In the textbook "Elementary Fluid Mechanics" by D.J. Acheson (Section 1.2), steady flow is defined by the equation $\frac{\partial \textbf{u}}{\partial t} = 0$, where $\textbf{u}(x, t)$ is the velocity field. Is it also true that during steady flow, the pressure $p$ has no explicit time-dependence, i.e. $\frac{\partial p}{\partial t} = 0$? If so, is this an additional condition that is part of the definition for steady flow, or is it something that can be deduced from the fact that $\frac{\partial \textbf{u}}{\partial t} = 0$ (perhaps using Euler's equations)?
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$\begingroup$ Steady flow means that nothing in the system varies with time. $\endgroup$– Chet MillerCommented Nov 7 at 11:52
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A steady flow is when all the conditions at any point in a stream are constant in time. This would not only include velocity, but also density, pressure, temperature, etc.
However, the equation involving $\partial_tp=\cdots$ is part of the closure condition and is usually included in the Euler equations as part of the energy equation (e.g., assuming ideal gas you have $e\sim p/(\gamma-1)$ for adiabatic index $\gamma$), so I'm not sure that 'deduced' is an appropriate word here, it's just following a definition at that point.
