1.What are fluid kinematics? kinematic descriptions of motion describe position, velocity, and accelerations (NOT FORCE) [ physical interpretation: what are we doing today? ] 2.What is Reynolds Transport Theorem (RTT)? RTT is a tool we use in fluid mechanics to relate different representations of fluid mechanics problems 3.Who cares !? if we can understand the premise of RTT, we can easily link fundamental derivations of the continuity and momentum equations (boring! (but necessary)) more excitingly, comprehension and application of the RTT sets up tools we can use to solve problems concerning anything from elevated municipal water reservoirs to fighter jets, satellites, oh, and jet skis (not boring!) FLUID KINEMATICS: THE REYNOLDS TRANSPORT THEOREM Fluid Mechanics FLUID KINEMATICS: THE REYNOLDS TRANSPORT THEOREM 1.The RTT allows us to move between system and control volume concepts in fluid mechanics 2.The Conservation of Mass and the Conservation of Linear Momentum can be derived from the RTT here B represents an extensive property, and b represents an intensive property B is directly proportional to the amount of the mass being considered for example: if we were to let B = mass, then it would follow that b = 1 b is independent of the amount of mass [EQN1] Fluid Mechanics FLUID KINEMATICS: THE REYNOLDS TRANSPORT THEOREM term 1: the time rate of change of an extensive parameter of the system, B, ex: mass, momentum [ physical interpretation ] 123 term 2: the rate of change of B within the CV as fluid flows through it term 3: the net flowrate of B across the entire control surface Fluid Mechanics FLUID KINEMATICS: THE REYNOLDS TRANSPORT THEOREM [ example ] if we assume we have a fixed CV and uniform inlet and outlet conditions, we can express RTT in the following simplified form neighbourhood (control volume) lets apply RTT to a simple example lets develop an expression to describe the time rate of change of the number of cars in a neighbourhood [EQN2] Fluid Mechanics FLUID KINEMATICS: THE REYNOLDS TRANSPORT THEOREM [ example (contd) ] we assume that at t = t o, the system of cars coincides with the neighbourhood such that we can express [EQN2] as neighbourhood (control volume) [EQN2] for our example, let us define the following: N = number of cars Nsys = number of cars in a system of cars Ncv = number of cars in the neighbourhood (CV) Fluid Mechanics FLUID KINEMATICS: THE REYNOLDS TRANSPORT THEOREM [ example (contd) ] now we can write that the rate at which the number of cars in the system changes with time is equal to the rate at which the number of cars in the neighbourhood changes with time, plus the net rate at which cars cross the neighbourhood boundary neighbourhood (control volume) we must sum the number of cars entering and exiting obtain the total number of cars in the neighbourhood Fluid Mechanics FLUID KINEMATICS: THE REYNOLDS TRANSPORT THEOREM [ example (contd) ] if we assume that cars are conserved (neither created nor destroyed), we can say neighbourhood (control volume) therefore, the final expression describing the rate of change of the number of cars in the neighbourhood is: Fluid Mechanics FLUID KINEMATICS: THE CONSERVATION OF MASS 1.We define a system as a collection of unchanging contents therefore the conservation of mass for a system is simply: where [EQN3] [EQN4] so we can say for a system and non-deforming CV that are coincident, the RTT allows us to write [EQN5] Fluid Mechanics FLUID KINEMATICS: THE CONSERVATION OF MASS 2.If the rate of change of the system mass = 0, then we express [EQN5] as: [EQN6] [EQN6] is known as the integral form of the continuity equation 12 term 1: the time rate of change of the mass of the contents of the control volume term 2: the net rate of mass through the control surface these two terms must sum to zero in order to conserve mass [ the continuity equation (integral form) ] Fluid Mechanics FLUID KINEMATICS: THE CONSERVATION OF MASS 3.We can use Gausss Divergence Theorem to write the integral form of continuity in a differential form, i.e., [ the continuity equation (differential form) ] [EQN7] we can also write: (as limits of the space integration are time independent) [EQN8] then: [EQN9] Fluid Mechanics FLUID KINEMATICS: THE CONSERVATION OF MASS [ the continuity equation (differential form) (contd) ] the integrand must be identically zero for the eqn to hold for ALL CVs, therefore we may write [EQN9] [EQN10] or, in Cartesian form (for incompressible flow) [EQN11] Fluid Mechanics FLUID KINEMATICS: THE CONTINUITY EQUATION (integral form) [ example ] GIVEN: REQD: Utilize the integral form of the continuity equation to find the velocity, V 1, of the flow entering the pipe Fluid Mechanics FLUID KINEMATICS: THE CONTINUITY EQUATION (integral form) [ example (contd) ] SOLU: (contd) combining (1) and (2) we solve: and we know A 1 =A 2, so: 3.But! we recall that air behaves as an ideal gas in this situation, and its volume and pressure are related by the ideal gas law: - (1) - (2) - (ans) Fluid Mechanics FLUID KINEMATICS: THE CONTINUITY EQUATION (differential form) [ example ] GIVEN: The following incompressible, steady flow field: 1.Lets write the continuity equation in differential form - (2) REQD: Determine the w velocity component such that the flow satisfies continuity SOLU: i.e.: - (1) 2.Now, sub the velocity component expressions (1) into (2) and solve for w then: - (3) Fluid Mechanics FLUID KINEMATICS: THE CONTINUITY EQUATION (differential form) [ example (contd) ] - (ans) Here we appreciate that the flow field cannot be explicitly defined without more information, for now this expression represents a family of flow fields that will satisfy continuity SOLU: (contd) 3.Now we integrate (3) with respect to z Fluid Mechanics