Page 439 - Introduction to Continuum Mechanics

Problems 423

630. Referring to Problem 6.29, consider a pipe having an elliptic cross section given by
2
2 2
2
y /a + z / b = 1. Assuming that

find A and B.
631. Referring to Problem 6.29, consider an equilateral triangular cross-section defined by
the planes z + &/(2VJ) = 0, z + V3> - fe/VJ = 0, z - VJy - b/VJ - 0. Assuming

find A and B.
"i •!
632. For the steady-state, time dependent parallel flow of water (density p = 10 Kg/ m ,
viscosity/* = 10 Ns/m ) near an oscillating plate, calculate the wave length for a) = 2cps.
633. The space between two concentric spherical shell is filled with an incompressible
Newtonian fluid. The inner shell (radius r,-) is fixed; the outer shell (radius r 0) rotates with an
angular velocity Q about a diameter. Find the velocity distribution. Assume the flow to be
laminar without secondary flow.
634. Consider the following velocity field in cylindrical coordinates:

(a) Show that v(r) = —, where A is a constant so that the equation of conservation of mass is
^
satisfied.

(b) If the rate of mass flow through a circular cylindrical surface of radius r and unit length is
Q m, determine the constant^ in terms QfQ m-
635. Given the following velocity field in cylindrical coordinates

(a) Show from the continuity equation tha

(b) In the absence of body forces, show that

where k and C are constants.