15 7 determne an expression for 1 for a rankine

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Unformatted text preview: gure 3.17b), assuming Problem 3.5 16 Describe the path lines and the streamlines. Note that path lines are found by following the motion of each paricle, that is, by solving the differential equations Problem 3.7 dx/dt = u(x, t) and dy/dt = vex, t), subject to x = Xo at t = O. 7. Determne an expression for 1/ for a Rankine vortex (Figure 3.17b), assuming that ue = U at r = R. 8. Take a plane polar element of fluid of dimensions dr and r de. Evaluate the right-hand side of Stokes' theorem f w. dA = f u. ds, and thereby show that the expression for vorticity in polar coordinates is i ( a aUr J Wz=- -(rue)-- . r ar ae Also, find the expressions for Wr and we in polar coordinates in a similar manner. 9. The velocity field of a certain flow is given by u = 2xy2 + 2xi, v = x2y, w = x2z. Consider the fluid region inside a spherical volume x2 + y2 + z2 = a2. Verify the validity of Gauss' theorem 17 Problem 3.7 18 dx/dt = u(x, t) and dy/dt = vex, t), subject to x = Xo at t = O. Problem 3.8 7. Determne an expression for 1/ for a Rankine vortex (Figure 3.17b), assuming that ue = U at r = R. 8. Take a plane polar element of fluid of dimensions dr and r de. Evaluate the right-hand side of Stokes' theorem f w. dA = f u. ds, and thereby show that the expression for vorticity in polar coordinates is i ( a aUr J Wz=- -(rue)-- . r ar ae Also, find the expressions for Wr and we in polar coordinates in a similar manner. 9. The velocity field of a certain flow is given by u = 2xy2 + 2xi, v = x2y, w = x2z. Consider the fluid region inside a spherical volume x2 + y2 + z2 = a2. Verify the validity of Gauss' theorem 19 Problem 3.8 20 and thereby show that the expression for vorticity in polar coordinates is i ( a aUr J Wz=- -(rue)-- . r ar ae Problem 3.9 Also, find the expressions for Wr and we in polar coordinates in a similar manner. 9. The velocity field of a certain flow is given by u = 2xy2 + 2xi, v = x2y, w = x2z. Consider the fluid region inside a spherical volume x2 + y2 + z2 = a2. Verify the validity of Gauss' the...
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