Heat Transfer in $^{3}\hbox {He}$–$^{4}\hbox {He}$ Mixtures in Cylindrical Geometry

Abstract

The paper presents the results of theoretical studies of the transport processes that take place in the newly proposed experiments on study of a vibrating quartz fork in superfluid $^{3}\hbox {He}$–$^{4}\hbox {He}$ mixtures. In addition to known mechanisms of energy loss from a vibrating quartz fork such as first sound radiation or interaction with thermal excitations, two more mechanisms specific for $^{3}\hbox {He}$–$^{4}\hbox {He}$ mixtures are proposed and studied in the paper. The relative contribution of these mechanisms: second sound and effective diffusion, is considered, and experimental conditions under which these mechanisms become effective are discussed.

Appendix

After Laplace transform of the system (1) and excluding w(r, t), the system of two equations is obtained

$$\begin{aligned} c(r,p)= & {} A_1 \Delta _{\mathrm{r}} c+B_1 \Delta _{\mathrm{r}} T \nonumber \\ T(r,p)= & {} A_2 \Delta _{\mathrm{r}} c+B_2 \Delta _{\mathrm{r}} T, \end{aligned}$$

(6)

where $\Delta _{\mathrm{r}}$ is Laplacian in cylindrical coordinates, p is a parameter of Laplace transform (inverse time). Here

$$\begin{aligned} A_1 (p)= & {} \frac{1}{p}\left( {\frac{u_{2N}^2 }{p}+D} \right) , \\ A_2 (p)= & {} \frac{1}{p}\left( {\frac{T_0 \bar{{S}}u_{2N}^2 }{c_0 C_{\mathrm{V}} }\frac{1}{p}+\frac{Dk_{\mathrm{T}} \rho _0 \zeta }{C_{\mathrm{V}} }} \right) , \\ B_1 (p)= & {} \frac{1}{p}\left( {\frac{c_0 \bar{{S}}\rho _{\mathrm{s}} }{\rho _0 \rho _{\mathrm{n}} }\frac{1}{p}+D\frac{k_{\mathrm{T}} }{T_0 }} \right) , \\ B_2 (p)= & {} \frac{1}{p}\left( {\chi +\frac{Dk_{\mathrm{T}}^2 \rho _0 \zeta }{T_0 C_{\mathrm{V}} }+\frac{u_{2\varepsilon }^2 }{p}} \right) . \end{aligned}$$

The system (6) can be transformed to the diagonal form

$$\begin{aligned} \left\{ {\begin{array}{l} F=\gamma \Delta _{\mathrm{r}} F \\ G=\eta \Delta _{\mathrm{r}} G \\ \end{array}} \right. \end{aligned}$$

(7)

where $F=T+\alpha _{1}c\, G=T+\alpha _2c$ are linear combinations, and

$$\begin{aligned} \gamma= & {} \frac{A_1 (p)+B_2 (p)+\sqrt{\left( {A_1 (p)-B_2 (p)} \right) ^{2}+4A_2 (p)B_1 (p)}}{2}, \\ \eta= & {} \frac{A_1 (p)+B_2 (p)-\sqrt{\left( {A_1 (p)-B_2 (p)} \right) ^{2}+4A_2 (p)B_1 (p)}}{2}, \\ \alpha _1= & {} \frac{A_2 (p)}{\gamma -A_1 (p)}, \quad \alpha _2 =\frac{A_2 (p)}{\eta -A_1 (p)}. \end{aligned}$$

The solution of system (5) can be written as

$$\begin{aligned} F= & {} F_1 I_0 \left( \sqrt{\gamma }r\right) +F_2 K_0 \left( \sqrt{\gamma }r\right) \nonumber \\ G= & {} G_1 I_0 \left( \sqrt{\eta }r\right) +G_2 K_0 \left( \sqrt{\eta }r\right) \end{aligned}$$

(8)

where $I_{0}$ and $K_{0}$ are modified Bessel functions.

Substitution of (6)–(8) gives the final result for concentration and temperature

$$\begin{aligned} c\left( r,t\right)= & {} \frac{1}{\alpha _1 -\alpha _2 }\left[ {-G_1 I_0 \left( \sqrt{\eta }r\right) -G_2 K_0 \left( \sqrt{\eta }r\right) +F_1 I_0 \left( \sqrt{\gamma }r\right) +F_2 K_0 \left( \sqrt{\gamma }r\right) } \right] \\ T\left( r,t\right)= & {} \frac{-\alpha _2 }{\alpha _1 -\alpha _2 }\left[ {F_1 I_0 \left( \sqrt{\gamma }r\right) +F_2 K_0 \left( \sqrt{\gamma }r\right) +\alpha _1 \left( {G_1 I_0 \left( \sqrt{\eta }r\right) +G_2 K_0 \left( \sqrt{\eta }r\right) } \right) } \right] \end{aligned}$$

Here $F_{i}, G_{i}\,(i=1,2)$ are constants depending on p that are found from the boundary conditions and finally give the results (3) of the paper.