1. NNPC FSTP ENGINEERS Physics Course Code: Lesson 7

2. Fluid Mechanics How Motion Takes Place in a Fluid Contents

3. Topics General Characteristics of Fluid Motion Bernoulli’s Equation

4. Fluid flow can be steady or nonsteady. When the fluid velocity v at any given point is constant in time, the fluid motion is said to be steady. That is, at any given point in a steady flow the velocity of each passing fluid particle is always the same. These conditions can be achieved at low flow speeds; a gently flowing stream is an example. Fluid flow can be rotational or irrotational. If the element of fluid at each point has no net angular velocity about that that point, the fluid is irrotational. Fluid flow can be compressible or incompressible General Characteristics of Fluid Flow

5. Fluid flow can be viscous or nonviscous. Viscosity in fluid motion is the analogue of friction in the motion of solids We shall confine our discussion of fluid dynamics to steady, irrotational, incompressible, nonviscous flow. In steady flow, the velocity v at a given point is constant in time. If we consider the point p within the fluid, since v at p does not change in time, every particle arriving at p will pass on with the same speed in the same direction. The same is true about the point Q and R. Hence if we trace out the path of the particle, that curve will be the path of every particle arriving at p. This curve is called a streamline. General Characteristics of Fluid Flow P Q R

6. If we assume a steady flow and select a finite number of streamlines to form a bundle, like the streamline pattern shown below, the tubular region is called a tube of flow. General Characteristics of Fluid Flow A 2,V 2 A 1,V 1 P Q

7. Let the speed be v 1 for fluid particle at p and v 2 for fluid particles at Q. Let A 1 and A 2 be the cross- sectional areas of the tubes perpendicular to the streamlines at the points p and Q respectively. At the time interval t a fluid element travels approximately the distance v t. Then the mass of fluid m 1 crossing A 1 in the time interval t is approximately, Or the mass flux is approximately General Characteristics of Fluid Flow

8. The mass flux at p is The mass flux at Q is Where  1 and  2 are the fluid densities at p and Q respectively. Because no fluid can leave through the walls of the tube and there are no sources or sinks where fluids can be created or destroyed in the tube. The mass crossing each section of the tube per unit time must be the same. In particular the mass flux at p must equal that at Q.  vA = constant General Characteristics of Fluid Flow

9. If the fluid is incompressible, as we shall henceforth assume, then or Av = constant. The product Av gives the volume flux or flow rate in General Characteristics of Fluid Flow

10. According to the continuity equation, the speed of fluid flow can vary along the paths of the fluid. The pressure can also vary; it depends on height as in the static situation and it also depends on the speed of flow. We shall derive an important relationship called Bernoulli’s equation that relates the pressure, flow speed, and height Bernoulli’s Equation

11. For flow of an ideal incompressible fluid. The equation is an ideal tool for analysing plumbing systems, hydroelectric generating stations and the flight of aeroplanes. The dependence of pressure on speed follows from the continuity equation. When an incompressible fluid flows along a flow tube, with varying cross section, its speed must change, and so an element of fluid must have an acceleration. If the tube is horizontal, the force that causes this acceleration has to be applied by the surrounding fluid. This means that the pressure must be different in regions of different cross section; if it were the same every where, the net force on every fluid element must be zero. When a horizontal flow tube narrows, and a fluid element speeds up, it must be moving towards a region of lower pressure in order to have a net forward force to accelerate it. Bernoulli’s Equation

12. Let us compute the work done by a fluid element during a time dt. The pressure at the two ends are p 1 and p 2 ; the force on the cross section at a is p 1 A 1 and the force at c is p 2 A 2. The net work dW done on the element by the surrounding fluid during this displacement is therefore The second term has a negative sign because the force at c opposes the displacement of the fluid. The work done would be equal to the change in the total mechanical energy (kinetic + gravitational potential energy) associated with the fluid element. Bernoulli’s Equation

13. The mechanical energy between b and c does not change. At the beginning of dt, the fluid between a and b has volume A 1 ds 1, mass  A 1 ds 1, and kinetic energy, At the end of dt, the fluid between c and d has kinetic energy The net change in kinetic energy dK during time dt is Bernoulli’s Equation

14. For the change in gravitational potential energy, at the beginning of dt, the potential energy for the mass between a and b is For the mass between c and d, The net change in potential energy dU during dt is Using the energy equation dW = dK + dU we obtain Bernoulli’s Equation

15. This is Bernoulli's equation. It states that the work done on a unit volume of fluid by the surrounding fluid is equal to sum of the changes in kinetic and potential energies per unit volume that occur during the flow. The first term on the right is the pressure difference associated with the change of speed of the fluid. The second term on the right is the additional pressure difference caused by the weight of the fluid and the difference in the elevation of the two ends. In a more convenient form,we can state the equation as Bernoulli’s Equation

16. The subscripts 1 and 2 refer to any two points along the flow tube so we can also write; Note that when the fluid is not moving, v 1 = v 2 = 0 and the equation reduces to the pressure relation we met before. Bernoulli’s Equation

17. APPLICATIONS OF BERNOULLIS EQUATIONS Bernoulli's equation is useful for describing a variety of phenomena. First we observe that if a fluid is at rest so that v = 0, P +  gh = constant. This is just the hydrostatic equation between pressure and height. For flow in a horizontal, constant height tube,

18. P 1,v 1, P 2, v 2 A1A1 A2A2 Thus pressure must be lower in a region in which a fluid is moving faster. Consider the flowmeter below:- A device designed to measure the speed of fluids in a pipe

19. Applying Bernoulli's equation, Since The pressure difference =  gh, so we can solve for v 1

20. EXAMPLE 2 Water flowing from a tank A1A1 P 1 =P A h A2A2 Consider points just outside and inside the hole and apply Bernoulli's equation

21. hence

22. If A 1  A 2  v U 2 term could be ignored

23. ASSIGNMENT GRP 4: WE CAN HARNESS ENERGY OF THE EARTH – GEOTHERMAL ALTERNATIVE! (15 mins)