5. Simple Cases of Bernoulli's equation

So, we start with the Bernoulli equation along a stream line. The equation relates pressure, velocity, and height along a streamline in fluid flow.

Bernoulli's equation states that in a fluid flowing in a streamline, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant. This means that if the velocity of the fluid increases, the pressure must decrease, and vice versa. The equation is generally expressed as: $P + \frac{1}{2} \rho V^2 + \rho g z = C$, where $P$ is the fluid pressure, $\rho$ is the density, $V$ is the fluid velocity, and $g z$ represents the potential energy per unit volume due to height.

Think of a mountain stream. As water flows down the mountain, it speeds up as it goes from a high elevation to a lower one. According to Bernoulli's principle, as its speed increases, the pressure of the water decreases. This is why you may feel a strong pull of water at a faster section of a stream.

Consider a scenario where the flow velocity V = 0. In this case, we can simplify Bernoulli's equation to analyze the pressure differences due to height.

In this scenario, when the velocity is zero (such as at the surface of a reservoir), the kinetic energy term $\frac{1}{2} \rho V^2$ drops out. Therefore, Bernoulli's equation becomes $P_1 + \rho g z_1 = P_2 + \rho g z_2$. This signifies that the pressure difference between two points in the fluid can be attributed solely to the difference in height, making it useful for calculating pressure at different elevations in static fluids.

Imagine two connected water tanks at different heights. Water at a higher elevation exerts more pressure downwards than at a lower elevation, allowing us to predict the pressure exerted on the walls of each tank according to their height difference.

Another scenario is when the pressure is constant, such as in a free jet.

In the case of a free jet, where pressure remains nearly constant (equal to atmospheric pressure), we can write the Bernoulli equation as: $z_1 + \frac{V_1^2}{2g} = z_2 + \frac{V_2^2}{2g}$. This equation allows for the calculation of the velocity of the fluid as it exits the jet by considering the height difference between two points.

Consider a fountain. Water is pushed up from a base and falls back down as a fountain jet. The pressure of the water is the same at the point of origin (the surface of the water in the fountain) and where it exits into the air, demonstrating how Bernoulli's principle operates between static and moving states.

To calculate the flow rate Q from a jet, we apply Bernoulli’s equation between two points to derive the velocity and then use the diameter of the jet.

By applying Bernoulli’s equation, we can determine the velocity of the fluid exiting the jet and subsequently calculate the flow rate. The flow rate can be expressed as $Q = A imes V$, where $A = \frac{\pi d^2}{4}$ is the cross-sectional area of the jet. Knowing the velocity allows us to find out how much fluid is flowing per second.

Think about a garden hose. When you put your thumb over the end of the hose, the water flows faster as it squeezes through a smaller opening. You can use the diameter of the hose and the faster flow rate to determine how much water comes out over time, just like calculating Q from the jet.

Bernoulli's equation has various applications, including in devices like pitot tubes and stagnation tubes for measuring fluid speeds.

Both the pitot tube and stagnation tube operate on the principle of measuring pressure differences to infer fluid velocity. The pitot tube, for example, measures static pressure and total pressure (which includes dynamic pressure) to calculate the speed of air or fluid flowing around it.

Consider how airplanes use pitot tubes to measure airspeed. Just like pressure differences in your vacuum cleaner allow it to lift dirt, airplanes measure air pressure to ensure they are flying safely and efficiently at the right speeds.

--