# 震动与热物理 Vibrations & Thermal Phys (JH) PHYS128001

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It can also be written
$$\frac{D \rho}{D t}+\rho \nabla \cdot \mathbf{u}=0$$
where a widely-used abbreviation for the differential operator is
$$\frac{D}{D t} \equiv \frac{\partial}{\partial t}+\mathbf{u} . \nabla$$
An important special case is that of an incompressible fluid in which there is no change of density following the motion of the fluid. If a small volume of fluid is labeled, then the fluid bearing that label always has the same density wherever it is in the future, although its neighbors may have different densities. The idea can be expressed in mathematical form by considering a short interval of time $\delta t$; the idea of incompressibility means that
$$\rho(\mathbf{r}+\mathbf{u} \delta t, t+\delta t)=\rho(\mathbf{r}, t)+O\left(\delta t^{2}\right)$$

## PHYS128001COURSE NOTES ：

$$\frac{\partial S}{\partial t}+\mathbf{u} \cdot \nabla S=\nabla . \kappa \nabla S+q$$
The special case when $\rho, c_{p}$, and $k$ are all constants reduces the equation to
$$\frac{\partial T}{\partial t}+\mathbf{u} . \nabla T=\kappa \nabla^{2} T+\frac{q}{\rho c_{p}}$$
If, in addition, the entire medium is moving with a constant velocity $U$ in the direction of the positive $x$ axis,
$$\frac{\partial T}{\partial t}+U \frac{\partial T}{\partial x}=\kappa \nabla^{2} T+\frac{q}{\rho c_{p}}$$