Page 57 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

42 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Concerning the equation of flow, if from the first form of div[a] we

take out the second derivative terms (the “dominant” terms), they could
be grouped into

pVv?v + (A +p) grad (div v).

According to the classification of the second order partial differential
equations, these equations are elliptic if the eigenvalues yz and A + 2p of
the associated quadratic form are positive. Consequently, in the steady
case, if p > 0 and A+ 2p > 0 the flow equations are of elliptic type. The
same property belongs to the energy equation if, by accepting for the
conduction heat the Fourier law, the thermal conduction coefficient is

positive. In the unsteady case the previous equations become parabolic.
Globally speaking, the whole system of equations would be elliptic-
hyperbolic in the steady case and parabolic-hyperbolic in the unsteady
case. If 4 = 0, then the elliptic and parabolic properties will be lost.
Concerning the initial and boundary conditions, the first ones specify
the flow parameters at t = to, being thus compulsory in the unsteady

case. As regards the boundary conditions, they imply some information
about the flow parameters on the boundary of the fluid domain and
they are always compulsory for determining the solution of the involved
partial differential equation in both steady and unsteady cases.
For a viscous fluid which “passes” along the surface of a rigid body, the
fluid particles “wet” the body surface, 1.e., they adhere. This molecular

phenomenon has been proved for all the continuous flows as long as the
Knudsen number (Ky) < 0,01.”
Due to this adherence the relative velocity between the fluid and the
surface of the body is zero or, in other terms, if Wg is the absolute
velocity of the body surface and v the absolute velocity of the fluid, we

should have Vsur face = Vs. If Vs = 0, that means the body surface is
at rest, then 1% = 0 and also v, = 0, t being a unit tangent vector on
the surface and n is the unit normal vector drawn to the surface.
These conditions are called the adherence or non-slip conditions, in
opposition with the s/ip conditions vy, = 0 and vz 4 0 whichcharacterize
the inviscid (ideal) fluid.
Obviously the presence of a supplementary condition (v_ = 0) for the
viscous fluids equations should not surprise because these are partial
differential equations of second order while the ideal fluids equations are

of first order.

l
“This number is an adimensional parameter defined by Ky = lL’ where /is the mean free
path and £ a reference length.