In this chapter, we describe axisymmetric irrotational

Chapter 10, Problem 58P

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Problem 58P

In this chapter, we describe  axisymmetric irrotational flow in terms of cylindrical coordinates r and z and velocity components u r and u z. An alternative description of axisymmetric flow arises if we use spherical polax coordinates and set the x-axis as the axis of symmetry. The two relevant directional components are now r and θ, and their corresponding velocity components are ur and u s. In this coordinate system, radial location r is the distance from the origin, and polar angle 8 is the angle of inclination between the radial vector and the axis of rotational symmetry (the x-axis), as sketched in Fig. P10-63; a slice defining the rθ-plane is shown. This is a type of two-dimensional flow because there are only two independent spatial variables, x and 6. In other words, a solution of the velocity and pressure fields in any rθ-plane is sufficient to characterize the entire region of axisymmetric irrotational flow. Write the Laplace equation for φ in spherical polar coordinates, valid in regions of axisymmetric irrotational flow. (Hint: You may consult a textbook on vector analysis.)

FIGURE P10-63