Fundamentals of Haemodynamics - Computational Hemodynamics: Theory, Modelling and Applications - page 73

Pressure is defined as a force per unit area, and if we consider the pressure force
in the x-direction then

FF

x x

P == which gives F = ⋅ P ( dy dz )

x x x

A dy dz ·

x

The change from P to P +∆ is based on the rate of change of P in the x -direction

multiplied by the distance of which the rate of change takes place. This is given as

x xx

dP dP

P =+ P dx and since F = ⋅ P ( dy dz ) F =+ ⋅ P dx ( dy dz )

xx +∆ x x x xx +∆ x

dx dx

The net pressure forcein the x -direction is then

dP

−=+ ⋅ − ⋅ F F P dx ( dy dz ) P ( dy dz )

(4.5)

xx +∆ x x x

dx

Dividing both sides by the volume dxdydz gives

F − == F

dP

xx +∆ x

f

(4.6)

x

dx dy dz ⋅ ⋅ dx

This means that the net pressure force per unit volume is equal to the pressure
gradient which suggests that the pressure gradient creates a net force to produce
fluid flow; and fluid flows from a higher pressure value to a lower pressure value.
In Fig. 4.5 the fluid element is moving from left to right where the pressure acting

Fig. 4.5 Net pressure force in the x-direction which drives the flow from left to right . In this case
the pressure is decreasing from left to right , and is a case of a negative pressure gradient