Introduction to biofluids

1. BME18R311
BIOFLUIDS AND DYNAMICS
GEORGE EBENEZER ARUL SAMUVEL

2. BIO MECHANICS
• Biological fluid mechanics (or biofluid mechanics) is the study of
the motion of biological fluids in any possible context.
1. blood circulation
2. hearth pumping
3. flow in the systemic arteries
4. flow in the pulmonary arteries
5. flow in the microcirculation

3. Use of biological fluid mechanics
• Pure physiology: understanding how animals, and in particular
humans, work.
• Pathophysiology: understanding why they might go wrong. In
other words, understanding
• the origins and development of diseases.
• Diagnosis: recognizing diseases from possibly non-traumatic
measurements.
• Cure: providing support to surgery and to the design of
prosthetic devices.

4. NEWTON LAWS

5. NEWTON LAWS
NEWTON'S FIRST LAW
Newton's First Law states that an object will remain at rest or in uniform motion in a straight line
unless acted upon by an external force. It may be seen as a statement about inertia, that objects will
remain in their state of motion unless a force acts to change the motion.
NEWTON’S SECOND LAW
Newton's Second Law as stated below applies to a wide range of physical phenomena, but it is not a
fundamental principle like the Conservation Laws. It is applicable only if the force is the net external
force.
NEWTON’S THIRD LAW
Newton’s Third Law All forces in the universe occur in equal but oppositely directed pairs. There are no
isolated forces; for every external force that acts on an object there is a force of equal magnitude but

6. Stress
• When solid bodies are deformed, internal forces get distributed
in the material. These are called stresses. Stress has the unit of
force per area.

7. Stress
• Lets assume,
Before the deformation, the surface is characterized by the area
dA and the normal vector N. After the deformation, these become
da and n.

8. Stress
• The equation can written as
An axially loaded bar.
An infinitesimal surface in the original and deformed configurations.
.

9. Stress
where F is the deformation gradient tensor and
.
where

10. Stress
Here
is called the second Piola-Kirchhoff stress tensor. This is a symmetric tensor that is
energy conjugate to the Green-Lagrange strain.
The first and second Piola-Kirchhoff stress tensors are related via:
This formula makes it possible to rewrite the momentum balance equation as:
which together with a constitutive relation of the form
will give a closed system of equations for the displacement vector.
Stress Components on a Rotated Plane
For the axially loaded bar, it is easy to think about the stress as a scalar number
and state that on this bar, only a normal stress exists. The full stress tensor is
where is the angle between the axis of the bar and the normal to the surface.

11. 1D Elasticity (axially loaded bar)
x
y
x=0 x=L
A(x) = cross section at x
b(x) = body force distribution
(force per unit length)
E(x) = Young’s modulus
u(x) = displacement of the bar
at x
x
1. Strong formulation: Equilibrium equation + boundary
conditions
Lxb
dx
d
 0;0

LxatF
dx
du
EA
xatu

 00Boundary conditions
Equilibrium equation
F

12. 3. Stress-strain (constitutive) relation : ε(x)E(x) 
E: Elastic (Young’s) modulus of bar
2. Strain-displacement relationship:
dx
du
ε(x) 

13. Problem definition
3D Elasticity
x
y
z
Surface (S)
Volume (V)
u
v
w
x
V: Volume of body
S: Total surface of the body
The deformation at point
x =[x,y,z]T
is given by the 3
components of its
displacement











w
v
u
u
NOTE: u= u(x,y,z), i.e., each
displacement component is a function
of position

14. 3D Elasticity:
EXTERNAL FORCES ACTING ON THE BODY
Two basic types of external forces act on a body
1. Body force (force per unit volume) e.g., weight, inertia, etc
2. Surface traction (force per unit surface area) e.g., friction

15. BODY FORCE
x
y
z Surface (S)
Volume (V)
u
v
w
x
Xa dV
Xb dV
Xc dV
Volume
element dV Body force: distributed
force per unit volume (e.g.,
weight, inertia, etc)











c
b
a
X
X
X
X
NOTE: If the body is accelerating,
then the inertia force
may be considered as part of X











w






 v
u
u
u
~
 XX

16. x
y
z
ST
Volume (V)
u
v
w
x
Xa dV
Xb dV
Xc dV
Volume
element dV
py
pz
px
Traction: Distributed
force per unit surface
area











z
y
x
p
p
p
TS
SURFACE TRACTION

17. 3D Elasticity:
INTERNAL FORCES
If I take out a chunk of material from the body, I will see that,
due to the external forces applied to it, there are reaction
forces (e.g., due to the loads applied to a truss structure, internal
forces develop in each truss member). For the cube in the figure,
the internal reaction forces per unit area(red arrows) , on each
surface, may be decomposed into three orthogonal components.
x
y
z
Volume (V)
u
v
w
x
Volume
element dV
x
y
z
tyz
tyx
txy
txz
tzy
tzx

18. 3D Elasticity





















zx
yz
xy
z
y
x
t
t
t




x
y
z
x
y
z
tyz
tyx
txy
txz
tzy
tzx
x, y and z are normal stresses.
The rest 6 are the shear stresses
Convention
txy is the stress on the face
perpendicular to the x-axis and points
in the +ve y direction
Total of 9 stress components of which
only 6 are independent since
xzzx
zyyz
yxxy
tt
tt
tt



The stress vector is therefore

19. Strains: 6 independent strain components





















zx
yz
xy
z
y
x







Consider the equilibrium of a differential volume element to
obtain the 3 equilibrium equations of elasticity
0
0
0



























c
zyzxz
b
yzyxy
a
xzxyx
X
zyx
X
zyx
X
zyx
tt
tt
tt

20. Compactly;
0 X
T




















































xz
yz
xy
z
y
x
0
0
0
00
00
00where
EQUILIBRIUM
EQUATIONS
(1)

21. Traction: Distributed
force per unit area











z
y
x
p
p
p
TS
n
nx
ny
nz
ST
TS
py
px
pz
If the unit outward normal to ST :











z
y
x
n
n
n
n
Then
zzyzyxxz
zyzyyxxy
zxzyxyxx
nnn
nnn
nnn
tt
tt
tt



z
y
x
p
p
p

22. nx
ny
ST
In 2D
dy
dx
ds
x
y
n
q q
x
y
n
ds
dy
n
ds
dx


q
q
cos
sin
TS
py
px
q
dy
dx
dsx
y
txy
txy
Consider the equilibrium of the wedge in
x-direction
yxyxxx
xyxx
xyxx
nnp
ds
dx
ds
dy
p
dxdydsp
t
t
t



Similarly
yyxxyy nnp t 

23. 2. Strain-displacement relationships:
x
w
z
u
y
w
z
v
x
v
y
u
z
w
y
v
x
u
zx
yz
xy
z
y
x


































24. Compactly; u



















































xz
yz
xy
z
y
x
0
0
0
00
00
00











w
v
u
u





















zx
yz
xy
z
y
x







(2)

25. x
y
A B
C
A’
B’
C’
v
u
dy
dx
dx
x
v


xd
x
u
u



dy
y
u


dy
y
v
v



x
u
x
v
tanβtanβββ)B'A'(C'angle
2
π
y
v
dy
dyvdy
y
v
vdy
AC
ACC'A'
x
u
dx
dxudx
x
u
udx
AB
ABB'A'
2121



















































xy
y
x



1
2
In 2D

26. 3. Stress-Strain relationship:
Linear elastic material (Hooke’s Law)
 D (3)
Linear elastic isotropic material
































2
21
00000
0
2
21
0000
00
2
21
000
0001
0001
0001
)21)(1(







E
D

27. PLANE STRESS: Only the in-plane stress components are nonzero
x
y
Area
element dA
Nonzero stress components xyyx t ,,
y
x
xyt
xyt
h
D
Assumptions:
1. h<<D
2. Top and bottom surfaces are free from
traction
3. Xc=0 and pz=0

28. PLANE STRESS Examples:
1. Thin plate with a hole
2. Thin cantilever plate
y
x
xyt
xyt

29. Nonzero strains: xyzyx  ,,,
Isotropic linear elastic stress-strain law


































xy
y
x
xy
y
x
E







t


2
1
00
01
01
1 2
Hence, the D matrix for the plane stress case is














2
1
00
01
01
1 2




E
D
 D
 yxz 


 


1
Nonzero stresses: xyyx t ,,
PLANE STRESS

30. PLANE STRAIN: Only the in-plane strain components are nonzero
Area
element dA
Nonzero strain components
y
x
xy
xy
Assumptions:
1. Displacement components u,v functions
of (x,y) only and w=0
2. Top and bottom surfaces are fixed
3. Xc=0
4. px and py do not vary with z
xyyx  ,,
x
y
z

31. PLANE STRAIN Examples:
1. Dam
2. Long cylindrical pressure vessel subjected to internal/external
pressure and constrained at the ends
1
Slice of unit
thickness
x
y
z
y
x
xyt
xyt
z

32. Nonzero stress:
Isotropic linear elastic stress-strain law
   




































xy
y
x
xy
y
x
E







t


2
21
00
01
01
211
Hence, the D matrix for the plane strain case is
  

















2
21
00
01
01
211 



E
D
 D
 z x y    
xyzyx t ,,,
Nonzero strain components: xyyx  ,,
PLANE STRAIN