# Extract body force from non-Newtonian Reynolds number

To simulate turbulent non-Newtonian channel flow I intend to use the power law viscosity model given by

$$\mu = K\gamma^{n-1}$$

with $$K$$ the flow consistency index, n the flow behavioural index and $$\gamma$$ the shear rate. I want to compare the simulation results with a turbulent newtonian channel flow simulation using the Reynolds number as non-dimensional number. Specifically I need to know what to input for $$K$$, $$n$$ and a body force $$g$$ in my simulation such to obtain a $$Re$$ number simmular to the Newtonian turbulent simulation.

As mentioned here defining a single non-Newtonian Reynolds number to non-dimensionalize all the parameters is not possible. There are however a few papers that simulate turbulent power-law fluids in a pipe which use the generalised Reynolds number $$Re_G = \frac{\rho U H}{K^{1/n} \tau_w ^{(n-1)/n}}$$ with $$U$$ the characteristic velocity, $$H$$ the characteristic length scale and $$\tau_w$$ the mean wall shear stress which can be calculated from the pressure gradient via $$\tau_w = H\frac{\partial P}{\partial z}$$ for channel flows.

Given that for a square channel the body force can be calculated from $$g = -\frac{1}{\rho}\frac{\partial P}{\partial z}$$ one would assume that this would be enough information to extract a body force $$g$$ given a $$Re$$, $$H$$, $$n$$ and $$K$$. The values I am however finding make no sense.

Am I missing something here?