Abstract
We model the flow of a bi-viscous non-Newtonian fluid in a porous medium by a square lattice where the links obey a piece-wise linear constitutive equation. We find numerically that the flow regime where the network transitions from all links behaving according to the first linear part of the constitutive equation to all links behaving according to the second linear part of the constitutive equation, is characterized by a critical point. We measure two critical exponents associated with this critical point, one of the being the correlation length exponent. We find that both critical exponents depend on the parameters of the model.
Highlights
The behavior of complex fluids when being inside a porous medium may be very different from that when they are not
One can mention the Carreau rheology which is Newtonian at low shear rate but behaves as a power law fluid above a certain shear rate [1]
By setting λy = 1/Ny we find from Equation (2) that the corresponding correlation length exponent is not the usual one, ν = 3/2 [11], but rather one that describes a correlated directed percolation problem
Summary
The behavior of complex fluids when being inside a porous medium may be very different from that when they are not This is a problem encountered in many biological or industrial applications ranging from impregnation of fibrous materials to immiscible multi-phase flow in porous media. Other examples are yield stress fluid that responds as a solid below a critical yield threshold They behave as a power law fluid [2]. Inertial effects can be described as a rheological change from a Newtonian fluid to a power law (quadratic or cubic) for a given large Reynolds number [3]. Another possible extension is the displacement of immiscible fluids in porous media. A non-zero amount of stress is required for a non-wetting phase to invade the smaller pore throats
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