Pressure Over Submerged Surfaces

Recalling our net force expression for fluids which is…
$F_{grav}+F_{pres}+F_{fric}=ma$ where:

• $F_{grav}=\rho dVg$
• $F_{pres}=-\nabla PdV$
• $F_{fric}=fdV$ from Newtons 2nd Law which simplifies to $\rho g-\nabla P+f_{fric}=\rho a=\rho \frac{DV}{Dt}$

Here, we start with two special cases:

• Hydrostatics → fluid is at rest
• Rigid body motion → no shear

Keeping in mind pressure acts normal to any surface!!!…

To compute this, we can compute a resulting force that is applied at the center of pressure buuuut what we will do is compute infinitesimal forces and integrate over the surface.

• Draw FBD
• Set up surface and coordinate systems
• Determine the differential force due to pressure on a surface element
• Add up forces and moments (force balance stuff)

For Inclined Surfaces

• We want to set up our coordinate system so that s grows from the bottom of the gate along the gate
• Write an expression for $d{F}_p$ which is our differential pressure perpendicular to our surface
• Now we integrate along the surface and solve for $F_p$

Curved Surfaces

Pressure acts normally to a given surface, $\theta$ goes from 0 to $\frac{\pi }{2}$, $p=pgRsin\theta +p_{atm}$ and components of our forces here are $d{F}_x=d{F}_p\cos \theta$ and $d{F}_y=d{F}_{p}sin\theta$

Using these relationships, we can derive the following for a circle…

• $F_{px}={\int }_0^{\frac{\pi }{2}}\left[\rho g\left(R\sin \theta \right)+p_{atm}\right]wR\cos \theta d\theta$
• $F_{py}={\int }_0^{\frac{\pi }{2}}\left[\rho g\left(R\sin \theta \right)+p_{atm}\right]wR\sin \theta d\theta$

Specific MTE 352 Example (Logs)

For a log shape…

• The log is circular
• For a length of w
• $h=H-R\sin \theta$
• $d{F}_p=\left[p_{atm}+\rho g\left(H-R\sin \theta \right)\right]wRd\theta$
• $d{F}_x={\int }_0^{2\pi }\left[p_{atm}+\rho g\left(H-R\sin \theta \right)\right]wR\cos \theta d\theta =0$ since $\cos \theta \sin \theta$ integrates to 0 → periodic, and ${\int }_0^{2\pi }\cos \theta$ integrates to 0
• $d{F}_y={\int }_0^{2\pi }-\left[p_{atm}+\rho g\left(H-R\sin \theta \right)\right]wR\sin \theta d\theta$ = $\rho g{R}^2\left[\frac{\theta }{2}-\frac{\sin 2\theta }{4}\right]$ from 0 to $2\pi$ = $\rho gw{R}^2\pi$
  • So this is just density x gravity x volume

Flat Plate

• The top and the bottom of the plate have different relationships
• Top: $F_y=-\rho ghwl$
• Bottom: $+\rho g\left(h+\Delta h\right)wl$
• where here, $p=\rho gh$, height is the distance form the surface of the fluid and w is the width fo the plate

Submerged Bodies

• A uniform pressure acting around a body creates no net force on the body
• If the fluid is static, the net force due to pressure acts vertically upward and is equal to the weight of the displaced fluid (Archimedes principle)

The buoyancy force is $F_B=\rho Vg$