Mathematical Models for FLUID MECHANICS

1. Mathematical Models for FLUID MECHANICS P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi Convert Ideas into A Precise Blue Print before feeling the same....

4. A path line is the trace of the path followed by a selected fluid particle

5. Few things to know about streamlines • At all points the direction of the streamline is the direction of the fluid velocity: this is how they are defined. • Close to the wall the velocity is parallel to the wall so the streamline is also parallel to the wall. • It is also important to recognize that the position of streamlines can change with time - this is the case in unsteady flow. • In steady flow, the position of streamlines does not change • Because the fluid is moving in the same direction as the streamlines, fluid can not cross a streamline. • Streamlines can not cross each other. • If they were to cross this would indicate two different velocities at the same point. • This is not physically possible. • The above point implies that any particles of fluid starting on one streamline will stay on that same streamline throughout the fluid.

6. A useful technique in fluid flow analysis is to consider only a part of the total fluid in isolation from the rest. This can be done by imagining a tubular surface formed by streamlines along which the fluid flows. This tubular surface is known as a streamtube. A Streamtube A two dimensional version of the streamtube The "walls" of a streamtube are made of streamlines. As we have seen above, fluid cannot flow across a streamline, so fluid cannot cross a streamtube wall. The streamtube can often be viewed as a solid walled pipe. A streamtube is not a pipe - it differs in unsteady flow as the walls will move with time. And it differs because the "wall" is moving with the fluid

7. Fluid Kinematics • The acceleration of a fluid particle is the rate of change of its velocity. • In the Lagrangian approach the velocity of a fluid particle is a function of time only since we have described its motion in terms of its position vector.

8. In the Eulerian approach the velocity is a function of both space and time; consequently, x,y,z are f(t) since we must follow the total derivative approach in evaluating du/dt.

10. Similarly for ay & az, In vector notation this can be written concisely

13. x Conservation laws can be applied to an infinitesimal element or cube, or may be integrated over a large control volume.

14. Basic Control-Volume Approach

15. Control Volume • In fluid mechanics we are usually interested in a region of space, i.e, control volume and not particular systems. • Therefore, we need to transform GDE’s from a system to a control volume. • This is accomplished through the use of Reynolds Transport Theorem. • Actually derived in thermodynamics for CV forms of continuity and 1st and 2nd laws.

16. Flowing Fluid Through A CV • A typical control volume for flow in an funnel-shaped pipe is bounded by the pipe wall and the broken lines. • At time t0, all the fluid (control mass) is inside the control volume.

17. The fluid that was in the control volume at time t0 will be seen at time t0 +dt as:          .

18. The control volume at time t0 +dt      . The control mass at time t0 +dt      . The differences between the fluid (control mass) and the control volume at time t0 +dt      .

19. II I • Consider a system and a control volume (C.V.) as follows: • the system occupies region I and C.V. (region II) at time t0. • Fluid particles of region – I are trying to enter C.V. (II) at time t0. III II • the same system occupies regions (II+III) at t0 + dt • Fluid particles of I will enter CV-II in a time dt. • Few more fluid particles which belong to CV – II at t0 will occupy III at time t0 + dt.

20. III II At time t0+dt II I At time t0 The control volume may move as time passes. III has left CV at time t0+dt I is trying to enter CV at time t0

21. Reynolds' Transport Theorem • Consider a fluid scalar property b which is the amount of this property per unit mass of fluid. • For example, b might be a thermodynamic property, such as the internal energy unit mass, or the electric charge per unit mass of fluid. • The laws of physics are expressed as applying to a fixed mass of material. • But most of the real devices are control volumes. • The total amount of the property b inside the material volume M , designated by B, may be found by integrating the property per unit volume, M ,over the material volume :

22. Conservation of B • total rate of change of any extensive property B of a system(C.M.) occupying a control volume C.V. at time t is equal to the sum of • a) the temporal rate of change of B within the C.V. • b) the net flux of B through the control surface C.S. that surrounds the C.V. • The change of property B of system (C.M.) during Dt is add and subtract

23. The above mentioned change has occurred over a time dt, therefore Time averaged change in BCMis

24. For and infinitesimal time duration • The rate of change of property B of the system.

25. Conservation of Mass • Let b=1, the B = mass of the system, m. The rate of change of mass in a control mass should be zero.

26. Conservation of Momentum • Let b=V, the B = momentum of the system, mV. The rate of change of momentum for a control mass should be equal to resultant external force.

27. Conservation of Energy • Let b=e, the B = Energy of the system, mV. The rate of change of energy of a control mass should be equal to difference of work and heat transfers.

35. Applications of Momentum Analysis This is a vector equation and will have three components in x, y and z Directions. X – component of momentum equation:

36. X – component of momentum equation: Y – component of momentum equation: Z – component of momentum equation: For a fluid, which is static or moving with uniform velocity, the Resultant forces in all directions should be individually equal to zero.

37. X – component of momentum equation: Y – component of momentum equation: Z – component of momentum equation: For a fluid, which is static or moving with uniform velocity, the Resultant forces in all directions should be individually equal to zero.

38. X – component of momentum equation: Y – component of momentum equation: Z – component of momentum equation: For a fluid, which is static or moving with uniform velocity, the Resultant forces in all directions should be individually equal to zero.

39. Vector equation for momentum: Vector momentum equation per unit volume: Body force per unit volume: Gravitational force:

40. Electrostatic Precipitators Electric body force: Lorentz force density The total electrical force acting on a group of free charges (charged ash particles) . Supporting an applied volumetric charge density. Where = Volumetric charge density = Local electric field = Local Magnetic flux density field = Current density

41. Electric Body Force • This is also called electrical force density. • This represents the body force density on a ponderable medium. • The Coulomb force on the ions becomes an electrical body force on gaseous medium. • This ion-drag effect on the fluid is called as electrohydrodynamic body force.

42. 0 Ideal Fluids….

43. Pressure Variation in Flowing Fluids • For fluids in motion, the pressure variation is no longer hydrostatic and is determined from and is determined from application of Newton’s 2nd Law to a fluid element.

44. Various Forces in A Flow field • For fluids in motion, various forces are important: • Inertia Force per unit volume : • Body Force: • Hydrostatic Surface Force: • Viscous Surface Force: • Relative magnitudes of Inertial Forces and Viscous Surface Force are very important in design of basic fluid devices.

45. Comparison of Magnitudes of Inertia Force and Viscous Force • Internal vs. External Flows • Internal flows = completely wall bounded; • Both viscous and Inertial Forces are important. • External flows = unbounded; i.e., at some distance from body or wall flow is uniform. • External Flow exhibits flow-field regions such that both inviscid and viscous analysis can be used depending on the body shape.

46. Ideal or Inviscid Flows Euler’s Momentum Equation X – Momentum Equation:

48. Euler’s Equation for One Dimensional Flow Define an exclusive direction along the axis of the pipe and corresponding unit direction vector Along a path of zero acceleration the pressure variation is hydrostatic

49. Pressure Variation Due to Acceleration For steady flow along l – direction (stream line) Integration of above equation yields

50. Momentum Transfer in A Pump • Shaft power Disc Power Fluid Power. • Flow Machines & Non Flow Machines. • Compressible fluids & Incompressible Fluids. • Rotary Machines & Reciprocating Machines.