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===Hydrostatic pressure===<!-- This section is linked from [[Water tower]] --><!-- target for redirect [[hHydrostic pressure]] --> {{See also|Vertical pressure variation}} In a fluid at rest, all frictional and inertial stresses vanish and the state of stress of the system is called ''hydrostatic''. When this condition of {{math|''V'' {{=}} 0}} is applied to the [[Navier–Stokes equations]] for viscous fluids or [[Euler equations (fluid dynamics)]] for ideal inviscid fluid, the gradient of pressure becomes a function of body forces only. The Navier-Stokes momentum equations are: {{Equation box 1 |indent=: |title='''Navier–Stokes momentum equation''' (''convective form'') |equation=<math> \rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = - \nabla [p - \zeta (\nabla\cdot\mathbf{u})] + \nabla \cdot \left\{ \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right\} + \rho\mathbf{g} .</math> |cellpadding |border |border colour = #FF0000 |background colour = #DCDCDC }} By setting the [[flow velocity]] <math> \mathbf u = \mathbf 0</math>, they become simply: <math> \mathbf 0 = - \nabla p + \rho\mathbf{g}</math> or: <math> \nabla p = \rho \mathbf{g}</math> This is the general form of Stevin's law: the [[pressure gradient]] equals the [[body force]] [[force density]] field. Let us now consider two particular cases of this law. In case of a [[conservative force|conservative]] body force with [[scalar potential]] <math> \phi</math>: <math> \rho \mathbf{g} = - \nabla \phi</math> the Stevin equation becomes: <math> \nabla p = - \nabla \phi</math> That can be integrated to give: <math> \Delta p = - \Delta \phi</math> So in this case the pressure difference is the opposite of the difference of the scalar potential associated to the body force. In the other particular case of a body force of constant direction along z: <math> \mathbf{g} = - g(x ,y,z) \hat k</math> the generalised Stevin's law above becomes: <math> \frac {\partial p}{\partial z}= - \rho(x,y,z) g(x, y, z)</math> That can be integrated to give another (less-) generalised Stevin's law: <math> p (x,y,z) - p_0(x,y) = - \int_0^z \rho(x,y,z') g(x, y, z') dz'</math> where: * <math>p</math> is the hydrostatic pressure (Pa), * <math>\rho</math> is the fluid [[density]] (kg/m<sup>3</sup>), * <math>g</math> is [[gravity|gravitational]] acceleration (m/s<sup>2</sup>), * <math>z</math> is the height (parallel to the direction of gravity) of the test area (m), * <math>0</math> is the height of the [[Pressure measurement#Absolute, gauge and differential pressures — zero reference|zero reference point of the pressure]] (m) * <math>p_0</math> is the hydrostatic pressure field (Pa) along x and y at the zero reference point For water and other liquids, this integral can be simplified significantly for many practical applications, based on the following two assumptions. Since many liquids can be considered [[Incompressible flow|incompressible]], a reasonable good estimation can be made from assuming a constant density throughout the liquid. The same assumption cannot be made within a gaseous environment. Also, since the height <math>\Delta z</math> of the fluid column between {{mvar|z}} and {{math|''z''<sub>0</sub>}} is often reasonably small compared to the radius of the Earth, one can neglect the variation of {{mvar|[[Gravity|g]]}}. Under these circumstances, one can transport out of the integral the density and the gravity acceleration and the law is simplified into the formula :<math>\Delta p (z) = \rho g \Delta z,</math> where <math>\Delta z</math> is the height {{math|''z'' − ''z''<sub>0</sub>}} of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called [[Simon Stevin|Stevin's]] law.<ref>{{cite book|last1=Bettini|first1=Alessandro|title=A Course in Classical Physics 2—Fluids and Thermodynamics|date=2016|publisher=Springer|isbn=978-3-319-30685-8|page=8}}</ref><ref>{{cite book|last1=Mauri|first1=Roberto|title=Transport Phenomena in Multiphase Flow|date=8 April 2015|publisher=Springer|isbn=978-3-319-15792-4|page=24|url=https://books.google.com/books?id=S3L0BwAAQBAJ&pg=PA24|access-date=3 February 2017}}</ref> One could arrive to the above formula also by considering the first particular case of the equation for a conservative body force field: in fact the body force field of uniform intensity and direction: <math> \rho \mathbf{g}(x,y,z) = - \rho g \hat k</math> is conservative, so one can write the body force density as: <math> \rho \mathbf{g} = \nabla (- \rho g z) </math> Then the body force density has a simple scalar potential: <math> \phi(z) = - \rho g z</math> And the pressure difference follows another time the Stevin's law: <math> \Delta p = - \Delta \phi = \rho g \Delta z</math> The reference point should lie at or below the surface of the liquid. Otherwise, one has to split the integral into two (or more) terms with the constant {{math|''ρ''<sub>liquid</sub>}} and {{math|''ρ''(''z''′)<sub>above</sub>}}. For example, the [[Pressure measurement#Absolute, gauge and differential pressures - zero reference|absolute pressure]] compared to vacuum is :<math>p = \rho g \Delta z + p_\mathrm{0},</math> where <math>\Delta z</math> is the total height of the liquid column above the test area to the surface, and {{math|''p''<sub>0</sub>}} is the [[atmospheric pressure]], i.e., the pressure calculated from the remaining integral over the air column from the liquid surface to infinity. This can easily be visualized using a [[pressure prism]]. Hydrostatic pressure has been used in the preservation of foods in a process called [[pascalization]].<ref>{{cite book|url=https://books.google.com/books?id=edPzm5KSMmYC|title=Understanding Food: Principles and Preparation|last=Brown|first=Amy Christian|publisher=Cengage Learning|year=2007|edition=3|isbn=978-0-495-10745-3|page=546}}</ref>
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