we can define the energy and entropy currents
What you defined were the rates of change ($\dot U$, $\dot S$). In the case of energy, which is conserved, the rate of change must correspond to the net flow. But entropy is different; it is not conserved.
Your example can be generalized to
$$T\dot S=\dot U-\sum_i F_i\frac{\partial x_i}{\partial t},$$
where $\dot S$ is the rate of system entropy increase and $F_j$ and $x_j$ are respectively generalized forces and displacements; examples include
- negative pressure and volume
- surface tension and surface area
- electric field and polarization
- voltage and charge
- magnetic field and magnetic moment
- force and distance
- stress and volumetric strain
- chemical potential and amount of matter
Along with energy, the generalized displacements above are conserved except for the electrical polarization and magnetic moment; thus, we ignore those forms of work for simplicity. By the divergence theorem expressed as $\frac{dA}{dt}=-\nabla\cdot\boldsymbol{J_A}+\dot\sigma =-\nabla\cdot\boldsymbol{J_A}$, where $\boldsymbol{J_A}$ is the flux of A and $\dot\sigma$ represents generation (absent for a conserved quantity), we obtain the continuity relation
$$T\dot S=-\nabla\cdot\boldsymbol{J_U}+\sum_iF_i\nabla\cdot\boldsymbol{J_i}.$$
From the chain rule, $A\nabla\cdot\boldsymbol{B}=-\boldsymbol{B}\nabla\cdot A+\nabla\cdot (A\boldsymbol{B})$; thus, letting $A=\frac{1}{T}$,
\begin{align}T\dot S&=T\left(\boldsymbol{J_U}\cdot\nabla\frac{1}{T}-\nabla\cdot\frac{\boldsymbol{J_U}}{T}-\sum_i\boldsymbol{J_i}\cdot\frac{F_i}{T}+\nabla\cdot\frac{\sum_iF_i\boldsymbol{J_i}}{T}\right)\\&=T\left[\boldsymbol{J_U}\cdot\nabla\frac{1}{T}-\sum_i\boldsymbol{J_i}\cdot\frac{F_i}{T}-\nabla\cdot\left(\frac{\boldsymbol{J_U}-\sum_iF_i\boldsymbol{J_i}}{T}\right)\right].\end{align}
Looking again at the divergence theorem, we infer a relation
$$T\dot S=T(\dot \sigma-\nabla\cdot \boldsymbol{J_S}),$$
where we identify
$$\dot\sigma =\boldsymbol{J_U}\cdot\nabla\frac{1}{T}-\sum_i\boldsymbol{J_i}\cdot\nabla\frac{F_i}{T}$$
as the entropy generation rate and $$\boldsymbol{J_S}=\frac{\boldsymbol{J_U}-\sum_iF_i\boldsymbol{J_i}}{T}$$
as the conserved entropy flux.
Applying the chain rule to the entropy generation rate gives
\begin{align}\dot\sigma&=\frac{1}{T^2}\left(-\boldsymbol{J_U}\cdot\nabla T+\sum_iF_i\boldsymbol{J_i}\cdot \nabla T-T\sum_i\boldsymbol{J_i}\cdot\nabla F_i\right)\\&=\frac{1}{T^2}\left[\left(-\boldsymbol{J_S}\cdot\nabla T\right)+T\sum_i\left(-\boldsymbol{J_i}\cdot\nabla F_i\right)\right].\end{align}
What this tells us is that entropy can be transferred in and out of a system through heating and through the entropy carried by matter, for example. This is $\boldsymbol{J_S}$.
In addition, entropy is generated whenever a generalized displacement (all $\boldsymbol{x_i}$ plus entropy itself) is driven by a gradient in any internal property (all $\boldsymbol{F_i}$ plus temperature). The direction of spontaneous flow is opposite the direction of the steepest gradient in the intensive property; in this way, Nature smooths out disturbances. This is $\boldsymbol{\dot\sigma}$.
Depending on how much entropy you transfer in and out and how much entropy you generate, $\boldsymbol{\dot S}$ can be positive or negative.