Analytical solutions for the flow of Carreau and Cross fluids in circular pipes and thin slits

Abstract

In this paper, analytical expressions correlating the volumetric flow rate to the pressure drop are derived for the flow of Carreau and Cross fluids through straight rigid circular uniform pipes and long thin uniform plane slits. The derivation is based on the application of Weissenberg-Rabinowitsch-Mooney-Schofield (WRMS) method to obtain flow solutions for generalized Newtonian fluids through pipes and our adaptation of this method to the flow through slits. The derived expressions are validated by comparing their solutions to the solutions obtained from direct numerical integration. They are also validated by comparison to the solutions obtained from the variational method which we proposed previously. In all the investigated cases, the three methods agree very well. The agreement with the variational method also lends more support to this method and to the variational principle which the method is based upon. We also compared the derived analytical solutions of Carreau and Cross fluids to the analytical solutions of power law fluids with comparable rheology and observed logical trends in the relation between the corresponding flow rates as a function of the applied pressure field.

Nomenclature

γ :

shear rate (s⁻¹)

γ _w :

shear rate at conduit wall (s⁻¹)

δ :

μ _o − μ _i (Pa.s)

λ :

characteristic time constant (s)

μ:

dynamic shear viscosity of generalized Newtonian fluid (Pa.s)

μ_i :

high-shear viscosity (Pa.s)

μ_o :

low-shear viscosity (Pa.s)

τ :

shear stress (Pa)

τ _w :

shear stress at conduit wall (Pa)

B :

slit half height (m)

f :

$\lambda ^{m}{\gamma _{w}^{m}}$

₂ F ₁ :

hypergeometric function

g :

1 + f

I :

definite integral expression (Pa³.s⁻¹ for pipe and Pa².s⁻¹ for slit)

k :

viscosity coefficient in power law model (Pa.s^{n -bar})

L :

length of conduit (m)

m :

indicial parameter in Cross model

n :

flow behavior index in Carreau model

n’ :

n − 1

$\overline {n}$ :

flow behavior index in power law model

Δp :

pressure drop (Pa)

Q :

volumetric flow rate (m³.s⁻¹)

r :

radius (m)

R :

tube radius (m)

v :

fluid velocity in the flow direction (m.s⁻¹)

W :

slit width (m)

z :

coordinate of slit smallest dimension (m)