Hydrostatic Pressure in a Static Fluid Element
Consider a static cubic fluid element of dimensions \( \delta x, \delta y, \delta z \) subjected to pressure \( p \) at its centroid.
Pressure Derivatives
The pressure gradients in the three directions are:
\[ \frac{\partial p}{\partial x} = 0,\quad \frac{\partial p}{\partial y} = 0,\quad \frac{\partial p}{\partial z} = -\rho g \]These can be compactly represented using the gradient operator:
\[ \nabla p = \rho g \hat{k} \]Where the gradient operator is defined as:
\[ \nabla = \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k} \]Integration for Constant Density
Assuming constant density \( \rho \), integrating the pressure gradient gives:
\[ p = -\rho g z + C \]At any point \( x \), if \( p = p_a \) at \( z = z_0 + h \), then:
\[ p_x = p_a + \rho g h \]Hydrostatic Pressure Insight
Thus, the hydrostatic pressure at any point in a static fluid depends only on the vertical depth below the free surface and the specific weight of the fluid. It is independent of the shape and size of the container.
This principle is used in Torricelli’s theorem via Bernoulli’s equation to determine the velocity at depth \( h \):
\[ v = \sqrt{2gh} \]Pressure Head
The height \( h \) is known as the pressure head, representing the height of a fluid column that produces a given pressure:
\[ h = \frac{p}{\rho g} \]Compressible Fluids
For compressible fluids, density is a function of pressure:
\[ \rho = f(p) \]Therefore, the pressure-depth relationship becomes:
\[ \int \frac{dp}{\rho g} = - \int dz \]