Fluid Static Definitions, Pressure at a point, Variation of Pressure in a static fluid

Fluid Static
Definitions, Pressure at a point, Variation of Pressure in a static fluid
Fluid Statics When the fluid velocity is zero, called the hydrostatic condition.
Definition of Pressure
Pressure is defined as the ratio of normal force to area at a point.
Figure 1: Pressure acting on the walls of a sphere
Pressure properties:
The reason that pressure is defined using a derivative is that pressure often varies from point to point.
Pressure is a scalar quantity; that is, it has magnitude only.
Pressure is not a force.
The pressure variation is due only to the weight of the fluid.


Pressure at a Point
Consider a small wedge of fluid at rest of size ?x, ?z, ?s and depth b into the paper.
There is no shear stress,
and pressure is assumed to be identical on each face (small element).
Since the element is at rest, summation of all forces must equal zero.
???Fx=0=Pxb?z-Pnb?s sin? ?
???Fz=0=Pzb?x-Pnb?s cos?-1/2 ? ?gb?x?z
From geometry: ?s sin?=?z, ?s cos?=?x
Fig. 2: Equilibrium of a small Fluid element at rest.
After substitution in above equations, one finds:
Px=Pn, Pz=Pn+1/2 ?g?z
This means:
There is no pressure change in the horizontal direction.

There is a vertical change in pressure proportional to

the density,
gravity and
depth change in the fluid (i.e. the weight of the column of the fluid above the point).
Note: in the limit as the fluid wedge shrinks to a point, ?z goes to zero, we have:
Px=Pz= Pn=P Thus, pressure in a static fluid is a point property.

Variation of a Pressure in Static Fluid
Assume the pressure vary arbitrarily in a fluid, p=p(x,y,z,t). Consider a fluid element of size ?x, ?y, ?z as shown in Fig. 3. The net force in the x?direction is given by:
dFx=Pdydz- (P+?P/?x dx)dydz=-?P/?x dxdydz