Hydrostatics, or fluid statics, is the study of fluids at rest. Important applications of hydrostatics include buoyancy and hydrostatic pressure.
Hydrostatic Forces
A static fluid in Earth’s atmosphere will only have two forces acting upon it:
- Gravity: Due to Earth’s gravitational field.
- Fluid Pressure: Due to air and fluid above a particular fluid particle at a given point.
The force due to pressure is:
\displaystyle\bm{F}=-\iiint_V(\nabla p) dV\llap{—}
\displaystyle\bm{F}=-\iint_{A_x}pdA_x\bm{\hat{i}}-\iint_{A_y}pdA_y\bm{\hat{j}}-\iint_{A_z}pdA_z\bm{\hat{k}}
\displaystyle\bm{F}=-\iint_{A_x}pdydz\bm{\hat{i}}-\iint_{A_y}pdxdz\bm{\hat{j}}-\iint_{A_z}pdxdy\bm{\hat{k}}
\displaystyle\bm{F}=-\iint_{A_x}\dfrac{\partial p}{\partial x}dxdydz\bm{\hat{i}}-\iint_{A_y}\dfrac{\partial p}{\partial y}dxdydz\bm{\hat{j}}-\iint_{A_z}\dfrac{\partial p}{\partial z}dxdydz\bm{\hat{k}}
\displaystyle\bm{F}=-\iiint_V(\nabla p) dV\llap{—}
Due to gravity fluid pressure varies with fluid depth. The effect of pressure can therefore be expressed as the follows:
\nabla p=\rho\bm{g}
Or more simply:
\dfrac{d p}{d z}=\rho g
From which we can conclude the familiar formula for pressure at a given depth in a fluid:
p=p_0+\rho gz
For the simplified 2D case of a submerged object, the following equation applies for the force due to fluid pressure:
\displaystyle F=\iint_A-(\rho g y) dA
The pressure force always acts normal to a surface, therefore we can determine the direction of the force independently from the magnitude of the force. To simplify analysis we define the point, (x_c, y_c), where the sum of all forces and moments due to fluid pressure can be described by a single equivalent force. This point is termed the centre of pressure.
x_c=\dfrac{1}{F}\displaystyle\iint_A xPdA
y_c=\dfrac{1}{F}\displaystyle\iint_A yPdA
Methodology for Plane Surfaces
It is assumed that plane surfaces in 2D will be parallel to the z axis (into the page). Therefore plane surfaces appear as a line with a given angle to the vertical axis. The plane could be any two dimensional shape, a triangle, circle, rectangle, or any 2D surface.
At this point to figure out the force due to fluid pressure we must simply:
- Calculate F
- Calculate (x_c,y_c)
- In your free-body diagram plot F at (x_c,y_c), in the direction perpendicular to the plane.
