= Fluid mechanics
{wiki}
<Fluid mechanics> studies the motion and <forces> of <liquids> and <gases>. Its central simplification is the <continuum approximation>: replace individual <molecules> by bulk properties defined at every position and time. The averaging region must contain many <molecules> while remaining small compared with the distances over which those properties vary. A <fluid element> is such a small material parcel, rather than a single <molecule>. This connects <fluid mechanics> to <continuum mechanics> and makes <calculus> useful for describing flow. https://ocw.mit.edu/courses/2-016-hydrodynamics-13-012-fall-2005/dc299d2ff05553b3515dccc23fc0f7d2_2005reading1.pdf[MIT, A. Techet, Hydrodynamics reading 1, pp. 2–3]{ref}
Three basic fields are the <velocity field> $\mathbf u(\mathbf x,t)$, the <mass density> $\rho(\mathbf x,t)$ and the <pressure> $p(\mathbf x,t)$. The first gives a parcel's instantaneous <velocity>; the second gives <mass> per unit <volume>. The <pressure> contributes an inward normal <traction> $-p\mathbf n$ across a surface with outward unit normal $\mathbf n$. Thus <pressure> is a <stress>, measured in $\mathrm{Pa}=\mathrm{N\,m^{-2}}$, and may vary with position even in <incompressible flow>. Additional <viscous stresses> can act tangentially. https://ocw.mit.edu/courses/2-016-hydrodynamics-13-012-fall-2005/dc299d2ff05553b3515dccc23fc0f7d2_2005reading1.pdf[MIT, A. Techet, Hydrodynamics reading 1, p. 5]{ref}
The guiding questions are what must be conserved and which <forces> matter. <Mass conservation> constrains the <velocity field>; <momentum conservation> determines its response to <pressure gradients>, <viscosity> and body <forces>. An <incompressible flow> model, a <Reynolds number> estimate and <Bernoulli's principle> answer different parts of a problem. Their appeal is that a few carefully stated assumptions can determine useful quantities before the entire <velocity field> is known.
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