Inviscid Flows
David Jon Furbish · Oxford University Press eBooks · 1997
This chapter covers an important step toward our development of dynamical equations of fluid motion. Herein we will develop explicit expressions for the forces that produce the fluid accelerations that we described kinematically in Chapter 7. In particular, we will consider the behavior of inviscid fluids. Viscous forces therefore are not involved; accelerations are wholly due to body forces and normal surface forces associated with fluid pressure. The results of our development are Euler’s equations, or the momentum equations for inviscid flow. One consequence of the inviscid assumption is that slip flow may occur at real boundaries, in contrast to the no-slip condition that occurs with real fluids. This is unrealistic for the viscous flows of interest in many geological problems. Nonetheless, situations exist in which viscous fluids can be treated as inviscid. Examples include fluids having small viscosity, and flows far from boundaries. The study of inviscid flow therefore is justified in its own right. A particularly important example involves the consideration of how velocity and pressure vary along a streamline, which leads to Bernoulli’s equation. Consider a rectangular control volume with edges of length dx, dy, and dz embedded within a local Cartesian coordinate system. This local system has an arbitrary orientation with respect to the Earth coordinate system; the x-axis is inclined at an angle α measured from the horizontal. Acceleration due to gravity g acts vertically, and the centroid of the control volume is at height h above a horizontal datum. The height h provides a measure of the position of the fluid within the gravitational field. Consider, now, forces acting on the control volume parallel to the x-axis. The weight W of fluid within the control volume possesses a component Wx parallel to the x-axis: . . . Wx = −ρg sin α dx dy dz . . . . . . (10.2) . . . where ρ is the fluid density, and the negative sign indicates that Wx acts in the direction of negative x.