8.4.2 Viscosity as a Function of Temperature

If you are modeling a problem that involves heat transfer, you can define the viscosity as a function of temperature . Five types of functions are available:

• piecewise-linear:

  
  

  
  

• piecewise-polynomial:

  
  

  $$\mbox{for} \; T_{{\rm min},1} \leq T < T_{{\rm max},1}: \; \mu (T) = A_1 + A_2 T + A_3 T^2 + ...$$ (8.4-2)

  $$\mbox{for} \; T_{{\rm min},2} \leq T < T_{{\rm max},2}: \; \mu (T) = B_1 + B_2 T + B_3 T^2 + ...$$ (8.4-3)

  
  

• polynomial:

  
  

  
  

• Sutherland's law

• power law

The power law described here is different from the non-Newtonian power law described in Section  8.4.5.

For one of the first three, select piecewise-linear, piecewise-polynomial, polynomial in the Viscosity drop-down list and then enter the data pairs ( $T_n , \mu_n$), ranges and coefficients, or coefficients that describe these functions Section  8.2. For Sutherland's law or the power law, choose sutherland or power-law respectively in the drop-down list and enter the parameters.

Sutherland Viscosity Law

Sutherland's viscosity law resulted from a kinetic theory by Sutherland (1893) using an idealized intermolecular-force potential. The formula is specified using two or three coefficients.

Sutherland's law with two coefficients has the form

where,

$\mu$	=	the viscosity in kg/m-s
$T$	=	the static temperature in K
$C_1$ and $C_2$	=	the coefficients
$Y_i$	=	the mass fraction of species $i$
$M_{w,i}$	=	the molecular weight of species $i$
$p_{\rm op}$	=	the Operating Pressure

For air at moderate temperatures and pressures, $C_1 = 1.458 \times 10^{-6}$ kg/m-s-K $^{1/2}$, and $C_2 = 110.4$ K.

Sutherland's law with three coefficients has the form

where,

$\mu$	=	the viscosity in kg/m-s
$T$	=	the static temperature in K
$\mu_0$	=	reference value in kg/m-s
$T_0$	=	reference temperature in K
$S$	=	an effective temperature in K (Sutherland constant)

For air at moderate temperatures and pressures, $\mu_0 = 1.716\times 10^{-5}$ kg/m-s, $T_0$ = 273.11 K, and $S$ = 110.56 K.

Inputs for Sutherland's Law

To use Sutherland's law, choose sutherland in the drop-down list to the right of Viscosity. The Sutherland Law dialog box will open, and you can enter the coefficients as follows:

1.   Select the Two Coefficient Method or the Three Coefficient Method.

Use SI units if you choose the two-coefficient method.

2.   For the Two Coefficient Method, set C1 and C2. For the Three Coefficient Method, set the Reference Viscosity $\mu_0$, the Reference Temperature $T_0$, and the Effective Temperature $S$.

Power-Law Viscosity Law

Another common approximation for the viscosity of dilute gases is the power-law form. For dilute gases at moderate temperatures, this form is considered to be slightly less accurate than Sutherland's law.

A power-law viscosity law with two coefficients has the form

where $\mu$ is the viscosity in kg/m-s, $T$ is the static temperature in K, and $B$ is a dimensional coefficient. For air at moderate temperatures and pressures, $B = 4.093\times 10^{-7}$, and $n = 2/3$.

A power-law viscosity law with three coefficients has the form

where $\mu$ is the viscosity in kg/m-s, $T$ is the static temperature in K, $T_0$ is a reference value in K, $\mu_0$ is a reference value in kg/m-s. For air at moderate temperatures and pressures, $\mu_0 = 1.716\times 10^{-5}$ kg/m-s, $T_0 = 273$ K, and $n = 2/3$.

The non-Newtonian power law for viscosity is described in Section  8.4.5.

Inputs for the Power Law

To use the power law, choose power-law in the drop-down list to the right of Viscosity. The Power Law dialog box will open, and you can enter the coefficients as follows:

1.   Select the Two Coefficient Method or the Three Coefficient Method.

Note that you must use SI units if you choose the two-coefficient method.

2.   For the Two Coefficient Method, set B and the Temperature Exponent $n$. For the Three Coefficient Method, set the Reference Viscosity $\mu_0$, the Reference Temperature $T_0$, and the Temperature Exponent $n$.