I think that the use of non-dimensional variables is usefull for modelling processes. But when material properties are not konstant, i.e. they change with the temperature, things become complicated.
Is there a theoretical based procedure to find the non-dimensional variables concerning with the material properties?
Hello Arturo,
You are very correct, scaling with a non-constant variable causes many problems. The primary concern is the viscosity (and all non-dim variables containing viscosity) as with non-newtonian flow strain rate hardening/softening takes place while in Newtonian fluids the viscous forces vary linearly with with strain rate.
To non-dimensionalize these variables it may be important to know what you intend on modeling.
v = velocity
L = length
rho_0 = density
eta_c = viscosity
g = gravity
tau = viscous stress
z= vertical height
e_dotij = tensor shear strain rate
Equation of Motion:
rho_0*(∂v/∂t + (v grad)V) = -grad P + ∂tau_ij/∂x_j + rho*g*z
*Assuming the flows stress and strain rate tensor can be described by the tensor shear strain rate and the viscous stress. An example of such a relation is:
e_dotij = C_1*tau^n1 + C_2*tau^n2 +...C_k*tau^nk
each term in this equation would then represent a flow mechanism, this is needed for describing non-newtonian flows that have different dominating effects at different stress ranges.
With newtonian non-dimensionalization we reach (only for the variables pertaining to viscosity):
t=t'*eta_0/(rho_0*g*L)
v = v'*rho_0*g*L^2/eta_0
e_dotij = e_dotij'*rho_0*g*L/eta_0
∂/∂t = (∂/∂t')*(rho_0*g*L/eta_0)
Characteristic viscosity (eta_c) is defined as:
tau_c/(2*e_dotij)
taking the dominating portion of e_dotij one can define eta_c as:
eta_c = (1/(2*C_c^(1/n_c))*(v/L)^(1/n_c)-1
substituting the the scaling of velocity into this equation yields:
eta_0 = 1/(2^n_0*C_0) * (L*g*rho_0)^(1-n_0)
With this new formulation, the required non-dim variables can then be substituted with eta_0:
t = t'/(2^(n0)*C_0*(L*g*rho_0)^(n_0)
v = v'*2^(n_0)*C_0*(L*g*rho_0)^(n_0)
e_dotij= e_dotij'*2^(n_0)*C_0*(L*g*rho_0)^(n_0)
∂/∂t = (∂/∂t')*2^(n_0)*C_0*(L*g*rho_0)^(n_0)
I know these equations are a bit difficult to follow when written as they are here, but if you have questions let me know I can perhaps spend some time and make them into a pdf where it is a bit easier to follow. I hope this helps!
David
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