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ON A PROBLEM FOR A SYSTEM OF DIFFERENTIAL EQUATIONS OF ENERGY AND MASS TRANSFER
P. V. Tsoi
The analytic theory of energy and mass transfer inside a porous body reduces to the solution of a system of differential equations of parabolic type (parabolic, in the sense of I. G. Petrovsky, systems) under various boundary conditions [1].
In this article we consider the problem of redistribution of fields of generalized charges \(u_k(x,t)\) \((k=1,2)\) (heat and mass of a bound substance) under imperfect contact of two semi-infinite conductors. At the contact boundary, the rates of two interrelated fluxes of generalized charges are investigated.
Mathematically, the problem is formulated as follows. It is necessary to determine the fields of distribution of generalized charges \(u_k(x,t)\), \(u_k^{(-)}(x,t)\) \((k=1,2)\), satisfying the systems
\[ \frac{\partial u_k}{\partial t} = a_{k1}^{2}\frac{\partial^{2}u_1}{\partial x^{2}} + a_{k2}^{2}\frac{\partial^{2}u_2}{\partial x^{2}} + \vartheta_k(x,t), \tag{1} \]
\[ u_k(x,0)=f_k(x),\quad (0<x<\infty,\ k=1,2), \tag{2} \]
\[ \frac{\partial u_k^{(-)}}{\partial t} = b_{k1}^{2}\frac{\partial^{2}u_1^{(-)}}{\partial x^{2}} + b_{k2}^{2}\frac{\partial^{2}u_2^{(-)}}{\partial x^{2}} + \vartheta_k^{(-)}(x,t), \tag{3} \]
\[ u_k^{(-)}(x,0)=f_k^{(-)}(x),\quad (-\infty<x<0,\ k=1,2). \tag{4} \]
At the interface of the two media \((x=0)\), the conjugation conditions for imperfect contact are satisfied:
\[ \lambda_k\left.\frac{\partial u_k}{\partial x}\right|_{x=0+} - \lambda_k^{(-)}\left.\frac{\partial u_k^{(-)}}{\partial x}\right|_{x=0-} = \chi_k(t), \tag{5} \]
\[ u_k(+0,t)-u_k^{(-)}(-0,t)=\mu_k(t). \tag{6} \]
Thus, at the contact boundary of two semi-infinite conductors, the distribution fields of the generalized charges and their derivatives undergo a finite nonstationary jump. For \(\chi_k(t)=\mu_k(t)=0\), the formulated problem represents a mathematical model of the contact problem for ideal contact and without phase transformations at the boundary \(x=0\).
Let
\[ \lambda_k^{(-)}\left.\frac{\partial u_k^{(-)}}{\partial x}\right|_{x=0-} = \varphi_k(t), \tag{7} \]
then from condition (5) for the right medium (conductor) we obtain
\[ \lambda_k\left.\frac{\partial u_k}{\partial x}\right|_{x=0+} = \varphi_k(t)+\chi_k(t). \tag{8} \]
The introduced functions \(\varphi_1(t)\), \(\varphi_2(t)\) represent the nonstationary specific fluxes of the generalized charges at the contact boundary on the left, and the functions \(\varphi_k(t)+\chi_k(t)\) \((k=1,2)\) are the specific fluxes on the right.
Suppose that \(\varphi_1(t)\), \(\varphi_2(t)\) are known functions (subsequently they will be determined); then the principal contact problem splits into two independent boundary-value problems of heat and mass transfer for a semi-infinite medium with boundary conditions of the second kind. The solution of system (1) under the boundary conditions (2), (8) can be obtained, for example, by the method of the Laplace integral transform with respect to the variable \(t\) and the Fourier cosine transform with respect to the variable \(x\) [5]. Another method can also be indicated [8]. For the existence and uniqueness of the solution of system (1) under the boundary conditions (2), (8), it is necessary to impose certain restrictions on the functions \(f_k(x)\), \(\vartheta_k(x,t)\), \(\varphi_k(t)\). The unknown functions \(u_k(x,t)\) are sought in the class for which, in the domain \(\Omega\{0\le x<\infty,\ t>0\}\), the Fourier integral transforms with respect to the spatial coordinate \(x\) and the Laplace transform with respect to \(t\) are applicable. Let the functions \(\vartheta_j(x,t)\), \(f_j(x)\), \(\varphi_j(t)\) belong to the class of functions \(L[\Omega]\), integrable in the domain \(\Omega\). In view of the restrictions adopted, all functions under consideration must be regular at infinity. These conditions are sufficient for determining the solution \(u_k(x,t)\).
The functions \(u_j(x,t)\) \((j=1,2)\) are determined by the following formula:
\[ \begin{aligned} u_j(x,t)=\sum_{k=1}^{2}\frac{1}{2c_k\sqrt{\pi}} \Bigg\{& (-1)^j\int_{0}^{t} \frac{m_{jk}\varphi_{11}(\tau)-n_{jk}\varphi_{12}(\tau)} {\sqrt{t-\tau}} \\ &\times \exp\left[-\frac{x^2}{4c_k^2(t-\tau)}\right]\,d\tau \\ &+\frac{(-1)^{j+1}}{\sqrt{t}}\int_{0}^{\infty} \bigl[m_{jk}f_1(\alpha)-n_{jk}f_2(\alpha)\bigr] \exp\left[-\frac{(x-\alpha)^2}{4c_k^2t}\right] \\ &\times\left[1+\exp\left(-\frac{\alpha x}{c_k^2t}\right)\right]\,d\alpha \\ &+(-1)^{j+1}\int_{0}^{t}\int_{0}^{\infty} \frac{m_{jk}\vartheta_1(\alpha,\tau)-n_{jk}\vartheta_2(\alpha,\tau)} {\sqrt{t-\tau}} \\ &\times \exp\left[-\frac{(x-\alpha)^2}{4c_k^2(t-\tau)}\right] \left[1+\exp\left(-\frac{\alpha x}{c_k^2(t-\tau)}\right)\right] \,d\tau\,d\alpha \Bigg\}, \end{aligned} \tag{9} \]
where
\[ \left. \begin{aligned} m_{1k}&=(a_{22}^{2}-c_k^{2})A_k,\quad n_{1k}=a_{12}^{2}A_k,\quad m_{2k}=(a_{11}^{2}-c_k^{2})A_k,\\ n_{2k}&=(a_{11}^{2}-c_k^{2})A_k,\quad A_k=(a_{11}^{2}+a_{22}^{2}-2c_k^{2})^{-1},\\ c_k^{2}&=\frac{(a_{11}^{2}+a_{22}^{2})+(-1)^{k+1} \sqrt{(a_{11}^{2}-a_{22}^{2})^{2}+4a_{12}^{2}a_{21}^{2}}}{2},\\ \varphi_{1j}(t)&=\frac{a_{j1}^{2}}{\lambda_1}\,[\varphi_1(t)+\chi_1(t)] +\frac{a_{j2}^{2}}{\lambda_2}\,[\varphi_2(t)+\chi_2(t)]. \end{aligned} \right\} \tag{10} \]
Solving system (3) under the boundary conditions (4)—(7), we obtain
\[ \begin{aligned} u_i^{(-)}(-z,t) &= \sum_{k=1}^{2}\frac{1}{2l_k\sqrt{\pi}} \Bigg\{ \int_{0}^{t} \frac{m_{ik}^{(-)}\varphi_{11}^{(-)}(\tau)-n_{ik}^{(-)}\varphi_{12}^{(-)}(\tau)} {\sqrt{t-\tau}} \\ &\qquad\qquad\qquad\times \exp\left[-\frac{z^2}{4l_k^2(t-\tau)}\right]\,d\tau +\frac{1}{\sqrt{t}}\int_{0}^{\infty} \bigl[m_{ik}^{(-)}f_1^{(-)}(-\alpha) \\ &\qquad\qquad\qquad -n_{ik}^{(-)}f_2^{(-)}(-\alpha)\bigr] \exp\left[-\frac{(z-\alpha)^2}{4l_k^2t}\right] \left[1+\exp\left(-\frac{\alpha x}{l_k^2t}\right)\right]\,d\alpha \\ &\qquad\qquad\qquad +\int_{0}^{t}\int_{0}^{\infty} \frac{ m_{ik}^{(-)}\vartheta_1^{(-)}(-\alpha,\tau) -n_{ik}^{(-)}\vartheta_2^{(-)}(-\alpha,\tau) } {\sqrt{t-\tau}} \\ &\qquad\qquad\qquad\times \exp\left[-\frac{(z-\alpha)^2}{4l_k^2(t-\tau)}\right] \left[1+\exp\left(-\frac{\alpha x}{l_k^2(t-\tau)}\right)\right]\,d\tau\,d\alpha \Bigg\}, \end{aligned} \tag{11} \]
where
\[ \left. \begin{aligned} m_{ik}^{(-)}&=(b_{11}^2-l_k^2)B_k,\qquad n_{ik}^{(-)}=b_{12}^2B_k,\qquad B_k=(b_{11}^2+b_{22}^2-2l_k^2)^{-1},\\ l_k^2&=\frac{(b_{11}^2+b_{22}^2)+(-1)^{k+1} \sqrt{(b_{11}^2-b_{22}^2)^2+4b_{12}^2b_{21}^2}}{2},\\ \varphi_{1i}^{(-)}(t)&=\sum_{k=1}^{2}\frac{b_{ik}^2}{\lambda_k^{(-)}}\,\varphi_k(t),\qquad (z=-x). \end{aligned} \right\} \tag{12} \]
The function \(u_2^{(-)}(-z,t)\) is determined analogously.
The presence of the exponential function in formula (9) ensures the uniform convergence of the improper integrals for a broad class of integrand functions \(f_k(x)\), \(\vartheta_j(x,t)\), \(\varphi_j(t)\). In those cases where these functions fall outside the class for which integral transforms are applicable, the solution (9) can be obtained by means of finite functions and a limiting passage. Introduce finite functions \(f_k^N(x)=f_k(x)\), \(\vartheta_k^N(x,t)=\vartheta_k(x,t)\) for \(x\le N\), and \(f_k^N(x)=\vartheta_k^N(x)=0\) for \(x>N\). Put \(f_k^N(x)\), \(\vartheta_k^N(x,t)\in L[\Omega]\). Following S. L. Sobolev, we shall call the solution \(u_k(x,t)=\lim_{N\to\infty}u_k^N(x,t)\) a “generalized solution.” The limits of applicability of formula (9) in the sense of generalized solutions can be considerably extended. Then the solution \(u_k^N(x,t)\) of system (1) under finite initial and boundary conditions (2)—(8) can be written by formula (9).
The solutions \(u_k(x,t)\), \(u_k^{(-)}(-z,t)\) include two unknown functions \(\varphi_1(t)\), \(\varphi_2(t)\). Thus, the basic problem will be completely solved if we determine these functions. Determination of these functions is also of physical interest, since they express the rates of redistribution of generalized charges at the contact boundary at time \(t\).
To determine the functions \(\varphi_1(t)\), \(\varphi_2(t)\), write the systems of equations (1)—(4) and the corresponding boundary conditions in matrix form:
\[ \frac{\partial u}{\partial t}=A\frac{\partial^2 u}{\partial x^2}+\vartheta(x,t),\quad (x\geqslant 0), \tag{13} \]
\[ \left. \begin{aligned} u(x,0)&=f(x),\\ \left.\frac{\partial u}{\partial x}\right|_{x=0+}&=\varphi(t)+\chi(t) \end{aligned} \right\} \tag{14} \]
and
\[ \frac{\partial u^{(-)}}{\partial t}=B\frac{\partial^2 u^{(-)}}{\partial x^2}+\vartheta^{(-)}(x,t),\quad (x\leqslant 0), \tag{15} \]
\[ \left. \begin{aligned} u^{(-)}(x,0)&=f^{(-)}(x),\\ \left.\frac{\partial u^{(-)}}{\partial x}\right|_{x=0-}&=\varphi^{(-)}(t) \end{aligned} \right\}, \tag{16} \]
where
\[ u(x,t)= \begin{Vmatrix} u_1(x,t)\\ u_2(x,t) \end{Vmatrix}, \quad A= \begin{Vmatrix} a_{11}^2 & a_{12}^2\\ a_{21}^2 & a_{22}^2 \end{Vmatrix}, \quad f(x)= \begin{Vmatrix} f_1\\ f_2 \end{Vmatrix}, \]
\[ \varphi(t)= \begin{Vmatrix} \dfrac{\varphi_1(t)}{\lambda_1}\\[6pt] \dfrac{\varphi_2(t)}{\lambda_2} \end{Vmatrix} \quad \text{etc.} \tag{17} \]
The column matrix \(u(x,t)\), satisfying equation (13) and conditions (14), is written by the formula
\[ \left. \begin{aligned} u(x,t)={}& \frac{1}{2\sqrt{\pi t}}\int_0^\infty \frac{1}{\sqrt{A}}\, \exp\left[-\frac{(x-\alpha)^2}{4At}\right]\times\\ &\times\left[E+\exp\left(-\frac{\alpha x}{At}\right)\right]f(\alpha)\,d\alpha+\\ &+\frac{1}{\sqrt{\pi}}\int_0^t \frac{\sqrt{A}}{\sqrt{t-\tau}}\, \exp\left[-\frac{x^2}{4A(t-\tau)}\right]\varphi(\tau)\,d\tau+\\ &+\frac{1}{2\sqrt{\pi}}\int_0^t\int_0^\infty \frac{1}{\sqrt{t-\tau}}\cdot\frac{1}{\sqrt{A}}\times\\ &\times \exp\left[-\frac{(x-\alpha)^2}{4A(t-\tau)}\right] \left[E+\exp\left(-\frac{\alpha x}{A(t-\tau)}\right)\right] \vartheta(\alpha,\tau)\,d\tau\,d\alpha \end{aligned} \right\}, \tag{18} \]
\[ E= \begin{Vmatrix} 1 & 0\\ 0 & 1 \end{Vmatrix}. \]
The matrix \(u^{(-)}(-z,t)\), \((z=-x)\), satisfying equation (15) and conditions (16), is determined by the formula
\[ \begin{aligned} u^{(-)}(-z,t)= {} & \frac{1}{2\sqrt{\pi t}}\int_{0}^{\infty}\frac{1}{\sqrt{B}}\, \exp\left[-\frac{(z-\alpha)^2}{4Bt}\right]\times \\ & {}\times \left[E+\exp\left(-\frac{z\alpha}{Bt}\right)\right]f^{(-)}(-\alpha)\,d\alpha+ \\ & {}+\frac{1}{\sqrt{\pi}}\int_{0}^{t}\frac{\sqrt{B}}{\sqrt{t-\tau}}\, \exp\left[-\frac{z^2}{4B(t-\tau)}\right]\varphi^{(-)}(\tau)\,d\tau+ \\ & {}+\frac{1}{2\sqrt{\pi}}\int_{0}^{t}\int_{0}^{\infty}\frac{1}{\sqrt{B}}\, \frac{1}{\sqrt{t-\tau}}\, \exp\left[-\frac{(z-\alpha)^2}{4B(t-\tau)}\right]\times \\ & {}\times \left[E+\exp\left(-\frac{\alpha z}{B(t-\tau)}\right)\right] \vartheta^{(-)}(-\alpha,\tau)\,d\tau\,d\alpha . \end{aligned} \tag{19} \]
To determine the matrices \(\varphi(t)\), \(\varphi^{(-)}(t)\), we use the conjugation condition (6), which in matrix form is written as follows:
\[ u(+0,t)-u^{(-)}(-0,t)=\mu(t). \tag{20} \]
Passing to the limit as \(x\to 0\) \((z\to 0)\) in formulas (18), (19) and substituting these limiting values into (20), we then have
\[ \int_{0}^{t}\frac{\Phi(\tau)}{\sqrt{t-\tau}}\,d\tau=F(t), \tag{21} \]
where
\[ \begin{aligned} \Phi(t)={}&\sqrt{A}\,\varphi(t)+\sqrt{B}\,\varphi^{(-)}(t),\\ F(t)={}&\frac{1}{\sqrt{t}}\int_{0}^{\infty} \left[\frac{1}{\sqrt{A}}\exp\left(-\frac{\alpha^2}{4At}\right)f(\alpha) -\frac{1}{\sqrt{B}}\times \right.\\ &\left. {}\times \exp\left(-\frac{\alpha^2}{4Bt}\right)f^{(-)}(-\alpha)\right]\,d\alpha+\\ &{}+\int_{0}^{t}\int_{0}^{\infty}\frac{1}{\sqrt{t-\tau}} \left\{\frac{1}{\sqrt{A}}\exp\left[-\frac{\alpha^2}{4A(t-\tau)}\right]\vartheta(\alpha,\tau)-\right.\\ &\left.{}-\frac{1}{\sqrt{B}}\exp\left[-\frac{\alpha^2}{4B(t-\tau)}\right]\vartheta^{(-)}(-\alpha,\tau)\right\}\,d\tau\,d\alpha-\\ &{}-\int_{0}^{t}\frac{\sqrt{A}}{\sqrt{t-\tau}}\chi(\tau)\,d\tau-\mu(t). \end{aligned} \tag{22} \]
\(\Phi(t)\), \(F(t)\) are one-column matrices.
Equation (21) is an Abel integral equation with respect to the column matrix \(\Phi(t)\); its solution is known [3]:
\[ \Phi(t)=\frac{1}{\pi}\frac{d}{dt}\int_0^t \frac{F(\tau)}{\sqrt{t-\tau}}\,d\tau . \tag{23} \]
To determine the functions \(\varphi_1(t)\), \(\varphi_2(t)\), we write the matrix equality (23) by rows.
Denote
\[ \gamma_k=\frac{1}{\lambda_k},\qquad \gamma_k^{(-)}=\frac{1}{\lambda_k^{(-)}},\qquad F(t)= \begin{Vmatrix} F_1(t)\\ F_2(t) \end{Vmatrix}, \]
then
\[ \Phi(t)=\sqrt{A}\,\varphi(t)+\sqrt{B}\,\varphi^{(-)}(t) = \begin{Vmatrix} d_{11}d_{12}\\ d_{21}d_{22} \end{Vmatrix} \cdot \begin{Vmatrix} \varphi_1(t)\\ \varphi_2(t) \end{Vmatrix}, \tag{24} \]
where
\[ d= \begin{Vmatrix} d_{11}d_{12}\\ d_{21}d_{22} \end{Vmatrix} = \sqrt{A}\, \begin{Vmatrix} \gamma_1&0\\ 0&\gamma_2 \end{Vmatrix} + \sqrt{B}\, \begin{Vmatrix} \gamma_1^{(-)}&0\\ 0&\gamma_2^{(-)} \end{Vmatrix}. \tag{25} \]
The elements of the matrix \(d\) and the functions \(F_1(t)\), \(F_2(t)\) are determined simply. For example, if \(c_k^2\), \(l_k^2\) are the roots of the characteristic determinants of the matrices \(A\), \(B\)—real and distinct, which are determined by formulas (10), (12)—then
\[ \begin{aligned} d={}& \left\{ \frac{c_1}{c_1^2-c_2^2} \begin{Vmatrix} a_{11}^2-c_2^2 & a_{12}^2\\ a_{21}^2 & a_{22}^2-c_2^2 \end{Vmatrix} - \frac{c_2}{c_1^2-c_2^2} \begin{Vmatrix} a_{11}^2-c_1^2 & a_{12}^2\\ a_{21}^2 & a_{22}^2-c_1^2 \end{Vmatrix} \right\} \begin{Vmatrix} \gamma_1&0\\ 0&\gamma_2 \end{Vmatrix} \\ &+ \left\{ \frac{l_1}{l_1^2-l_2^2} \begin{Vmatrix} b_{11}^2-l_2^2 & b_{12}^2\\ b_{21}^2 & b_{22}^2-l_2^2 \end{Vmatrix} - \frac{l_2}{l_1^2-l_2^2} \begin{Vmatrix} b_{11}^2-l_1^2 & b_{12}^2\\ b_{21}^2 & b_{22}^2-l_1^2 \end{Vmatrix} \right\} \begin{Vmatrix} \gamma_1^{(-)}&0\\ 0&\gamma_2^{(-)} \end{Vmatrix}. \end{aligned} \tag{26} \]
The elements of the column matrix \(F(t)\) are determined analogously. Writing equality (23) by rows, we then have
\[ \left. \begin{aligned} d_{11}\varphi_1(t)+d_{12}\varphi_2(t)&=\Gamma_1(t)\\ d_{21}\varphi_1(t)+d_{22}\varphi_2(t)&=\Gamma_2(t) \end{aligned} \right\}, \tag{27} \]
where
\[ \Gamma_k(t)=\frac{1}{\pi}\frac{d}{dt}\int_0^t \frac{F_k(\tau)}{\sqrt{t-\tau}}\,d\tau . \tag{28} \]
Hence
\[ \varphi_1(t)=\frac{\Gamma_1(t)d_{22}-\Gamma_2(t)d_{12}}{d_{11}d_{22}-d_{12}d_{21}}, \qquad \varphi_2(t)=\frac{\Gamma_2(t)d_{11}-\Gamma_1(t)d_{21}}{d_{11}d_{22}-d_{12}d_{21}}. \tag{29} \]
Thus, we have determined the unknown functions \(\varphi_1(t)\), \(\varphi_2(t)\). The main problem is completely solved.
Let us consider a special case. Suppose that at the initial moment of time (before contact) there were constant fields of distribution of the generalized charges, i.e. \(f_k(x)=u_k^0=\mathrm{const}\), \(f_k^{(-)}(x)=v_k^0=\mathrm{const}\). In addition, let us set \(\vartheta_k(x,t)=\vartheta_k^{(-)}(x,t)=0\), \(\chi_k(t)=\mu_k(t)=0\) (ideal contact without internal sources).
From formula (22) we obtain
\[ F(t)=\frac{\sqrt{\pi}}{2}\,(u^0-v^0) = \frac{\sqrt{\pi}}{2}\,\Delta u = \frac{\sqrt{\pi}}{2} \begin{Vmatrix} \Delta u_1\\ \Delta u_2 \end{Vmatrix}. \tag{30} \]
Consequently,
\[ \Gamma_k(t)=\frac{1}{\pi}\frac{d}{dt}\int_0^t \frac{F_k(\tau)}{\sqrt{t-\tau}}\,d\tau =\frac{\Delta u_k}{\sqrt{\pi t}}, \tag{31} \]
where
\[ \Delta u_k=u_k^0-v_k^0. \]
Substituting the values (31) into (29), we then obtain
\[ \varphi_1(t)=\frac{d_{22}\Delta u_1-d_{12}\Delta u_2}{(d_{11}d_{22}-d_{12}d_{21})\sqrt{\pi t}}, \qquad \varphi_2(t)=\frac{d_{11}\Delta u_2-d_{12}\Delta u_1}{(d_{11}d_{22}-d_{12}d_{21})\sqrt{\pi t}}. \tag{32} \]
For “uncoupled” transfers \((a_{12}=a_{21}=0,\ b_{12}=b_{21}=0)\), after some simplifications we obtain
\[ \varphi_1(t)=\frac{u_1^0-v_1^0}{(a_{11}\gamma_1+b_{11}\gamma_1^{(-)})\sqrt{\pi t}}. \tag{33} \]
In the theory of systems of differential equations of parabolic type it is known that the solution of system (1) under the boundary conditions (2), (8) is well posed. Consequently, the limiting transitions in formulas (22), (29), (31), and (32) lead to solutions of the corresponding problems under simplified conditions. For example, expression (33) represents the value of the specific heat flux at the contact boundary of two semi-infinite bodies, if \(u_1^0, v_1^0\) are temperature fields. Formula (33) is known in the literature on heat conduction [2], [7].
References
- Lykov A. V., Mikhailov Yu. A. Theory of Heat and Mass Transfer. Gosenergoizdat, 1963.
- Lykov A. V. Theory of Heat Conduction. Gostekhizdat, 1952.
- Mikhlin S. G. Lectures on Linear Integral Equations. Fizmatgiz, 1959.
- Marnevskaya L. A. In: Thermophysics in Foundry Production. Publishing House of the Academy of Sciences of the BSSR, 1963, p. 69.
- Tsoi P. V. IFZh, No. 1, 1961; IFZh, No. 4, 1961.
- Eidelman S. D. Parabolic Systems. Publishing House “Nauka,” 1964.
- Tsoi P. V. News of Higher Educational Institutions of the USSR, Energetics, No. 9, 1964.
- Tsoi P. V. Int. J. Heat Mass Transfer, Vol. 7, 1964.
Received by the editors
December 19, 1964
Tajik Polytechnic Institute