Bernoulli's Equation & Fluid Dynamics Cheat Sheet

A printable reference covering Bernoulli's equation, continuity, pressure, density, flow rate, and viscosity for grades 11-12.

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Bernoulli's equation and fluid dynamics describe how liquids and gases move and how pressure changes within flowing fluids. This cheat sheet helps students connect pressure, speed, height, density, and flow rate in common physics problems. It is especially useful for solving pipe flow, tank drainage, lift, and pressure difference questions. The main goal is to recognize when ideal-fluid models apply and how to use them correctly. The most important ideas are conservation of mass and conservation of energy in a moving fluid. The continuity equation $A_1v_1 = A_2v_2$ shows that fluid speeds up when it moves through a narrower region. Bernoulli's equation $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ shows how pressure, kinetic energy density, and gravitational potential energy density trade off along a streamline. Real fluids also involve viscosity, turbulence, and energy loss, so ideal equations must be used with care.

Key Facts

Density is mass per unit volume, given by $\rho = \frac{m}{V}$ .
Pressure is force per unit area, given by $P = \frac{F}{A}$ .
Volume flow rate is given by $Q = \frac{\Delta V}{\Delta t} = Av$ for steady flow through cross-sectional area $A$ .
For an incompressible fluid in steady flow, the continuity equation is $A_1v_1 = A_2v_2$ .
Bernoulli's equation for ideal steady flow along a streamline is $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ .
In a horizontal pipe, Bernoulli's equation becomes $P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$ .
Torricelli's law for fluid leaving a hole below the surface is $v = \sqrt{2gh}$ when the tank surface speed is negligible.
The Reynolds number $Re = \frac{\rho vL}{\eta}$ helps predict whether flow is laminar or turbulent.

Vocabulary

Fluid: A fluid is a substance, such as a liquid or gas, that can flow and change shape to fit its container.
Pressure: Pressure is the normal force exerted per unit area, calculated with $P = \frac{F}{A}$ .
Flow rate: Flow rate is the volume of fluid passing a point each second, calculated with $Q = Av$ .
Streamline: A streamline is a path that shows the direction a small fluid element follows in steady flow.
Viscosity: Viscosity is a measure of a fluid's internal resistance to flow.
Turbulence: Turbulence is irregular, swirling fluid motion that often occurs at high speeds or around obstacles.

Common Mistakes to Avoid

Using Bernoulli's equation across different streamlines, which is wrong because $P + \frac{1}{2}\rho v^2 + \rho gh$ is constant only along the same streamline for ideal flow.
Forgetting height changes, which is wrong because the term $\rho gh$ can be significant when the fluid moves vertically.
Assuming higher speed means higher pressure, which is wrong in ideal horizontal flow because increasing $v$ usually lowers static pressure $P$ .
Mixing gauge pressure and absolute pressure, which is wrong because all pressure terms in one Bernoulli equation must use the same pressure reference.
Applying ideal-fluid equations to strongly viscous or turbulent flow without checking conditions, which is wrong because friction and eddies can cause energy loss not included in Bernoulli's equation.

Practice Questions

1 Water flows through a pipe with area $A_1 = 0.040\ \text{m}^2$ at speed $v_1 = 3.0\ \text{m/s}$ . If the pipe narrows to $A_2 = 0.010\ \text{m}^2$ , what is $v_2$ ?
2 In a horizontal pipe, water speeds up from $v_1 = 2.0\ \text{m/s}$ to $v_2 = 6.0\ \text{m/s}$ . Using $\rho = 1000\ \text{kg/m}^3$ , find $P_1 - P_2$ .
3 A tank has a small hole $1.25\ \text{m}$ below the water surface. Using $g = 9.8\ \text{m/s}^2$ , estimate the exit speed with $v = \sqrt{2gh}$ .
4 Explain why Bernoulli's equation predicts lower pressure where an ideal fluid moves faster, and describe one real-world situation where viscosity or turbulence limits this prediction.

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