Pressure at a point within a fluid

Consider a fluid at rest as shown in Fig. 2.2. From around the point of interest, P in the fluid let us pull out a small wedge of dimensions dx x dz x ds . Let the depth normal to the plane of paper be b. In some of the derivations we chose z to be the vertical coordinate. This is consistent with the use of z as the elevation or height in many applications involving atmosphere or an ocean. Let us now mark the surface and body forces acting upon the wedge.

Figure 2.2 : Pressure at a point

 

The surface forces acting on the three faces of the wedge are due to the pressures, $p_x$, $p_z$ and $p_n$ as shown. These forces are normal to the surface upon which they act. We follow the usual convention that compression pressure is positive in sign. We again remind ourselves that since the fluid is at rest there is no shear force acting. In addition we have a body force, the weight, W of the fluid within the wedge acting vertically downwards.

Summing the horizontal and the vertical forces we have,

$$\Sigma F_x =0;~ \texttt{ie.}, p_x ~ dz~b ~-~ p_n ~ b ~ ds ~ sin \theta = 0$$

$$\Sigma F_z =0;~ \texttt{ie.}, p_z ~ dx~b ~-~ p_n ~ b ~ ds ~ cos \theta~-~W ~~= ~~0$$

$$\texttt{i.e.,} = p_z ~ dx~b ~-~ p_n ~ b ~ ds ~ cos \theta~-~{1 \over 2} \rho~g ~b ~dx~ dz ~~= ~~0$$ (2.1)

Noting that

$$ds~sin\theta~=~dz,~~\texttt{and }~~~ ds~cos\theta~=~dx$$ (2.2)

we have after simplification,

$$p_x~=~p_n,~~~\texttt{and}~~~p_z~=~p_n~+~{1 \over 2}~ \rho~g~dz$$ (2.3)

We note that the pressure in the horizontal direction does not change, which is a consequence of the fact that there is shear in a fluid at rest. In the vertical direction there is a change in pressure proportional to density of the fluid, acceleration due to gravity and difference in elevation.

Now if we take the limit as the wedge volume decreases to zero, i.e., the wedge collapses to the point P, we have,

$$p_x~=~p_z~=p_n~=~p$$ (2.4)

This equation is known as Pascal's Law. It is important to note that it is valid only for a fluid at rest. In the case of a moving fluid, pressures in different directions could be different depending upon fluid accelerations in different directions. Hence, for a moving fluid pressure is defined as an average of the three normal stresses acting upon the fluid element.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney