we take into account (6), (27), (35), (34) and, using the properties
of Fourier series [4], for \(-h<z<h\) we represent
\[ \psi_k(z)=\sum_{j=1}^{\infty} b_{j,k} f_j(z), \qquad b_{j,k}=\frac{2}{h}\int_0^h \psi_k(z) f_j(z)\,dz . \tag{37} \]
It is obvious that, for the expansion coefficients (33), taking into account (32), (34), [4], the formula
\[ b_j(s)=\frac{2}{h}\int_0^h Q_0(z,s) f_j(z)\,dz \]
is valid.
Taking (37), (27) into account, for the temperature (6), for \(-h<z<h\) we obtain the representation
\[ T=e^{-az}\operatorname{Re}\sum_{j=1}^{\infty} f_j(z)(\Phi_j+p_j), \tag{38} \]
where \(p_j=\sum_{k=0}^{n} b_{j,k}\Lambda^k q\). Taking (38), (33) into consideration, we satisfy the boundary condition (31) on the surface \(\Pi\) by solving the following plane problems on the contour \(K\) with respect to the functions \(\Phi_j\) (7):
\[ \Phi_j=b_j(s)-p_j, \qquad j=1,2,3,\ldots \tag{39} \]
The plane problems (39) can be solved by known methods [3, 5]. The problem is solved analogously when \(Q_0(-z,s)=-Q_0(z,s)\). The functions \(f_j(z)\), \(\psi_k(z)\), satisfying equations (28), (29), in this case must be odd.
2. Problems for a hollow sphere. The hollow sphere under consideration is bounded by two spherical surfaces whose radii are \(h\) and \(H\) \((h<H)\), having one common center. Let \(\rho\) be the distance of a point of the hollow sphere to the center of the spheres, and let \(\varphi\) and \(\theta\) be the angles characterizing the position of a point on the sphere of radius \(\rho\) \((-\pi<\varphi\leqslant\pi,\ -\pi\leqslant\theta\leqslant\pi)\). Since
\[ \rho=\sqrt{x^2+y^2+z^2}, \qquad \varphi=\operatorname{arc\,tg}\frac{y}{x}, \qquad \theta=\operatorname{arc\,tg}\left(\frac{1}{z}\sqrt{x^2+y^2}\right), \]
\[ x=\rho\sin\theta\cos\varphi,\qquad y=\rho\sin\theta\sin\varphi,\qquad z=\rho\cos\theta, \tag{40} \]
\[ \frac{\partial}{\partial x} = \frac{1}{\rho}\left( x\frac{\partial}{\partial\rho} - \frac{\sin\varphi}{\sin\theta}\frac{\partial}{\partial\varphi} + \cos\theta\cos\varphi\frac{\partial}{\partial\theta} \right), \]
\[ \frac{\partial}{\partial y} = \frac{1}{\rho}\left( y\frac{\partial}{\partial\rho} + \frac{\cos\varphi}{\sin\theta}\frac{\partial}{\partial\varphi} + \cos\theta\sin\varphi\frac{\partial}{\partial\theta} \right), \]
\[ \frac{\partial}{\partial z} = \cos\theta\frac{\partial}{\partial\rho} - \frac{\sin\theta}{\rho}\frac{\partial}{\partial\theta}, \]
then in the spherical coordinate system (40) equation (1) takes the form
\[ N(T)=\lambda\Delta T + \frac{\partial\lambda}{\partial\rho}\frac{\partial T}{\partial\rho} + \frac{1}{\rho^2\sin^2\theta}\frac{\partial\lambda}{\partial\varphi}\frac{\partial T}{\partial\varphi} + \frac{1}{\rho^2}\frac{\partial\lambda}{\partial\theta}\frac{\partial T}{\partial\theta} =0, \tag{41} \]
where \(\Delta\) is the Laplace operator. Let \(\lambda=\mu_0(\rho)\mu_1(\varphi)\mu_2(\theta)\), where \(\mu_0,\mu_1,\mu_2\) are differentiable functions of the variables \(\rho,\varphi,\theta\), respectively. Then equation (41) can be transformed to the form
\[ N(T)=\frac{\lambda}{\rho^2}\left[\Delta_2+\rho^2\frac{\partial^2}{\partial \rho^2}+\omega_0(\rho)\frac{\partial}{\partial \rho}\right]T=0, \tag{42} \]
where
\[ \Delta_2=\frac{1}{\sin^2\theta}\left[\frac{\partial^2}{\partial \varphi^2} +\frac{\mu_1'(\varphi)}{\mu_1\varphi}\frac{\partial}{\partial \varphi}\right] +\frac{\partial^2}{\partial \theta^2} +\omega_1(\theta)\frac{\partial}{\partial \theta}, \tag{43} \]
\[ \omega_0(\rho)=2\rho+\rho^2\mu_0'(\rho):\mu_0(\rho),\qquad \omega_1(\theta)=\operatorname{ctg}\theta+\mu_2'(\theta):\mu_2(\theta). \tag{44} \]
We shall assign to the temperature the form
\[ T=\operatorname{Re}\left[\sum_{k=0}^{n} f_k(\rho)\Lambda_\alpha^{k}\Phi\right]. \tag{45} \]
Here \(\Phi\) denotes a function of \(\varphi,\theta\) satisfying the equation
\[ \Lambda_\alpha^{\,n+1}\Phi=0; \tag{46} \]
\(\Lambda_\alpha\) is the differential operator
\[ \Lambda_\alpha=\Delta_2+\alpha; \tag{47} \]
\(\alpha\) is an arbitrary complex constant; \(f_k(\rho)\) is a function of \(\rho\) that is a solution of the equation
\[ \rho^2 f_k''(\rho)+\omega_0(\rho)f_k'(\rho)-\alpha f_k(\rho)+f_{k-1}(\rho)=0. \tag{48} \]
It is not difficult to verify that, under conditions (46), (47), (48), the temperature (45) satisfies equation (42).
Let us consider the solution of problems A′ and B′ under the following generalized boundary conditions on spherical surfaces:
\[ \begin{aligned} \text{problem A′}\qquad &\gamma_1 T+\gamma_2\frac{\partial T}{\partial \rho}=\operatorname{Re} q &&\text{for } \rho=H,\\[6pt] &\gamma_1' T+\gamma_2'\frac{\partial T}{\partial \rho}=0 &&\text{for } \rho=h; \end{aligned} \tag{49} \]
\[ \begin{aligned} \text{problem B′}\qquad &\gamma_1 T+\gamma_2\frac{\partial T}{\partial \rho}=0 &&\text{for } \rho=H,\\[6pt] &\gamma_1' T+\gamma_2'\frac{\partial T}{\partial \rho}=\operatorname{Re} p &&\text{for } \rho=h. \end{aligned} \]
Here \(\gamma_1,\gamma_2,\gamma_1',\gamma_2'\) are known constants, while \(q\) and \(p\) are prescribed functions of the variables \(\varphi,\theta\), which are assumed to be subject to the conditions
\[ \Lambda_\beta^{\,n+1}q=0,\qquad \Lambda_\delta^{\,n+1}p=0, \tag{50} \]
where \(\Lambda_\sigma\) is the differential operator (see (47)), and \(\beta,\delta\) are prescribed complex constants.
If the function \(f_k(\rho)\) is made to satisfy the conditions
\[ \begin{gathered} \text{for problem A'} \qquad \gamma_1 f_0(H)+\gamma_2 f'_0(H)=1,\\ \gamma_1 f_\nu(H)+\gamma_2 f'_\nu(H)=0,\qquad \gamma'_1 f_k(h)+\gamma'_2 f'_k(h)=0;\\ \text{for problem B'} \qquad \gamma_1 f_k(H)+\gamma_2 f'_k(H)=0,\\ \gamma'_1 f_0(h)+\gamma'_2 f'_0(h)=1,\qquad \gamma'_1 f_\nu(h)+\gamma'_2 f'_\nu(h)=0, \end{gathered} \tag{51} \]
where \(\nu=1,2,\ldots,n\), and, taking into account (45) and \(\Lambda^0_\alpha=1\), one sets
\[ \text{for problem A'} \quad \alpha=\beta,\ \Phi=q; \qquad \text{for problem B'} \quad \alpha=\delta,\ \Phi=p, \tag{52} \]
then the boundary conditions (49) will be satisfied. It is not difficult to verify, taking (48) into account, that the function \(f_k(\rho)\) in principle makes it possible to fulfill the conditions (51). Thus the problems (49) may be regarded as solved.
In the particular case \(\mu_0(\rho)=\rho^c\), where \(c\) is a constant and \(\alpha\ne -\left(\dfrac{1+c}{2}\right)^2\), the solution of equation (48) can be represented in the form
\[ f_k(\rho)=c_k\rho^{\alpha_1}+e_k\rho^{\alpha_2}+v_k(\rho). \tag{53} \]
Here \(c_k,e_k\) are arbitrary constants;
\[ \alpha_1=-\frac{1+c}{2}+\sqrt{\left(\frac{1+c}{2}\right)^2+\alpha},\qquad \alpha_2=-\frac{1+c}{2}-\sqrt{\left(\frac{1+c}{2}\right)^2+\alpha}; \]
\[ v_0(\rho)=0, \]
and \(v_k(\rho)\) \((k=1,2,\ldots,n)\) is a particular solution of the equation
\[ \rho^2 v''_k(\rho)+(2+c)\rho v'_k(\rho)=\alpha v_k(\rho)-v_{k-1}(\rho)-c_{k-1}\rho^{\alpha_1}-e_{k-1}\rho^{\alpha_2}. \]
Solving the systems of equations (51), we obtain the constants \(c_k,e_k\) in (53). We give the values of these constants. For problem A', \(c_0=a_1a_5,\ e_0=-a_3a_5\); for problem B', \(c_0=-a_2a_5,\ e_0=a_1a_5\), while for \(k=1,2,\ldots,n\), for both problems one obtains:
\[ c_k=a_5[a_2\psi_k(h)-a_4\varphi_k(H)],\qquad e_k=a_5[a_3\varphi_k(H)-a_1\psi_k(h)]. \]
Here the following notation is used:
\[ a_1=\left(\gamma_1+\frac{\alpha_1}{H}\gamma_2\right)H^{\alpha_1},\qquad a_2=\left(\gamma_1+\frac{\alpha_2}{H}\gamma_2\right)H^{\alpha_2}, \]
\[ a_3=\left(\gamma'_1+\frac{\alpha_1}{h}\gamma'_2\right)h^{\alpha_1},\qquad a_4=\left(\gamma'_1+\frac{\alpha_2}{h}\gamma'_2\right)h^{\alpha_2},\qquad a_5=(a_1a_4-a_2a_3)^{-1}, \]
\[ \varphi_k(\rho)=\gamma_1 v_k(\rho)+\gamma_2 v'_k(\rho),\qquad \psi_k(\rho)=\gamma'_1 v_k(\rho)+\gamma'_2 v'_k(\rho). \]
References
- V. I. Makhovikov, Izv. AN SSSR, OTN, Energetics and Transport, No. 3, 1963.
- A. V. Lykov, Theory of Heat Conduction. Gostekhizdat, Moscow, 1952.
- B. M. Budak, A. A. Samarskii, A. N. Tikhonov, Collection of Problems in Mathematical Physics. Gostekhizdat, Moscow, 1956.
- G. P. Tolstov, Fourier Series. Gostekhizdat, Moscow, 1951.
- N. N. Lebedev, I. P. Skalskaya, Ya. S. Uflyand, Collection of Problems in Mathematical Physics. Gostekhizdat, Moscow, 1955.
Received by the editors
January 11, 1965.