Parallel-Plate Viscometer: Symbolic Computations of Dynamic Viscosity
​Housam Binous

Here, we determine the dynamic viscosity 𝜇 of a Newtonian and non-Newtonian fluids using a parallel-plate viscometer. ​In a parallel-plate viscometer, the fluid is sheared between two circular plates separated by a gap ℎ, with the top plate rotating at angular velocity ω. The shear rate varies linearly with radius:

γ
(r)= ω r/h, so the torque on the plate is obtained by integrating the local shear stress over the plate surface.​​For a Newtonian fluid: torque
T=
π
4
R
ωμ
2h
, and viscosity is directly
μ=
2hTMeasured
π
4
R
w
.​​​For a non-Newtonian power-law fluid (τ=κ*γ^n): ​1- Measure torque 𝑇 at known ω, 𝑅, ℎ. 2- Use torque formula to solve for κ. We assume that the flow index 𝑛 is known or determined independently (by doing multiples measurements of torque 𝑇 at different angular velocities ω).3- Apply Rabinowitsch correction to get true wall shear rate (the apparent wall shear rate at the plate edge is corrected using the Rabinowitsch formula as follows :
[

γ
]
wall
=
(3+n)
4
[

γ
]
apparent
).4- Compute true viscosity at wall:
μ=
n-1
Rw(n+3)
4h
κ
.​Newtonian fluids: no correction needed (set
n=1
). Conversely, for Non-Newtonian fluids: Rabinowitsch correction ensures accurate viscosity measurement.​A more general Rabinowitsch approach (valid for any non-Newtonian constitutive equation) given in Ref. 1 is the following:​​
μ=
3TMeasuredh
2π
4
R
w
3
4
+
1
4
(ln(T))
(ln(w))

​​Note that a good temperature control is essential for the exact experimental determination of the dynamic viscosity.​​Nomenclature:​μ (dynamic viscosity in Pa·s)ω (angular velocity of top plate in rad/s)h (gap between plates in m)R (radius of plates in m)r (radial coordinate in m)κ (consistency index)n (Power-law index)​​Reference:​J. O. Wilkes, Fluid Mechanics for Chemical Engineers, 2nd edition, NJ: Prentice Hall, 2006.
Newtonian fluid case
In[]:=
(*shearrateasafunctionofradius*)​​γdot=wr/h
Out[]=
rw
h
In[]:=
(*shearstress*)​​τ=μγdot
Out[]=
rwμ
h
In[]:=
(*differentialtorqueforanannularring*)​​dT=2τPir^2dr
Out[]=
2drπ
3
r
wμ
h
In[]:=
(*Totaltorque*)​​Ttotal=Integrate[dT/dr,{r,0,R}]
Out[]=
π
4
R
wμ
2h
In[]:=
(*viscosityμ*)​​μExpr=μ/.Solve[Ttotal==TMeasured,μ][[1]]
Out[]=
2hTMeasured
π
4
R
w
non-Newtonian fluid case
In[]:=
(*Non-NewtonianPower-LawFluid*)​​(*Shearstress:τ=κ*γ^n*)​​τPL=κ(γdot)^n
Out[]=
n
rw
h
κ
In[]:=
(*Differentialtorque*)​​dTPL=2PiτPLr^2dr
Out[]=
2drπ
2
r
n
rw
h
κ
In[]:=
(*Totaltorque*)​​TtotalPowerLaw=Refine[Integrate[dTPL/dr,{r,0,R}],n>0]
Out[]=
2π
3
R
n

Rw
h

κ
3+n
In[]:=
(*Apparentwallshearrate*)​​γdotApp=γdot/.r->R
Out[]=
Rw
h
In[]:=
(*Rabinowitschcorrection*)​​γdotTrue=((3+n)/4)γdotApp
Out[]=
(3+n)Rw
4h
In[]:=
(*Trueviscosityatwall*)​​ηWall=κ(γdotTrue/.r->R)^(n-1)
Out[]=
1-n
4
-1+n
(3+n)Rw
h
κ
(*anotherderivationofTrueviscosityexpression*)
In[]:=
T[w_]:=Refine[Integrate[2Piκ(γdotTrue/.r->R)^nr^2,{r,0,R}],n>0]
In[]:=
T[w]//Simplify
Out[]=
1
3
1-2n
2
n
(3+n)
π
3
R
n
Rw
h
κ
In[]:=
Simplify[D[T[w],w]]w/T[w]//Simplify
Out[]=
n
In[]:=
3T[w]h
2π
4
R
w(3/4+1/4n)
//FullSimplify
Out[]=
1-n
4
-1+n
(3+n)
-1+n
Rw
h
κ