LECTURE 7
ECTURE
L 7
STRESS IN FLUIDS. CONSTITUTIVE RELATION AND
TRESS IN FLUIDS ONSTITUTIVE RELATION AND
S . C
NEWTONIAN FLUID.
EWTONIAN FLUID
N .
MATHEMATICAL MODEL FOR INTERNAL FORCES IN FLUIDS. STRESS TENSOR.
MATHEMATICAL MODEL FOR INTERNAL FORCES IN FLUIDS. STRESS TENSOR.
According to Cauchy hypothesis, the surface (or interface) reaction force acting between
two adjacent portions of a fluid can be characterized by its surface vector density called the
stress.
x3
Thus, for an infinitesimal piece dA of the interface
dF = �dA
ś�W�1 �� ś�W�2, we have (see figure)
W�2�
n
dF =� � dA and FW� ��W�1 =� �dA
��
2
W�1�
ś�W�1��ś�W�2
0
The stress vector � is not a vector field: it depends not
only on the point x but also on the orientation of the
surface element dA or  equivalently  on the vector n
normal (perpendicular) to dA at the point x.
x2
x1
From the 3rd principle of Newton s dynamics (action-reaction principle) we have
�(x,n) =� -��(x,-�n)
d
A
We will show that the value of stress vector � can be expressed by means of a tensor field. To
this aim, consider a portion of fluid in the form of small tetrahedron as depicted in the figure
below.
The front face D�ABC belongs to the plane which
x3
is describes by the following formula
(n,x) �� njx =� h , h  small number.
j
C
The areas of the faces of the tetrahedron are S, S1,
n=[n1,n2,n3]
S2 and S3 for D�ABC, D�OBC, D�AOC and
D�ABO, respectively. Obviously, S :� O(h2).
-e2
-e1
0
Moreover, the following relations hold for
j = 1,2,3:
a� B
Sj =� Scos[S�(n,ej)] =� S��(n,ej) =� Snj
A
-e3 D
x2
x1
The volume of the tetrahedron is VW� :� O(h3).
x3
The momentum principle for the fluid contained inside
the tetrahedron volume reads
C
d
r�vdx =� F +� Fsurf
n=[n1,n2,n3]
vol
��
{� {�
dt
W�
total volume total surface
1�4�2�4�3�
-e2
-e1
force
force
time derivative
of the momentum
0
We need to calculate the total surface force Fsurf .
a� B
We have:
A
-e3 D
x2
on D�ABC: �(x,n) =� �(0,n) +� O(h)
x1
D�ABC
Fsurf =� S�(0,n) +� O(h3)
on D�OBC: �(x,-�e1) =� -��(x,e1) =� -��(0,e1) +� O(h)
D�OBC
Fsurf =� -�S1 �(0,e1) +� O(h3) =� -�Sn1�(0,e1) +� O(h3)
on D�AOC: �(x,-�e2) =� -��(x,e2) =� -��(0,e2) +� O(h)
D�AOC
Fsurf =� -�S2 �(0,e2) +� O(h3) =� -�Sn2 �(0,e2) +� O(h3)
on D�AOB: �(x,-�e3) =� -��(x,e3) =� -��(0,e3) +� O(h)
D�AOB
Fsurf =� -�S3 �(0,e3) +� O(h3) =� -�Sn3 �(0,e3) +� O(h3)
When the above formulas are inserted to the equation of motion we get
d
r�vdx =� Fvol +� S[�(0,n) -� nj �(0,ej)] +� O(h3)
��
{�
1�4�4�4�2�4�4�4�4�
4� 3�
dt
x3
W�
O(h3)
1�4�2�4�
3�
O(h2 )
O(h3)
C
When h �� 0 the above equation reduces to
n=[n1,n2,n3]
-e2
-e1
�(0,n) -� nj �(0,ej) =� 0
0
In general case, the vertex O is not the origin of the coordinate
system and the field of stress is time dependent. Hence, we
a� B
A
can write
-e3 D
x2
x1
�(t,x,n) =� nj �(t,x,ej)
In the planes oriented perpendicularly to the vectors e1, e2 or e3, the stress vector can be
written as
�(t,x,ej) =� X�ij(t,x)ei
Thus, the general formula for the stress vector takes the form
�(t,x,n) =� nj�(t,x,ej) =� X�ij(t,x)njei �� ś(t,x)n
We have introduced the matrix ś which represents the stress tensor. The stress tensor
depends on time and space coordinates, i.e., we actually have the tensor field.
Note that the stress tensor X� can be viewed as the linear mapping (parameterized by t and x)
between vectors in 3-dimensional Euclidean space
X� : E3 '� w =� wjej a� X�ijwjei ��E3
In particular
X� (n) �� śn =� X�ijnjei =� �
i.e., the action of X� on the normal vector n at some point of the fluid surface yields the
stress vector � at this point.
It is often necessary to calculate the normal and tangent stress components at the point of
some surface.
Normal component is equal �n =� (n��Łn)n �� (n,Łn) n
1� 3�
4�2�4�
inner (scalar)
product
Tangent component can be expressed as
�t� =� � -�s�nn =�s�ijnjei -�(s�kmnknm)niei =� [s�ijnj -�(s�kmnknm)ni]ei
1�4�4�4�4�2�4�4�4�4�3�
(�� )�i
t�
or, equivalently (verify!) as �t� =� n��(���n)
CONSTITUTIVE RELATION
CONSTITUTIVE RELATION
The constitutive relation for the (simple) fluids is the relation between stress tensor ś and
the deformation rate tensor D. This relation should be postulated in a form which is frame-
invariant and such that the stress tensor is symmetric.
Let s remind two facts:
�� The velocity gradient �v can be decomposed into two parts: the symmetric part D
called the deformation rate tensor and the skew-symmetric part R called the (rigid)
rotation tensor.
�v =� D +� R
�� Tensor D can be expressed as the sum of the spherical part DSPH and the deviatoric part
DDEV
D =� DSPH +� DDEV
1 1(���v)I
where DSPH =� trD��I =�
3 3
ć�
ś�vi +� ś�v j �� ś�vk d�ij
1divv��I �� (DDEV)ij =� 1 1
and DDEV =� D -� -�
�� ��
�� ��
3 2 ś�x ś�xi ł� 3ś�xk
j
Ł�
The general constitutive relation for a (simple) fluid can be written in the form of the matrix
 polynomial
ś =�P (D) =� Ł0 +� c0I +� c1D +� c2D2 +� c3D3 +�...
where the coefficients are the function of 3 invariants of the tensor D, i.e.
ck =� ck[I1(D),I2(D),I3(D)].
Consider the characteristic polynomial of the tensor D
pD(l�) =� det[D -� l�I] =� -�l�3 +� I1l�2 -� I2l� +� I3.
The Cayley-Hamilton Theorem states that the matrix (or tensor) satisfies its own
characteristic polynomial meaning that
pD(D) =� -�D3 +� I1D2 -� I2D +� I3 =� 0
Thus, the 3rd power of D (and automatically all higher powers) can be expressed as a linear
combinations of I, D and D2.
Hence, the most general polynomial constitutive relation is given by the 2nd order formula
Ł =�P (D) =� Ł0 +� c0I +� c1D +� c2D2
NEWTONIAN FLUIDS
NEWTONIAN FLUIDS
The behavior of many fluids (water, air, others) can be described quite accurately by the
linear constitutive relation. Such fluids are called Newtonian fluids.
For Newtonian fluids we assume that:
�� c0 is a linear function of the invariant I1,
�� c1 is a constant,
�� c2 =� 0.
If there is no motion we have the Pascal Law: pressure in any direction is the same. It means
that the matrix ś should correspond to a spherical tensor and
0
ś0n =� -�p �� ś0 =� -�pI
The constitutive relation for the Newtonian fluids can be written as follows
2
ś =� -�pI +�z� (Ń���v)I +� 2m� DDEV =� -�pI +� (z� -� m�)(Ń���v) I +� 2m� D
3
{�
{� {� {�
1�4�4� 3�
4�2�4�4�4�
c1
I1(D)
ś0 ś0
c0
where
ó� ź - (shear) viscosity (the physical unit in SI is kg/m"s)
ó� ś - bulk viscosity (the same unit as ź) ; usually z� <�<� m� and can be assumed zero.
The constitutive relation can be written in the index notation
��
��
ś�vk ł� ś�vi ś�v j ł�
2
ę� ś�
s�ij =� -�p +�(z� -� m�) d�ij +� m� +�
ę�
3
ś�xk ś� ś�x ś�xi ś�
ę�
ę� ś�
j
�� ��
�� ��
ś�v
j
For an incompressible fluid we have ���v �� div v �� =� 0 and the constitutive relation
ś�x
j
reduces to the simpler form
ś =� -�pI +� 2m�D
or, in the index notation
�� ł�
ś� ś�
ę� ś�
s�ij =� -�pd�ij +� m� vi +� v
ś�xj ś�xi jś�
ę�
�� ��
Example: Calculate the tangent stress in the wall shear layer.
The velocity field is defined as follows:
v1(x1, x2 ) =� Uwall x2 / H , v2(x1, x2 ) �� 0
and the pressure is constant. At the bottom wall, the
normal vector which points outwards is n =� [0,-�1].
Then
ś�v1 ś�v1 ś�v2
1
�� (ś�x +� )ł� 0
��0ł� �� ł�
ś�x1 2 ś�x1
2
� =� śn =� -�p n +� 2m�Dn =� +� 2m� =�
{� ę�1 (ś�x +� ) ś�
ś�v1 ś�v2 ś�v2
ę�pś� ę�
=�[0,-�1]
�� �� ��-�1ś�
��
ś�x1 ś�x2
��2 ��
2
ś�v1
ś�v1
1
�� 0 ł�
0
��-�m� ł�
��0ł� �� ł� ��-�m�Uw / Hł�
2 ś�x2
ś�x2
=� +� 2m� =� =�
ę�1 ś�
ś�v1
ę� ś�
ę�pś� ę� ę� ś�
0 p
p
�� �� ��-�1ś� �� ��
��
�� ��
��2 ś�x2 ��
According to the action-reaction principle, the tangent stress at the bottom wall is
m�Uw
ś�
t�wall =� m� v1 =�
ś�x2 wall
H