Fundamentals of Haemodynamics - Computational Hemodynamics: Theory, Modelling and Applications - page 78

( PA − P A )( + τ A − τ A ) 0 =

xx x xx +∆ rr r rr +∆

(2 P π r dr − P 2 )(2 π τπ τ π r dr + r dx − 2 )0 r dx =

x xx +∆ r rr +∆

Cancelling out 2π and grouping the x -terms and r -terms gives

( PP − )( τ r − τ ) 0 r

x xx +∆ r rr +∆

r + =

dx dr

Taking the limits as dx and dr → 0

dP d () 0 τ r

r + =

dx dr

Substituting Newton's law of viscosity formula (

τ µ =− /) du dy

we get

dP µ d du

= r

dx r dr dr

which is rewritten as

2

dP d u µ du

(4.9)

= µ +

2

dx r dr

dr

This is the general equation for a steady laminar incompressible pipe flow in the
fully developed region. Its general solution is

1 dP

2

u r () = r + Aln r () + B

(4.10)

4 µ dx

To obtain the velocity profile we define the boundary conditions which are:

I. At the centre of the pipe r = 0, u = U max which means A must be equal to zero for
the equation to balance.
II. Combining the above condition with the wall boundary condition of r = R ( R is the
distance from pipe centre to pipe wall), u = 0 means that

2

r

ur () = U 1 −

(4.11)

max

2

R