# Fluid Mechanics-Bernoulli and Energy Equations

• Joseph95
In summary, the conversation discusses determining the force needed to maintain a constant speed of 4 mm/s on a piston with 4 small holes in a sealed water-filled cylinder. Using the Bernoulli equation, the pressure in the right compartment is found to be 56624.8 Pa, and the force required is calculated to be 40 N. The symbols and equations used are the Bernoulli equation, P1/ρ + V1^2/2 + gz1 = P2/ρ + V2^2/2 + gz2, and the flow rate equation, Q1 = Q2 V1.A1=V2.A2.
Joseph95

## Homework Statement

A well-fitting piston with 4 small holes in a sealed water-filled cylinder,shown in Fig is pushed to the right at a constant speed of 4 mm /s while the pressure in the right compartment remains constant at 50 kPa gage. Disregarding the frictional effects,determine the force F that needs to be applied to the piston to maintain this motion.

## Homework Equations

Bernoulli Equation: P1/ρ + V1^2/2 + gz1 = P2/ρ + V2^2/2 + gz2

Flow Rate : Q1 = Q2 V1.A1=V2.A2

## The Attempt at a Solution

I set two points.The first point (1) is back of the piston and the other point (2) front of the piston.So applied Bernoulli Equation:

P1/ρ + V1^2/2 +gz1 = P2/ρ + V2^2 / 2 + gz2 No elevation difference z1=z2

P1/ρ + V1^2/2 = P2/ρ + V2^2/2 V1=4 mm/s =0.004 m/s
V2= V1.A1/A2 = 0.004 x 0.0113 / 0.0000031 m/s
V2=3.64 m/s but there are 4 small holes so V2/4= 14.58/ 4 m/s=3.64 m/s

P1= P2 + ρV2^2/2 (V1 can be neglected)

P1= 50 kPa + 1000 (3.64)^2 /2 Pa = 56624.8 Pa

F/πDr^2/4=56624.8 Pa

F=40 N

I'm not following this. (1) is back of piston and (2) is front of piston. So why is V1 so small and v2 so big ?
Joseph95 said:
F/πDr^2/4=56624.8 Pa

F=40 N
What does it say here ? Could you explain the symbols and place appropriate brackets ?

## 1. What is the Bernoulli equation in fluid mechanics?

The Bernoulli equation is a fundamental equation in fluid mechanics that describes the relationship between the pressure, velocity, and elevation of a fluid in a steady flow. It states that the sum of the pressure, kinetic energy, and potential energy per unit volume of a fluid remains constant along a streamline.

## 2. How is the Bernoulli equation derived?

The Bernoulli equation is derived from the conservation of energy principle, which states that energy cannot be created or destroyed, only converted from one form to another. In fluid mechanics, this means that the total energy per unit volume of a fluid remains constant in a steady flow.

## 3. What is the role of the energy equation in fluid mechanics?

The energy equation is another important equation in fluid mechanics that is derived from the first law of thermodynamics. It takes into account the energy changes due to work, heat transfer, and changes in internal energy of a fluid. This equation is often used in conjunction with the Bernoulli equation to analyze fluid flow problems.

## 4. What are the assumptions made in the Bernoulli equation?

The Bernoulli equation makes several assumptions about the fluid flow, including: the flow is steady, inviscid (no viscosity), incompressible, and the fluid is flowing along a streamline. These assumptions may not hold true in all fluid flow situations, but the Bernoulli equation is still a useful approximation in many cases.

## 5. How is the Bernoulli equation applied in real-world situations?

The Bernoulli equation is commonly used in the analysis of fluid flow in pipes, nozzles, and other engineering applications. It can be used to determine the pressure drop or flow rate through a pipe, or the velocity and pressure at a specific point in a fluid flow. The equation is also used in the design of aircraft wings and other aerodynamic systems.

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