Stating Bernoulli’s Equation
Bernoulli’s Equation
The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant:
\(P+\cfrac{1}{2}{\mathrm{\rho v}}^{2}+\rho \text{gh}=\text{constant,}\)
where \(P\) is the absolute pressure, \(\rho \) is the fluid density, \(v\) is the velocity of the fluid, \(h\) is the height above some reference point, and \(g\) is the acceleration due to gravity. If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli’s equation becomes
\({P}_{1}+\cfrac{1}{2}{\mathrm{\rho v}}_{1}^{2}+\rho {\mathrm{gh}}_{1}={P}_{2}+\cfrac{1}{2}{\mathrm{\rho v}}_{2}^{2}+\rho {\mathrm{gh}}_{2}\text{.}\)
Bernoulli’s equation is a form of the conservation of energy principle. Note that the second and third terms are the kinetic and potential energy with \(m\) replaced by \(\rho \). In fact, each term in the equation has units of energy per unit volume. We can prove this for the second term by substituting \(\rho =m/V\) into it and gathering terms:
\(\cfrac{1}{2}{\mathrm{\rho v}}^{2}=\cfrac{\frac{1}{2}{\text{mv}}^{2}}{V}=\cfrac{\text{KE}}{V}\text{.}\)
So \(\cfrac{1}{2}{\mathrm{\rho v}}^{2}\) is the kinetic energy per unit volume. Making the same substitution into the third term in the equation, we find
\(\rho \mathrm{gh}=\cfrac{\mathrm{mgh}}{V}=\cfrac{{\text{PE}}_{\text{g}}}{V},\)
so \(\rho \text{gh}\) is the gravitational potential energy per unit volume. Note that pressure \(P\) has units of energy per unit volume, too. Since \(P=F/A\), its units are \({\text{N/m}}^{2}\). If we multiply these by m/m, we obtain \(\text{N}\cdot {\text{m/m}}^{3}={\text{J/m}}^{3}\), or energy per unit volume. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.
Making Connections: Conservation of Energy
Conservation of energy applied to fluid flow produces Bernoulli’s equation. The net work done by the fluid’s pressure results in changes in the fluid’s \(\text{KE}\) and \({\text{PE}}_{\text{g}}\) per unit volume. If other forms of energy are involved in fluid flow, Bernoulli’s equation can be modified to take these forms into account. Such forms of energy include thermal energy dissipated because of fluid viscosity.
The general form of Bernoulli’s equation has three terms in it, and it is broadly applicable. To understand it better, we will look at a number of specific situations that simplify and illustrate its use and meaning.
This lesson is part of:
Fluid Dynamics and Applications