Thermal conduction force under standing and quasistanding temperature field

Thermal conduction force plays a crucial role in manipulating the local thermal conductivity of crystals. However, due to the diffusive nature of thermal conduction, investigating the force effect is challenging. Recently, researchers have explored the force effect based on the wavelike behavior of thermal conduction, specifically second sound. However, their focus has been primarily on the progressive case, neglecting the more complex standing temperature field case. Additionally, establishing a connection between the results obtained from the progressive case and the standing case poses a challenging problem. In this study, we investigate the force effect of standing and quasistanding temperature fields, revealing distinct characteristics of thermal conduction force. Moreover, we establish a link between the progressive and standing cases through the quasistanding case. Our findings pave the way for research in more intricate scenarios and provide an additional degree of freedom for manipulating the local thermal conductivity of dielectric crystals.

Figure 1

Schematic of thermal conduction force under standing and quasistanding temperature field. (a) An adiabatic spherical particle in the presence of plane standing temperature field is depicted. The particle center is the origin of the Cartesian coordinate system $(x,y,z)$, and $\theta$ denotes the polar angle. (b) A particle with radius $a$ in the presence of Bessel standing second sound is shown, with $\delta$ representing the half cone angle. (c) The local thermal conductivity of dielectric crystals can be tuned by manipulating the interaction between the impurity particle (the blue sphere) and the second sound (the pink wave). ${\omega }_1$ and ${\omega }_2$ are the vibrating angular frequencies of the impurity particle before and after interacting with the second sound, and ${\kappa }_1$ and ${\kappa }_2$ are the corresponding local thermal conductivities of the crystal. The region containing the impurity particle is zoomed in for clarity.

Figure 2

Numerical results of the reduced second sound radiation force $Y_{st}$ under the conditions of (a) plane standing second sound as a function of $qa$ (where $q$ and $a$ are the second sound wave number and the particle radius) with the temperature ratio $T_1/T_0=0.02,0.06,0.10$ (where $T_1$ is the second sound amplitude, and $T_0$ is the background temperature), (b) zeroth-order Bessel standing second sound as a function of $qa$ and $\delta$ (unit: degree) with $T_1/T_0=0.10$, and (c) first-order Bessel standing second sound as a function of $qa$ and $\delta$ with $T_1/T_0=0.10$. The area enclosed by the red line is where $Y_{st}<0$.

Figure 3

Numerical results of the reduced second sound radiation force $Y_{st}$ under the conditions of (a) second-order Bessel standing second sound as a function of $qa$ and $\delta$ (unit: degree), (b) third-order Bessel standing second sound as a function of $qa$ and $\delta$, and (c) fourth-order Bessel standing second sound as a function of $qa$ and $\delta$, with $T_1/T_0=0.10$. The area enclosed by the red line is where $Y_{st}<0$.

Figure 4

Numerical results of the second sound radiation force $F$ under the condition of plane standing second sound as a function of $qa$ with $T_1/T_0=0.10$.