Bernoulli

Cheatsheet Content

### Introduction Bernoulli's principle states that for an inviscid incompressible fluid in steady flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. This cheatsheet outlines its derivation. ### Assumptions The derivation of Bernoulli's equation relies on several key assumptions: - **Inviscid Flow:** No viscosity (frictionless fluid). - **Incompressible Flow:** Density ($\rho$) is constant. - **Steady Flow:** Velocity ($\vec{v}$), pressure ($P$), and density ($\rho$) at any point do not change with time. - **Flow Along a Streamline:** The equation applies along a single streamline. - **No Heat Transfer:** Adiabatic process. - **No Shaft Work:** No pumps or turbines doing work on the fluid. ### Euler's Equation of Motion The derivation starts from Euler's equation of motion for a fluid, which is essentially Newton's second law for a fluid element. For steady, inviscid flow along a streamline, Euler's equation can be written as: $$ \frac{dP}{\rho} + v dv + g dz = 0 $$ Where: - $P$ is the pressure - $\rho$ is the fluid density - $v$ is the fluid velocity - $g$ is the acceleration due to gravity - $z$ is the elevation ### Integration Along a Streamline To obtain Bernoulli's equation, we integrate Euler's equation along a streamline. Given the assumptions (incompressible flow means $\rho$ is constant), we can integrate each term: 1. **Pressure Term:** $$ \int \frac{dP}{\rho} = \frac{P}{\rho} $$ (Since $\rho$ is constant) 2. **Velocity Term:** $$ \int v dv = \frac{v^2}{2} $$ 3. **Gravity Term:** $$ \int g dz = g z $$ (Since $g$ is constant) Summing these integrated terms and setting them equal to a constant (the constant of integration): $$ \frac{P}{\rho} + \frac{v^2}{2} + g z = \text{Constant} $$ ### Bernoulli's Equation This constant holds true for any two points (1 and 2) along the same streamline: $$ \frac{P_1}{\rho} + \frac{v_1^2}{2} + g z_1 = \frac{P_2}{\rho} + \frac{v_2^2}{2} + g z_2 $$ This is the most common form of Bernoulli's equation. Each term represents a form of energy per unit mass: - $P/\rho$: Pressure energy per unit mass - $v^2/2$: Kinetic energy per unit mass - $g z$: Potential energy per unit mass #### Alternative Forms - **In terms of pressure (multiplying by $\rho$):** $$ P + \frac{1}{2}\rho v^2 + \rho g z = \text{Constant} $$ Here, each term represents pressure: - $P$: Static pressure - $\frac{1}{2}\rho v^2$: Dynamic pressure - $\rho g z$: Hydrostatic pressure - **In terms of head (dividing by $g$):** $$ \frac{P}{\rho g} + \frac{v^2}{2g} + z = \text{Constant} $$ Here, each term represents a "head" or height: - $P/\rho g$: Pressure head - $v^2/2g$: Velocity head - $z$: Elevation head ### Applications Bernoulli's equation is fundamental in fluid mechanics and is used in various applications, including: - **Aerodynamics:** Explaining lift on an airplane wing. - **Venturi Meter:** Measuring fluid flow rate. - **Orifice Plate:** Similar to Venturi meter for flow measurement. - **Pipe Flow:** Analyzing pressure and velocity changes in piping systems.