/sci/ - Science & Math

>>11310439
>>11310483
Okay, so imagine you have a large, still body of water, like a lake. Now, select a differential element in the middle of the lake, some depth down. Choose this element to be a cube with its edges aligned with the Cartesian axes to make analysis simple (z dir. points up, and x and y are horizontal directions). Now, we know for an experimental fact that pressure changes with depth, we just don't know how. We also know there are some forces on this differential cube and that the cube is in equilibrium. (Forces in the horizontal directions balance obviously by symmetry, nothing interesting to look at.) As far as forces along the z dir. go, there is a pressure on the bottom face, [math]
P [/math], pressure on the top face [math] P+\text{d}P [/math], and the body force of gravity [math] W=\gamma\text{ d}V [/math] where gamma is specific weight and dV is the volume of the cube. Now, since we have equilibrium, we are ready to apply Newton's laws.
[eqn] \sum F_z=P\text{ d}A-(P+\text{d}P)\text{d}A-\gamma\text{ d}V=0 \implies \text{d}P\text{ d}A=-\gamma\text{ d}V [/eqn]
But, since [math] \text{d}A=\text{d}x\text{d}y [/math] and [math] \text{d}V=\text{d}x\text{d}y\text{d}z [/math], we get
[eqn]\text{d}P=-\gamma\text{ d}z\implies\frac{\partial P}{\partial z}=-\gamma [/eqn]
which says that the hydrostatic pressure at a depth is proportional to that depth and to the density of the fluid, and nothing else. That's the symbolic "proof" that comes from Newton's laws, but you are also welcome to confirm this with experiment. It is counter-intuitive, but nature doesn't care.
>>11310705
Use a psychrometric chart. https://www.ashrae.org/File%20Library/Technical%20Resources/Bookstore/UP3/SI-1.pdf If the process is adiabatic, you will be following that straight, diagonal line of constant enthalpy.
>>11309543
>do tobacco leaves just have a natural resin
that's called "tar," darling. isn't smoking fun?