Total Force

Adding the surface and body forces acting on the fluid element, we have total force,

$$dF=dF_S+dF_B~=~\left( -\nabla p~+~\rho~g \right) dx~dy~dz$$ (2.10)

On a unit volume basis this equation becomes

$${{dF} \over {d\forall}}=-\nabla p + \rho~g$$ (2.11)

Thus we have obtained an expression for force upon a unit volume of a fluid element at rest. To extend this to the case of a moving fluid, one has to include normal and shear stresses due to viscosity in addition to the one given above. They are together balanced by inertia forces. Coming back to fluid at rest, the net force given by Eq.2.11 should be equal to zero. Accordingly,

$$~-\nabla p + \rho~g~ =~0$$ (2.12)

The above equation consists of three separate equations, one for each direction and each of them must be equal to zero. Thus for x, y and z directions we have,

$$- {{\partial p \over \partial x}}+\rho~g_x~=~0; ~~~~ - {{\partia... ...\partial y}}+\rho~g_y~=~0;~~~~ - {{\partial p \over \partial z}}+\rho~g_z~=~0.$$ (2.13)

If the coordinate system be so selected as to align one of x, y or z with acceleration due to gravity, g the equations simplify considerably. The natural selection is to have z-direction align with -g, such that $g_z=-g$ . Consequently, $g_x~=~g_y~=~0$. Then we have,

$${\partial p \over \partial x} =0; ~~~~{\partial p \over \partial z} =0 ~~\texttt{and} ~~ {\partial p \over \partial z} =-\rho g;$$ (2.14)

The above equation shows that pressure in a static fluid does not vary in x or y direction. It varies only in the z-direction. This enables one to write,

$${dp \over dz}=-\rho g=-\gamma$$ (2.15)

The above equation is a fundamental equation in Fluid Statics. It defines the manner in which pressure varies with height or elevation and finds many applications. Mainly it enables one to determine atmospheric pressures at different elevations above the sealevel. Then we employ the same equation to determine pressure at various depths of an ocean. The other application is in Manometry , which forms the basis of a class of pressure measuring instruments.

A close look at the equation reveals that the pressure gradient is a function of density, $\rho$ and acceleration due to gravity, $g$. The latter one, $g$ is almost a constant and therefore it is the variation in density with elevation that influences the pressure values. Density is constant for incompressible fluids and varies with pressure and temperature for compressible fluids. Therefore it is necessary to consider these two types of fluids separately.

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