and that the streamline, other than , that passes through this stagnation point satisfies
The flow pattern (7.161) can be reinterpreted as that which results when a blunt obstacle lying to the right of the previously specified streamline is placed in a uniform stream of velocity . Show that the obstacle in question has the asymptotic (i.e., as ) thickness . Demonstrate that the pressure distribution over the surface of the obstacle is
where is the fluid mass density, and the pressure at infinity. Show that the maximum pressure, , on the surface occurs at , and that the minimum pressure, , occurs at . Finally, demonstrate that these are, respectively, the maximum and minimum pressures attained in the whole flow pattern.
where , are the angles that , make with . being a general point. In addition, show that the cone defined by the equation
divides the streamlines issuing from into two sets, one extending to infinity, and the other terminating at . (Milne-Thompson 2011.)
and . Show that the streamline (other that ) that passes through these points satisfies
and also passes through the points and , where
The flow pattern (7.162) can be reinterpreted as that which results when an axisymmetric solid body of oval cross-section , lying inside the previously specified streamline, is placed in a uniform stream of velocity . Such an obstacle is known as a Rankine solid.
is a possible stream function for an axisymmetric, incompressible, irrotational flow pattern, and find the corresponding velocity potential. (Milne-Thompson 2011.)
where , are spherical coordinates whose origin coincides with the center of the sphere. If the sphere is divided into two halves by a diametral plane lying in the - plane, show that the resultant force between the two parts is less than it would have been if the fluid were at rest, the pressure at infinity remaining the same, by an amount , where is the fluid density. (Milne-Thompson 2011.)
the center of the sphere being instantaneously at the origin. (Milne-Thompson 2011.)
where is real and positive. Show that the stream function
can be interpreted as that of incompressible irrotational flow, with mean speed , through a circular hole of radius in an infinite plane wall, corresponding to .