MATHEMATICAL PHYSICS

L. I. RUBINSHTEIN

ON THE HEATING AND MELTING OF A SOLID BODY BY FRICTION

(Presented by Academician S. L. Sobolev on 16 X 1961)

In the paper of the same title by S. S. Grigoryan \((^1)\), the problem of the heating and melting of a solid body streamed around by a high-velocity flow of a viscous liquid is considered in a formulation admitting a self-similar solution. Here the same problem is considered in a formulation that does not admit a self-similar solution. Using the method developed by us earlier \((^{2,3})\), we reduce the problem to a system of nonlinear functional equations of Volterra type, solvable for small values by iteration.

§ 1. Let a solid body filling the strip \(0 < x^* < l^*\) border, along the plane \(x^* = l^*\), on a viscous incompressible liquid filling the half-space \(x^* > l^*\). We shall assume that the solid body moves with velocity \(v_0 = \mathrm{const}\) parallel to the plane \(x^* = l^*\), and that, owing to viscous dissipation of energy, the temperature at the boundary \(x^* = l^*\) reaches, at the instant \(t_0^* < 0\), the melting temperature of the solid body. In the subsequent process three phases will participate: the solid—filling the region \(0 < x^* < y^*(t^*)\), the melt—the region \(y^*(t^*) < x^* < l^*\), and the surrounding liquid—the region \(l^* < x^* < \infty\). We assume that the state of the system is known for \(t^* \leq 0\), with \(y^*(0) < (0,l^*)\).

We shall assign the index \(i = 1\) to the solid phase, \(i = 2\) to the melt, and \(i = 3\) to the surrounding liquid. Let \(\vartheta_i, w_i, k_i^*, c_{pi}, \rho_i, a_i^{2*}\), and \(\nu_i\) denote, respectively, the temperature, velocity, coefficient of thermal conductivity, specific heat, density, coefficient of thermal diffusivity, and viscosity of the \(i\)-phase. Let, further, \(\lambda^*\) be the latent heat of fusion per unit mass. We shall take the melting temperature to be zero. We assume \(k_i^*, c_{pi}, \rho_i, a_i^{2*}, \nu_i\), and \(\lambda^*\) to be constant, \(\rho_1 = \rho_2 = \rho\). Let \(\psi_i^*(x^*)\) be the initial temperature of the \(i\)-phase, \(f^*(t^*)\) the temperature at the boundary \(x^* = 0\), and \(\varphi_i^*(x^*)\) the initial velocity in the \(i\)-phase. Put

\[ T = \max \left\{ |\psi_i^*(x^*)|;\ |f^*(t^*)|;\ i = 1,2,3 \right\} \tag{1.1} \]

and introduce dimensionless quantities

\[ v_i = \frac{w_i}{v_0}; \qquad u_i = \frac{\vartheta_i}{T}; \qquad f = \frac{f^*}{T}; \qquad \psi_i = \frac{\psi_i^*}{T}; \qquad \varphi_i = \frac{\varphi_i^*}{v_0}; \qquad x = \frac{x^*}{l^*}; \]

\[ t = \frac{a_1^{2*} t^*}{l^{*2}}; \qquad y = \frac{y^*}{l^*}; \qquad l = \frac{y^*(0)}{l^*}; \qquad a_i^2 = \frac{a_i^{2*}}{a_1^{2*}}; \qquad b_i^2 = \frac{\nu_i}{a_1^{2*}}; \tag{1.2} \]

\[ \gamma_i = \frac{\nu_i}{c_{pi} a_1^{2*}} \cdot \frac{v_0}{T}; \qquad \lambda_i = \rho_i \nu_i; \qquad k_i = \frac{k_i^* T}{\lambda \rho a_1^{2*}} . \]

In what follows, the index referring to the liquid phases will be omitted. The unknowns \(u_1, u, v\), and \(y\) are to be determined from the conditions

\[ \frac{\partial^2 u_1}{\partial x^2} = \frac{\partial u_1}{\partial t} \quad \text{for } 0 < x < y(t); \qquad u_1 \big|_{x=0} = f(t); \qquad u_1 \big|_{x=y(t)} = 0; \]

\[ u_1 \big|_{t=0} = \psi_1(x); \tag{1.3\(_1\)} \]

\[ b^2 \frac{\partial^2 v}{\partial x^2} = \frac{\partial v}{\partial t} \quad \text{for } y(t) < x < \infty; \qquad x \ne 1; \qquad v \big|_{t=0} = \varphi(x); \qquad v \big|_{x=y(t)} = 0; \]

\[ v \ \text{and} \ \lambda \frac{\partial v}{\partial x} \ \text{are continuous for } x = 1; \tag{1.3\(_2\)} \]

\[ a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t} - \gamma^2 \left( \frac{\partial v}{\partial x} \right)^2 \quad \text{for } y(t)<x<\infty;\quad x \ne 1; \]

\[ u\big|_{x=y(t)}=0;\qquad u\big|_{t=0}=\psi(x); \tag{1,3_3} \]

\[ u \text{ and } k\frac{\partial u}{\partial x} \text{ are continuous for } x=1 \]

\[ \frac{dy}{dt} = k_1 \frac{\partial u_1}{\partial x}\bigg|_{x=y(t)-0} - k_2 \frac{\partial u_2}{\partial x}\bigg|_{x=y(t)+0}; \qquad y(0)=l \subset (0,1). \tag{1,3_4} \]

We assume that

\[ f(t) \le -\delta <0;\qquad \psi_1(x)\le 0,\quad \psi(x)\ge 0 \quad \text{near } x=l. \tag{1,4} \]

§ 2. Let \(w^*(x,t)\) be a solution, regular at infinity, of the equation

\[ \alpha^2(x)\frac{\partial^2 w^*}{\partial x^2} = \frac{\partial w^*}{\partial t} - F(x,t); \qquad x\ne 1, \tag{2,1} \]

satisfying the conditions

\[ w^* \text{ and } \mu(x)\frac{\partial w^*}{\partial x} \text{ are continuous for } x=1. \tag{2,2} \]

Here

\[ \alpha^2(x)= \begin{cases} \alpha_1^2=\mathrm{const}, & x<1;\\ \alpha_2^2=\mathrm{const}, & x>1; \end{cases} \qquad \alpha(x)= \begin{cases} \mu_1=\mathrm{const}, & x<1;\\ \mu_2=\mathrm{const}, & x>1. \end{cases} \tag{2,3} \]

By the fundamental solution \(g(x,\xi,t-\tau\mid \alpha^2;\mu)\), corresponding to problem (2,1), (2,2), we shall mean the solution of the adjoint equation

\[ \alpha^2(\xi)\frac{\partial^2 g}{\partial \xi^2} + \frac{\partial g}{\partial \tau} = -\alpha^2(\xi)\delta(t-\tau), \qquad \tau<t, \tag{2,4_1} \]

regular at infinity and satisfying the conjugation conditions

\[ \alpha^2(\xi)\frac{\partial g}{\partial \xi} \text{ and } \mu^*(\xi)\alpha^2(\xi)g \text{ are continuous for } \xi=1,\quad x\ne 1. \tag{2,4_2} \]

Here \(\mu^*=\mu_2\) for \(\xi<1\) and \(\mu=\mu_1\) for \(\xi>1\); \(\delta(x)\) is the Dirac \(\delta\)-function. We must have

\[ \begin{aligned} w^*(x,t) ={}& \int_{x_1(0)}^{x_2(0)} w^*(\xi,0)\, g\left(x,\xi,t\mid \alpha^2;\mu\right)\,d\xi \\ &+ \int_0^t \alpha^2(\xi)\frac{\partial}{\partial \xi}w^*(\xi,\tau)\, g\left(x,\xi,t-\tau\mid \alpha^2;\mu\right) \bigg|_{\xi=-x_1(\tau)}^{x_2(\tau)} \,d\tau \\ &- \int_0^t w^*(\xi,\tau) \left[ \alpha^2(\xi)\frac{\partial}{\partial \xi} g\left(x,\xi,t-\tau\mid \alpha^2;\mu\right) - \frac{d\xi}{d\tau} g\left(x,\xi,t-\tau\mid \alpha^2;\mu\right) \right]_{\xi=x_1(\tau)}^{x_2(\tau)} \,d\tau \\ &+ \int_0^t d\tau \int_{x_1(\tau)}^{x_2(\tau)} F(\xi,\tau)\, g\left(x,\xi,t-\tau\mid \alpha^2;\mu\right)\,d\xi, \end{aligned} \tag{2,5} \]

if \(\chi_i(t)\) are differentiable for \(t>0\), \(\chi_1(t)<x<\chi_2(t)\), and \(1\subset(\chi_1,\chi_2)\).

Put

\[ g_{11}\left(x,\xi,t\mid \alpha^2;\mu\right) = (2\alpha_1\sqrt{\pi t})^{-1} \exp\left(-\frac{(x-\xi)^2}{4\alpha_1^2 t}\right), \]

\[ g_{12}\left(x,\xi,t\mid \alpha^2;\mu\right) = \frac{\mu_1\alpha_2-\mu_2\alpha_1}{\mu_1\alpha_2+\mu_2\alpha_1} \,g_{11}\left(2-x,\xi,t\mid \alpha^2;\mu\right), \tag{2,6} \]

\[ g_{13}\left(x,\xi,t\mid \alpha^2;\mu\right) = \frac{2\alpha_1\mu_2}{\mu_1\alpha_2+\mu_2\alpha_2} \, \frac{1}{2\alpha_2\sqrt{\pi t}} \exp\left\{ -\left(\frac{x-1}{\alpha_1}-\frac{\xi-1}{\alpha_2}\right)^2 \frac{1}{4t} \right\}. \]

Next, define \(g_{14}, g_{15}\), and \(g_{16}\) by the expressions \(g_{13}, g_{12}\), and, respectively, \(g_{11}\), replacing in them \(\alpha_1\) and \(\mu_1\) by \(\alpha_2\) and \(\mu_2\), and conversely. Finally, set:

\[ g_i(x,\xi,t\mid \alpha^2;\mu)= \begin{cases} g_{11}-(-1)^i g_{12}, & \text{for } -\infty < x;\ \xi < 1,\\ \zeta_i g_{13}, & \text{for } -\infty < x < 1;\ 1<\xi<\infty,\\ \zeta_i^{-1} g_{14}, & \text{for } -\infty < \xi < 1;\ 1<x<\infty,\\ (-1)^i g_{15}+g_{16}, & \text{for } 1<x;\ \xi<\infty. \end{cases} \]

Here

\[ \zeta_1=1;\qquad \zeta_2=\alpha_1/\alpha_2. \tag{2,7} \]

By direct verification it is easy to see that \(g_1(x,\xi,t-\tau\mid \alpha^2;\mu)\) is the fundamental solution corresponding to the problem (2,1), (2,2)\(^*\). We note that \(g_1\) and \(g_2\) are related by

\[ \frac{\partial g_1}{\partial x}=-\frac{\partial g_2}{\partial \xi};\qquad \alpha^2(\xi)\frac{\partial^2 g_1}{\partial x\,\partial \xi} = \frac{\partial g_2}{\partial t}. \tag{2,8} \]

Let, finally, \(G_i(x,\xi,t)\) be the Green’s functions of the first and second boundary-value problems for the heat-conduction equation on the half-line, i.e.

\[ G_i(x,\xi,t)= \left\{ \exp\left[-\frac{(x-\xi)^2}{4t}\right] + (-1)^i \exp\left[-\frac{(x+\xi)^2}{4t}\right] \right\} (2\sqrt{\pi t})^{-1}. \tag{2,9} \]

It is obvious that for a one-layer problem (i.e. for the case \(\alpha_1=\alpha_2\)), \(g_i\) in (2,7) may be replaced by \(G_i\). In addition, \(G_i\) satisfy (2,10).

§ 3. Put

\[ w(t)=\left.\frac{\partial v}{\partial x}\right|_{x=y(t)+0};\qquad z_1(t)=\left.\frac{\partial v_1}{\partial x}\right|_{x=y(t)-0};\qquad z_2(t)=\left.\frac{\partial u}{\partial x}\right|_{x=y(t)+0}. \tag{3,1} \]

We reduce problem (1,3) to a system of Volterra-type functional equations with respect to \(w, z, z_1\), and \(y\). Below we shall write

\[ \frac{\partial v}{\partial x}=v_1(x,t); \]

\[ g_i(x,\xi,t\mid a^2;k)=g_i(x,\xi,t);\qquad g_i(x,\xi,t\mid b^2;\lambda)=g_i^*(x,\xi,t). \tag{3,2} \]

Using (2,7), the boundary conditions (1,3), and the notation (2,11), (3,1), and (3,2), we find that the following equalities must hold:

\[ u_1= \int_0^t f(\tau)\frac{\partial}{\partial \xi}G_1(x,0,t-\tau)\,d\tau + \int_0^l \psi_1(\xi)G_1(x,\xi,t)\,d\xi + \]

\[ + \int_0^t z_1(\tau)G_1(x,y(\tau),t-\tau)\,d\tau; \tag{3,3_1} \]

\[ v=-b_2^2\int_0^t w(\tau)g_1^*(x,y(\tau),t-\tau)\,d\tau + \int_l^\infty \varphi(\xi)g_1^*(x,\xi,t)\,d\xi; \tag{3,3_2} \]

\[ u=-a_2^2\int_0^t z(\tau)g_1(x,y(\tau),t-\tau)\,d\tau + \int_l^\infty \psi(\xi)g_1(x,\xi,t)\,d\xi + \]

\[ + \int_0^t d\tau \int_{y(\tau)}^\infty \gamma^2(\xi)v_1^2(\xi,\tau)g_1(x,\xi,t-\tau)\,d\xi. \tag{3,3_3} \]

Differentiating (3,3\(_2\)) and using (2,10) and (3,2), we obtain

\[ v_1(x,t)=\varphi(l)g_2^*(x,l,t)+I_{0,1}(x,t)+I_{0,2}(x,t\mid y,w), \tag{3,4} \]

where

\[ I_{0,1}=\int_l^\infty \dot{\varphi}(\xi)g_2^*(x,\xi,t)\,d\xi;\qquad I_{0,2}=b_2^2\int_0^t w(\tau)\frac{\partial}{\partial \xi}g_2^*(x,y(\tau),t-\tau)\,d\tau. \tag{3,5} \]

\[ \rule{0.22\linewidth}{0.4pt} \]

\(^*\) \(g_1^*\) is easily found by using the Laplace–Carson transform. In the region \(-\infty<x<\infty;\ 1<\xi<\infty\), the construction of \(g_1^*\) is given in (4).

Equality \((3,3_2)\) determines \(v\) for \(x>y(t)\). At the same time \(v_1(x,t)\) is defined only for \(x>y(t)\). We continue \(v_1\) to the half-line \(x \leq y(t)\) by means of the equalities

\[ v_1(y(t),t)=\varphi(l)g_2^*(y(t),l,t)+\frac12 w(t)+I_{0,1}(y(t),t)+ \]
\[ +I_{0,2}(y(t),t\mid y,w); \tag{3,5_1} \]

\[ v_1(x,t)=\varphi(l)g_2^*(x,l,t)+w(t)+I_{0,1}(x,t)+I_{0,2}(x,t\mid y,w) \]
\[ \text{for } x<y(t). \tag{3,5_2} \]

By virtue of the theorem on the jumps of the thermal potential of a double layer and of the definition of \(g_1^*\) and \(g_2^*\), the function \(v_1(x,t)\) so defined will be continuous at \(x=y(t)\), if \(w(t)\) is continuous.

We shall write

\[ 2I_{0,1}(y(t),t)=I_{1,1}(t\mid y);\qquad 2I_{0,2}(y(t),t\mid y,w)=I_{1,2}(t\mid y,w). \tag{3,6} \]

Then from \((3,1)\), \((3,2)\), and \((3,5_1)\) it follows that

\[ w(t)=2\varphi(l)g_2^*(y(t),l,t)+I_{1,1}(t\mid y)+I_{1,2}(t\mid y,w). \tag{3,7} \]

Analogously we find

\[ z_1(t)=2\{[\psi_1(0)-f(0)]G_2(y(t),0,t)-\psi_1(l)G_2(y(t),l,t)+ \]
\[ +I_{2,1}(t\mid y)+I_{2,2}(t\mid y,z_1)\}, \tag{3,8} \]

\[ z(t)=2\{\psi(l)g_2(y(t),l,t)+I_{3,1}(t\mid y)+I_{3,2}(t\mid y,z)+I_{3,3}(t\mid y,v_1)\}. \tag{3,9} \]

Here

\[ I_{2,1}=\int_0^l \psi_1(\xi)G_2(y(t),\xi,t)\,d\xi, \]

\[ I_{2,2}=-\int_0^t z_1(\tau)\frac{\partial}{\partial \xi}G_2(y(t),y(\tau),t-\tau)\,d\tau, \tag{3,10} \]

\[ I_{3,1}=\int_l^\infty \psi(\xi)g_2(y(t),\xi,t)\,d\xi; \]

\[ I_{3,2}=a_2^2\int_0^t z(\tau)\frac{\partial}{\partial \xi}G_2(y(t),y(\tau),t-\tau)\,d\tau; \]

\[ I_{3,3}=-\int_0^t d\tau \int_{y(\tau)}^\infty \gamma^2(\xi)v_1^2(\xi,\tau)\frac{\partial}{\partial \xi}g_2(y(t),\xi,t-\tau)\,d\xi. \]

To the system \((3,4)\)—\((3,10)\) we adjoin condition \((1,3)\), which, by virtue of \((3,1)\), is written in the form

\[ y(t)=l+\int_0^t [k_1z_1(\tau)-kz(\tau)]\,d\tau. \tag{3,11} \]

This completes the required reduction.

The theorem of existence and uniqueness is proved locally under the assumption that \(f(t)\) is twice differentiable, while \(\psi(x)\) and \(\psi_1(x)\) are three times differentiable, with \(\psi_1(0)=f(0)\); \(\varphi(l)=\psi(l)=\psi_1(l)=0\); \(0<l<1\). The equivalence theorem is proved in the same way as in \((^2)\).

Computing Center
of the Latvian State University
named after Peter Stuchka

Received
27 IX 1961

CITED LITERATURE

\(^{1}\) S. S. Grigoryan, Prikl. matem. i mekh., 22, no. 5 (1958).
\(^{2}\) L. I. Rubinshtein, On Certain Nonlinear Problems Generated by the Fourier Equation, Dissertation, Moscow State University, 1957.
\(^{3}\) L. I. Rubinshtein, DAN, 142, no. 3 (1962).
\(^{4}\) B. M. Budak, A. A. Samarskii, A. N. Tikhonov, Collection of Problems in Mathematical Physics, Moscow, 1956.