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Fluid Flow - Equation of continuity and Bernoulli's Principle

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Fluid Flow: Introduction

- Fluid in physics refers to liquids and gases. To study fluid flow we have to make the following assumptions:
  • We consider fluids to be incompressible
  • We assume that they have little or no internal friction or viscosity

Streamline and Turbulent Flow

- The path followed by a small element of a moving fluid is called a line of flow.
- A streamline is a curve whose tangent at any point is in the direction of the fluid velocity at that point.
A streamline flow occurs when all elements of a fluid passing a particular point follow the same path or line of flow as the elements that passed through that point previously.
- A streamline flow is achieved only when the speed is low.
- Turbulent flow generally occurs when the speed is high and where there are sharp bends along the path of the fluid.

Fluid Flow - Form 2 Physics

Equation of continuity

- Consider a fluid flowing (streamline flow) through a horizontal pipe with different cross-sectional areas as shown.

Fluid Flow - Form 2 Physics

- Let the cross-sectional area in both sections be A1 and A2 and the corresponding speeds of the fluid be V1and V2 respectively.
- The volume of fluid flowing per second in each section is given by;
V = A L = A v t = A v. Where L, v t and v is the distance moved in one second.
- Since the volumes in each section is the same, then,
A1 V1 = A2 V2, hence A v = constant.
- The above equation is known as the equation of continuity.
Since A1 > A2, then V2 > V1. i.e. the speed increases when a tube narrows.
- The quantity (A v) is called volume efflux i.e. volume flowing per second.

Practice Example 1

A horizontal pipe of cross-sectional area 50 cm2 carries water at the rate of 0.20 litres per second. Determine the speed:
a) Of the speed of water in the pipe.
b) When the tube narrows to 20 cm2 at another point.
a) Volume efflux = 0.20 l per second = A v
From V (volume) = A v, then v = V / A = 0.20 × 10-3 / 50 x 10-4 = 0.04 m/s.
b) Since A1/v1 = A2/v2 then v2 = (0.05 x 0.04) / 0.02 = 0.1 m/s.

Bernoulli's Principle

- Daniel Bernoulli (1700 – 1782) explained the variation of pressure exerted by a moving fluid when its speed is changed. The pressure is lower where the speed is higher.

Fluid Flow - Form 2 Physics

- Bernoulli's principle states that For a fluid flowing through a tube, the sum of the pressure, the kinetic energy per unit volume and the potential energy per unit volume of the fluid is a constant.
- Mathematically expressed as;

Fluid Flow - Form 2 Physics

Bernoulli's effect

- When air is blown through the tunnel formed, the area marked ‘T’ collapses inwards showing that pressure outside is more than the one inside the tunnel.
- The pressure inside the tunnel decreases as the air through it increases in speed.

Fluid Flow - Form 2 Physics

Applications of Bernoulli's principle

1. Car carburetor

– Inside the carburetor the air passage is partially constricted at the point where petrol mixes with air hence air intake increases the speed of air while decreasing the pressure inside for petrol to vaporize quickly before it gets to the cylinder where combustion occurs.

Fluid Flow - Form 2 Physics

2. Horizontal pipe.

– For a streamline flow through a pipe the term ρ g h is eliminated from the Bernoulli's equation leaving,
P + ½ρ v2 = constant, indicating that pressure in liquid is greatest when speed is least.
- When this is combined with the equation of continuity, the pressure is then greatest when the pipe is widest hence the following observation.

Fluid Flow - Form 2 Physics

3. Dynamic lift.

- when air is blown at the top a flat sheet of paper the ends of the paper moves upward and this because the speed of air on top of the paper is greater than below and according to Bernoulli’s principle the pressure on top lowers and the pressure below becomes sufficient enough to produce a force which moves the paper upwards.
- This is what is referred to as the dynamic lift since it is caused by motion.
- The upward force is equal to the product of the pressure difference and the area of the surface lifted.
- It is applied in the taking off of airplanes, the trajectory of a spinning ball, paint sprayer and Bunsen burner among others.

Fluid Flow - Form 2 Physics

Fluid Flow - Form 2 Physics


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