Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 102

gradient is negative, i.e. a deceleration. We describe the fluid motion sweeping past
points in space by the advection term in the momentum equations. If the geometry
between points A, B, and C is fixed and does not vary, and in the absence of any
other forces (e.g., gravity, thermal etc.) then the advection becomes zero.

5.2.2.2 Interpreting the Pressure Term

During fluid flow, the pressure term is a relative measure of the local intensity of
material in moving. It is a stress force that acts normal to the control surface of the
fluid element. When these forces are summed, the net pressure in each coordinate
determines the fluid motion. Thus it is not the pressure itself which causes a net
pressure but rather the pressure gradient, (e.g., −∂ P / ) across the fluid element.
A positive pressure gradient, i.e. increasing pressure, slows the fluid down, while a
negative pressure gradient i.e. decreasing pressure, accelerates it.
To illustrate this, let us consider a steady flow through a bifurcation where we
assume that at the bifurcation, the geometry has a circular shape shown in Fig. 5.8 .
From fluid dynamics we can use the known flow variation along the approaching
stagnation streamline (A-B) which is

2

R
x

ux U ()= 1 − (5.15)

∞

2

If we neglect the viscosity term in the momentum equation so that we can focus on
the pressure term, then we have an inviscid flow. The diffusion terms are therefore
zero. Along the stagnation streamline, the vertical velocity component v is zero. The
x- momentum equation becomes

∂ u ∂ u ∂ P

ρ + u =−

(5.16)

∂ t ∂ x ∂ x

local convection pressure

Fig. 5.8 Schematic of
a steady flow through a
bifurcation