Friction

On solid:

{\dot{m}}_{i n} = {\dot{m}}_{o u t} ⟶ V_{o u t} = V_{i n} = V

h_{T, o u t} + w = h_{T, i n} + q

\frac{P_{o u t}}{ρ} + C T_{o u t} + w = \frac{P_{i n}}{ρ} + C T_{i n} + q

where $w$ is the work of friction forces and $q$ is the heat due to friction

s_{o u t} \geq s_{i n} + \frac{q}{T}

f_{p u s h} + f_{f r i c t i o n} = m a = 0

(P_{i n} - P_{o u t}) π R^{2} = f_{f r i c t i o n}

(P_{i n} - P_{o u t}) π R^{2} L = W_{f r i c t i o n}

where $π R^{2} L$ is the volume of CV

\frac{P_{o u t}}{ρ} + C T_{o u t} + w = \frac{P_{i n}}{ρ} + C T_{i n} + q \Rightarrow T_{o u t} = T_{i n} + \frac{q}{C}

s_{o u t} \geq s_{i n} + \frac{q}{T} \Rightarrow C \ln (\frac{T_{o u t}}{T_{i n}}) \geq \frac{q}{T} \Rightarrow T_{o u t} \approx T_{i n} + \frac{q}{C}