Fluid Statics Equation

The fundamental equation of fluid statics describes the spatial variation of hydrostatic pressure p in the continuous mass of a fluid.

Derivation: Consider a fluid element at rest of given mass with volume V and bounded by the surface S.
 

The fluid element stays at equilibrium under the action of the following two forces

(i) The Resultant Body Force



(ii) The Resultant Surface Force
 

    


For the fluid element to be in equilibrium , we have



The equation is valid for any volume of the fluid element, no matter how small, thus we get
     
 
 
This is the fundamental equation of fluid statics.

Fundamental Fluid Static Equations in Scalar Form

Considering gravity as the only external body force acting on the fluid element, Eq. (3) can be expressed in its scalar components with respect to a cartesian coordinate system (see Fig. 1) as



From Eqs (3a)-(3c), it can be concluded that the pressure p is a function of z only.

Thus, Eq. (3c) can be re-written as,

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