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 shows that fluid speeds up when it moves through a narrower region. Bernoulli's equation 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 .
- Pressure is force per unit area, given by .
- Volume flow rate is given by for steady flow through cross-sectional area .
- For an incompressible fluid in steady flow, the continuity equation is .
- Bernoulli's equation for ideal steady flow along a streamline is .
- In a horizontal pipe, Bernoulli's equation becomes .
- Torricelli's law for fluid leaving a hole below the surface is when the tank surface speed is negligible.
- The Reynolds number 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 .
- Flow rate
- Flow rate is the volume of fluid passing a point each second, calculated with .
- 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 is constant only along the same streamline for ideal flow.
- Forgetting height changes, which is wrong because the term can be significant when the fluid moves vertically.
- Assuming higher speed means higher pressure, which is wrong in ideal horizontal flow because increasing usually lowers static pressure .
- 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 at speed . If the pipe narrows to , what is ?
- 2 In a horizontal pipe, water speeds up from to . Using , find .
- 3 A tank has a small hole below the water surface. Using , estimate the exit speed with .
- 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.
Understanding Bernoulli's Equation & Fluid Dynamics
Pressure in a fluid comes from countless molecular collisions with a surface. At a point in still water, this pressure acts equally in every direction. It increases with depth because deeper water supports the weight of more water above it.
This is why dams are made thicker near the bottom and why ears feel pressure during a deep swim. In moving water, it helps to separate this ordinary pressure from the motion of the water.
A pressure gauge attached to the wall of a pipe measures the push on that wall. It does not directly measure the speed of the fluid passing by.
A useful way to picture flow is to follow a small packet of fluid along one path. That path is called a streamline. If the pipe changes width, the same amount of liquid must pass each cross section in the same time during steady flow.
The fluid therefore moves faster in a narrow section. This creates the basis of a Venturi meter, which uses pressure readings at wide and narrow sections to find flow speed. Spray bottles and some laboratory gas burners use a related effect.
Fast air can create a lower pressure region that draws liquid upward. These examples work best when the flow is smooth and the geometry guides the fluid in a controlled way.
Height changes deserve careful attention. When water rises through a pipe, it gains gravitational potential energy. Unless an outside pump adds energy, some other part of its energy must decrease.
Its pressure may fall, its speed may fall, or both may happen. For a tank with a small outlet, compare the calm water surface with the hole. Both locations are exposed to air, so their air pressures cancel.
The surface moves slowly because its area is much larger than the hole area. The drop in height then explains the outlet speed. This result predicts the initial speed well, but the water level falls over time, so the speed gradually decreases.
Real fluid problems often differ from the ideal model because fluid layers rub against one another and against pipe walls. This internal resistance is viscosity. Thick oils have greater viscosity than water, so they need more pressure to maintain the same flow through a narrow tube.
Near a wall, a fluid can move very slowly while fluid near the center moves faster. At low speeds, these layers can slide in an orderly pattern called laminar flow. At higher speeds or around rough surfaces, swirling motion can develop.
This is turbulence, and energy is transferred into random motion and heating. When solving a problem, first identify whether the fluid is nearly incompressible, whether the flow is steady, and whether two points lie on the same streamline. Then choose a reference height, label every pressure clearly, and check whether the answer has sensible units and direction.