# [[Introduction to Fluid Mechanics - Fox, McDonald.pdf#page=63|Introduction to Fluid Mechanics, 48-56]] - Alternative way of expressing density of a substance is to compare it to an accepted reference value, typically the maximum density of water ($1000kg/m^3$ at 4C) thus the specific gravity $SG=\frac{\rho}{\rho_{H_{2}O}}$ - For a static fluid, only normal force is present - pressure - The only body force that must be considered in most engineering problems is due to gravity. For a differential fluid element, the body force is $ d\vec{F}_{B}=\vec{g}dm=\vec{g}\rho dV=\rho \vec{g}dxdydz $ - Net surface force acting on element given by $ d\vec{F}_{S}=-\left( \frac{\partial p}{\partial x}\hat{i}+ \frac{\partial p}{\partial y}\hat{j}+\frac{\partial p}{\partial z}\hat{k} \right)dxdydz=-\nabla p~dxdydz $ - Physically the gradient of pressure is the negative of the surface force per unit volume due to pressure - Net force $d\vec{F}$ given by $ d\vec{F}=d\vec{F}_{S}+d\vec{F}_{B}=(-\nabla p+\rho \vec{g})dxdydz=(-\nabla p+\rho \vec{g})dV;~ \frac{d\vec{F}}{dV}=-\nabla p+\rho \vec{g} $ - For a static fluid, $\vec{a}=0$ so $\frac{d\vec{F}}{dV}=\rho \vec{a}=0$ - Assuming that the coordinate system is chosen w/ the z-axis directed vertically upward, $g_{x}=g_{y}=0$ and $g_{z}=-g$ then $\boxed{ \frac{dp}{dz}=-\rho g=-\gamma }$ - Pressure values must be stated w/ respect to a reference level. If the reference level is a vacuum, pressures are termed *absolute* - As height changes, $\boxed{ \Delta p=\rho gh }$ for incompressible fluids - Pressure differences between 2 points in static incompressible fluid can be determined by measuring elevation difference between two points, device used called manometers 1. Any two points at the same elevation in a continuous region of the same liquid are at the same pressure 2. Pressure increases as one goes *down* a liquid column - See [[Introduction to Fluid Mechanics - Fox, McDonald.pdf#page=68|Example 3.1, 3.2]] # Lecture Notes - $\lambda=\frac{k_{B}T}{\sqrt{ 2 }\pi d^2\rho}$ where $k_{B}$ is Boltzmann's constant, $d$ is molecular diameter - $T\propto\lambda$ $p,d\propto \frac{1}{\lambda}$ - $k_{N}= \frac{\lambda}{L}$, continuum for $k_{N}\ll{1}$ - Specific gravity $SG=\frac{\rho}{\rho_{H_{2}O}}$ - Specific weight $\gamma=\rho g$ - Stresses $\sigma_{n}=\frac{F_{N}}{A_{0}},\tau_{T}=\frac{F_{T}}{A_{0}}$ -> $p$ is $\sigma_{n}$ acting - Viscosity $\mu$ - Absolute pressure is due to fluid eq. of state ($p=\rho RT$) - Gauge pressure $p_{gauge}=p_{abs}-p_{ref}$, $p_{ref}$ is often $p_{atm}=1atm=101.325kPa$ > > Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity (in our cases). Normal force due to gravity affects the liquid in all directions. See pg. 02 for further derivation of below formulas > $ > \boxed{ \frac{d\vec{F}}{dV}=\rho \vec{g}-\nabla \vec{p} } > $ > $ > \boxed{ \rho \vec{g}=\nabla p\vec{v} } ~~~ \boxed{ \frac{dp}{dz}=-\rho g=-\gamma } > $ > *assuming $g$ acts in $-\hat{k}$* - Barometers measure pressure relative to a vaccum - Incompressible: density does not vary -> $\Delta p=p-p_{0}=-\rho g(z-z_{0})=\rho g\Delta h$ - **Pressures do not vary horizontally!!** - For convoluted setup, use $\Delta p=g\sum_{i}\rho_{i}h_{i}$ (see [[Introduction to Fluid Mechanics - Fox, McDonald.pdf#page=72&selection=0,0,4,3|Example 3.3]] )