FLUID KINEMATICS 4 5 VORTICITY AND ROTATIONALITY Another

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FLUID KINEMATICS

FLUID KINEMATICS

4– 5 ■ VORTICITY AND ROTATIONALITY Another kinematic property of great importance to the

4– 5 ■ VORTICITY AND ROTATIONALITY Another kinematic property of great importance to the analysis of fluid flows is the vorticity vector, defined mathematically as the curl of the velocity vector Vorticity is equal to twice the angular velocity of a fluid particle The direction of a vector cross product is determined by the righthand rule. The vorticity vector is equal to twice the angular velocity vector of a rotating fluid particle. 2

CONTINUITY EQUATION The continuity equation is based on the principle of conservation of mass.

CONTINUITY EQUATION The continuity equation is based on the principle of conservation of mass. Consider two cross-sections of a pipe as shown in Fig Let, A 1 , V 1 and ρ1 = Area of the pipe, Velocity of the fluid and Density of the fluid at section 1 -1 and A 2, V 2, ρ2 are corresponding values at sections 2 -2. The total quantity of fluid passing through section 1 -1= ρ1 A 1 V 1 and, the total quantity of fluid passing through section 2 -2 = ρ2 A 2 V 2

From the law of conservation of mass (theorem of continuity), we have: ρ1 A

From the law of conservation of mass (theorem of continuity), we have: ρ1 A 1 V 1 = ρ2 A 2 V 2. . (3. 22) In case of incompressible fluids, ρ1 = ρ2 and the continuity eqn. reduces to: A 1 V 1 = A 2 V 2 Rate of flow (or discharge) is defined as the quantity of a liquid flowing per second through a section of pipe or a channel. It is generally denoted by Q. Let us consider a liquid flowing through a pipe. Discharge, Q = Area x average velocity i. e. , Q=A. V If area is in m 2 and velocity is in m/s, then the discharge, 4 Q = m 2 x m/s = m 3/s.

Example A pipe (1) 450 mm in diameter branches into two pipes (2 and

Example A pipe (1) 450 mm in diameter branches into two pipes (2 and 3) of diameters 300 mm and 200 mm respectively as shown in Fig. 5. 15. If the average velocity in 450 mm diameter pipe is 3 m/s find. (i) Discharge through 450 mm diameter pipe, (i) Velocity in 200 mm diameter pipe if the average velocity in 300 mm pipe is 2. 5 m/s. Solution. Diameter, D 1 = 450 mm = 0. 45 m i. e. 5

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CIRCULATION AND VORTICITY

CIRCULATION AND VORTICITY

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The vorticity

The vorticity

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Example 5. 28. The velocity components for a fluid flow are: u = a

Example 5. 28. The velocity components for a fluid flow are: u = a + by – cz, v = d – bx – ez, w = f + cx – ey where a, b, c, d, e and f are arbitrary constants. (i) Show that it is a possible case of fluid flow. (ii) Is the fluid flow irrotational? If not, determine the vorticity and rotation. Solution. Given: u = a + by – cz, v = d – bx – ez, w = f + cx – ey. . . Velocity components. (i) Possible case of fluid flow? Continuity equation is given as: 12

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