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Steady flow is the condition in which the flow velocity profile does not vary with time.

Mathematically this is translated to $\frac{\partial \mathbf{v}}{\partial t} = 0$, for example in the derivation of Bernoulli's principle for the Navier-Stokes equation.

Why is it not $\frac{d\mathbf{v}}{dt}$?

I know the mathematical differences between the two differente operators, but what is the physical difference?

if the flow profile is constant in time, shouldn't the total derivative of the vector field $\mathbf{v}$ be constant everywhere?

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if the flow profile is constant in time, shouldn't the total derivative of the vector field v be constant everywhere?

No. Total derivative $d\mathbf v/dt$ can be written as $\partial \mathbf v/\partial t + \mathbf v \cdot \nabla \mathbf v$. Value of this derivative at point $\mathbf x$ gives acceleration of a fluid particle when it is at $\mathbf x$. Steadiness does not mean that acceleration of particles that pass through $\mathbf x$ vanishes. Most steady flows have particles accelerating (often they move curvilinearly), so this term is not zero. The steadiness refers to the fact that the velocity attached to the point $\mathbf x$ is constant in time, hence $\partial \mathbf v/ \partial t$ is zero.

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  • $\begingroup$ So why does incompressible flow <=> $\frac{d\rho}{dt} = 0$? $\endgroup$ – SuperCiocia Apr 20 '14 at 18:22
  • $\begingroup$ I do not see any relation to incompressible flow. $\endgroup$ – Ján Lalinský Apr 20 '14 at 22:16
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A Steady flow is one in which the condition (Velocity,pressure and cross-sectional) may differ from point to point but do not change with time.

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