If I'm correct Bernoulli's principle says that for a given streamline $\vec{\gamma}(t) \overset{\mathrm{def}}{=} \vec{\gamma}_0 + \int_0^t\vec{v}(\vec{ \gamma}(\tau))\mathrm{d}\tau$ in an incompressible, inviscid fluid

$$ p(\vec{\gamma}(t)) + \frac{1}{2}\rho |\vec{v}(\vec{\gamma}(t))|^2+\rho \vec{g} \cdot \vec{\gamma(t)} = \mathrm{const} $$

for all $t\ge0$. I can make sense of the pressure (or rather gradients thereof) as being what causes the velocity to vary (disregarding gravity). However, the other way round is less clear to me. How do differences in the velocity give rise to pressure gradients? Actually, what is the analytical expression for/the model of pressure most commonly assumed when talking about Bernoulli's principle? I'm struggling quite a bit intuitively to think of a model for pressure for a material like an incompressible, inviscid fluid which is incompressible but (as long as the volume isn't changed) aribtrarily deformable, and at arbitrary rates, too...!


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