CE 6303- MECHANICS OF FLUIDS UNIT II 1. Define circulation and Write its expression. [N/D-14] If V is the fluid velocity on a small element of a defined curve, and dl is a vector representing the differential length of that small element, the contribution of that differential length to circulation is dΓ: where θ is the angle between the vectors V and dl. 2. Write Euler’s equation. [N/D-14] Energy is the basis of Euler’s equation. Energy can neither be created nor destroyed. Euler’s equation is derived by the use of Newton’s second law of motion. According to Newton’s second law of motion, the time rate of change of momentum of a fluid mass in any direction is equal to the sum of all the external forces in that direction, i.e., F = d(M V)/dt where F is the force, M is the mass and V is the velocity. 3. Write the continuity equation in three dimensional differential form for compressible fluids [A/M-15] Conservation of mass is the basis for continuity equation. Mass can neither be created nor destroyed. In a given control volume, the net mass rate of inflow into the control volume is equal to the rate of change of mass. 4. State the impulse momentum principle [A/M-15] The impulse-momentum theorem states that the change in momentum of an object equals the impulse applied to it. The impulse-momentum theorem is logically equivalent to Newton's second law of motion (the force law). 5. Define flow net [M/J-16] A flow net is a graphical representation of two-dimensional steady-state groundwater flow through aquifers. Construction of a flow net is often used for solving groundwater flow problems where the geometry makes analytical solutions impractical. 6. Distinguish between streamline and streak line. [M/J-16] Streamlines and streak lines are field lines in a fluid flow. They differ only when the flow changes with time, that is, when the flow is not steady. Considering a velocity vector field in three-dimensional space in the framework of continuum mechanics, we have that: Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction in which a massless fluid element will travel at any point in time. Streak lines are the loci of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streak line. 7. What is mean by Absolute pressure and Gauge pressure? [M/J-11] Absolute Pressure: It is defined as the pressure which is measured with the reference to absolute vacuum pressure. Gauge Pressure: It is defined as the pressure which is measured with the help of a pressure measuring instrument, in which the atmospheric pressure is taken as datum. The atmospheric pressure on the scale is marked as zero. 8. Define Manometers. [A/M-10] Manometers are defined as the devices used for measuring the pressure at a point in a fluid by balancing measuring the column of fluid by the same or another column of fluid. 1. Simple M 2. Differential M 9. Write down the types of fluid flow. [A/M-12] The fluid flow is classified as : 1. Steady and Unsteady flows. 2. Uniform and Non – uniform flows. 3. Laminar and turbulent flows. 4. Compressible and incompressible flows. 5. Rotational and irrotational flows 6. One, two and three dimensional flows. 10. Differentiate rotational flow from irrotational flow. [M/J-09] If the fluid elements undergo rotations during the flow, thenthe flow is rotational. The orientations of the fluid elements are changed. If there is no rotation or no change in the orientation, then the flow is irrotational. 11. What is the basis of continuity equation? [N/D-14] Conservation of mass is the basis for continuity equation. Mass can neither be created nor destroyed. In a given control volume, the net mass rate of inflow into the control volume is equal to the rate of change of mass. 12. Define – Stream Function [M/J-15] A stream function, ψ, is defined as a function of x and y (and t if the flow is unsteady) such that when differentiated with respect to y, it yields the velocity component in the x-direction, u. When differentiated with respect to x, it yields the -ve velocity component in the y-direction, v. Hence by definition: u = /y and v = -/x 13. What is a current meter? [N/D-12] Current meter is an instrument, which is used to measure the velocity of flow at a point in a cross section of open channels. This instrument works on the basis of conversion of kinetic energy into mechanical energy. Every current meter has to go with a chart known as ‘Calibration Chart’. 1. In a two – two dimensional incompressible flow, the fluid velocity components are given by u = x – 4y and v= - y – 4x. Show that velocity potential exists and determine its form. Find also the stream function. [A/M-12] [N/D-14] 2. A ventrimeter of inlet diameter 300 mm and throat diameter 150 mm is inserted in vertical pipe carrying water flowing in the upward direction. A differential mercury manometer connected to the inlet and throat gives a reading of 200 mm. Find the discharge if the co-efficient of discharge of meter is 0.98. [M/J-15] [N/D-14] 3. Find the density of a metallic body which floats at the interface of mercury of sp. Gr 13.6 and water such that 40% of its volume is sub-merged in mercury and 60% in water. [M/J-13][A/M-15] 4. An oil of specific gravity 0.8 is flowing through a horizontal venturimeter having a inlet diameter 200 mm and throat diameter 100 mm. The oil – mercury differential manometer shows a reading of 250 mm, calculate the discharge of oil through the venturimeter. Take Cd = 0.98. [N/D-11] [A/M-15] 5. (a) Water flows through a pipe AB 1.2m diameter at 3m/s and then passes through a pipe BC 1.5m diameter. At C, the pipe branches. Branch CD is 0.8m in diameter and carries one- third of the flow in AB. The flow velocity in branch CE is 2.5m/s. Find the volume rate of flow in AB, the velocity in BC, the velocity in CD and diameter of CE. [N/D-11] [A/M-12] [M/J-16] 6. (b)State Bernoulli’s theorem for steady flow of an incompressible fluid. Derive an expression for Bernoulli’s equation from first principle and state the assumptions made for such a derivation. [N/D-11] [A/M-12] [M/J-16]