Volumetric Flow Rate for Pipe Flow of non-Newtonian Fluids using Symbolic Calculations​
​Authors: Housam Binous and Ahmed Bellagi
Reference:
​
H . Binous, Introducing non-Newtonian fluid mechanics computations with Mathematica® in the undergraduate curriculum, Chemical Engineering Education, 41 (1), pp . 59 - 64, 2007.
​
Volumetric Flow Rate for Pipe Flow of a Bingham Fluid​​
Q=
ΔP
4
a
π
8
μ
0
L
1-
4
τ
0
3
τ
R
+
1
3
4
τ
0
τ
R
with
τ
R
=
ΔPR
2L
​​
Calculation of shear stress using boundary condition:
τ
rz
(0)=0
In[]:=
sol1=DSolve[D[r
τ
rz
[r],{r,1}]PGr,
τ
rz
[r],r]
Out[]=

τ
rz
[r]
PGr
2
+

1
r

In[]:=
τ
rz
[r]=sol1[[1,1,2]]/.C[1]0
Out[]=
PGr
2
Calculation of velocity
v
z
(r)
between
r=
-2
τ
0
PG
and
r=a
using boundary condition:
v
z
(a)=0
In[]:=
sol2=DSolve[D[
v
z
[r],{r,1}](
τ
rz
[r]+
τ
0
)/μ,
v
z
[r],r]
Out[]=

v
z
[r]

1
+
PG
2
r
2
+2r
τ
0
2μ

In[]:=
v
z
[r]=sol2[[1,1,2]]/.C[1]-
PG
2
a
4μ
-
a
τ
0
μ
Out[]=
-
2
a
PG
4μ
-
a
τ
0
μ
+
PG
2
r
2
+2r
τ
0
2μ
Calculation of Q1 flow rate between r=
-2
τ
0
PG
and r=a where velocity is given by
v
z
(r)
In[]:=
Q1=Integrate[2Pir
v
z
[r],{r,-2
τ
0
/PG,a}]
Out[]=
-
4
a
PGπ
8μ
-
3
a
π
τ
0
3μ
+
2
a
π
2
τ
0
PGμ
+
4aπ
3
τ
0
2
PG
μ
+
10π
4
τ
0
3
3
PG
μ
Calculation of constant velocity
v
0
between r=0 and r=
-2
τ
0
PG
using boundary condition:
v
0
=
v
z
(a)
In[]:=
v
0
=1/μPG(r^2-a^2)/4+
τ
0
/μ(r-a)/.r-2
τ
0
/PG
Out[]=
τ
0
-a-
2
τ
0
PG
μ
+
PG-
2
a
+
4
2
τ
0
2
PG
4μ
Calculation of Q2 flow rate between r=0 and r=
-2
τ
0
PG
where velocity is constant equal to
v
0
In[]:=
Q2=Integrate[2Pir
v
0
,{r,0,-2
τ
0
/PG}]
Out[]=
4π
2
τ
0
τ
0
-a-
2
τ
0
PG
μ
+
PG-
2
a
+
4
2
τ
0
2
PG
4μ
2
PG
Total Flow Rate is equal to Q1+Q2
In[]:=
Q=Q1+Q2//Expand
Out[]=
-
4
a
PGπ
8μ
-
3
a
π
τ
0
3μ
-
2π
4
τ
0
3
3
PG
μ
Volumetric Flow Rate for Pipe Flow of a Power Law Fluid​​
Q=
3+
1
n
a
nπ
1
n

ΔP
2kL

1+3n
​​
Clearing previous shear stress and velocity expressions
In[]:=
τ
rz
[r]=.
In[]:=
v
z
[r]=.
Calculation of shear stress using boundary condition:
τ
rz
(0)=0
In[]:=
sol3=DSolve[D[r
τ
rz
[r],{r,1}]PGr,
τ
rz
[r],r]
Out[]=

τ
rz
[r]
PGr
2
+

1
r

In[]:=
τ
rz
[r]=sol1[[1,1,2]]/.C[1]0
Out[]=
PGr
2
Calculation of velocity
v
z
(r)
between
r=0
and
r=a
using boundary condition:
v
z
(a)=0
In[]:=
sol4=DSolve[D[
v
z
[r],{r,1}]-(-
τ
rz
[r]/k)^(1/n),
v
z
[r],r]
Out[]=

v
z
[r]-
-1/n
2
r
1
n
-
PGr
k
1+
1
n
+

1

In[]:=
v
z
[r]=sol4[[1,1,2]]/.C[1]
-1/n
2
na
1
n
-
PGa
k

1+n
Out[]=
-1/n
2
an
1
n
-
aPG
k
1+n
-
-1/n
2
r
1
n
-
PGr
k
1+
1
n
Calculation of Q flow rate between
r=0
and
r=a
where velocity is given by
v
z
(r)
In[]:=
Q=Integrate[2Pir
v
z
[r],{r,0,a}]//Simplify