Non-Newtonian fluids don't follow simple rules like water. They can get thicker or thinner when you stir them. Scientists use math to figure out how these weird fluids behave.
These math models help engineers design stuff like pipes and pumps for tricky fluids. They look at things like how much force it takes to get the fluid moving and how it changes when you mix it faster.
Rheological Models for Non-Newtonian Fluids
Purpose of rheological models
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Mathematical equations describe relationship between shear stress and shear rate in fluids
Newtonian fluids have linear relationship with constant viscosity (water, honey)
Non-Newtonian fluids have nonlinear relationship with varying viscosity (ketchup, shampoo)
Accurately predict and characterize flow behavior of complex fluids
Aid in design and optimization of processes involving non-Newtonian fluids (extrusion, pumping)
Enable selection of appropriate equipment and operating conditions (pipe diameter, pump type)
Power-law model for viscosity
Power-law model (Ostwald-de Waele model) described by equation: τ = K γ ˙ n \tau = K \dot{\gamma}^n τ = K γ ˙ n
τ \tau τ = shear stress
K K K = consistency index, measures fluid's viscosity at shear rate of 1 s− 1 ^{-1} − 1
γ ˙ \dot{\gamma} γ ˙ = shear rate
n n n = flow behavior index, characterizes degree of non-Newtonian behavior
Shear-thinning fluids (pseudoplastic) have n < 1 n < 1 n < 1 , viscosity decreases with increasing shear rate (blood, paint, polymer solutions)
Shear-thickening fluids (dilatant) have n > 1 n > 1 n > 1 , viscosity increases with increasing shear rate (cornstarch suspensions, some colloidal dispersions)
Bingham plastic model for yield stress
Bingham plastic model used for fluids exhibiting yield stress, minimum stress required to initiate flow
Model described by equation: τ = τ 0 + μ p γ ˙ \tau = \tau_0 + \mu_p \dot{\gamma} τ = τ 0 + μ p γ ˙
τ \tau τ = shear stress
τ 0 \tau_0 τ 0 = yield stress
μ p \mu_p μ p = plastic viscosity, slope of shear stress-shear rate curve above yield stress
γ ˙ \dot{\gamma} γ ˙ = shear rate
Fluids following Bingham plastic model do not flow until applied stress exceeds yield stress (toothpaste, mayonnaise, some drilling fluids)
Herschel-Bulkley vs other models
Herschel-Bulkley model combines Power-law and Bingham plastic models, incorporating yield stress and shear-thinning/thickening behavior
Model described by equation: τ = τ 0 + K γ ˙ n \tau = \tau_0 + K \dot{\gamma}^n τ = τ 0 + K γ ˙ n
τ \tau τ = shear stress
τ 0 \tau_0 τ 0 = yield stress
K K K = consistency index
γ ˙ \dot{\gamma} γ ˙ = shear rate
n n n = flow behavior index
Compared to Power-law model:
Herschel-Bulkley includes yield stress term (τ 0 \tau_0 τ 0 ), Power-law does not
When τ 0 = 0 \tau_0 = 0 τ 0 = 0 , Herschel-Bulkley reduces to Power-law model
Compared to Bingham plastic model:
Herschel-Bulkley incorporates shear-thinning/thickening behavior through flow behavior index (n n n )
Bingham plastic assumes constant plastic viscosity
When n = 1 n = 1 n = 1 , Herschel-Bulkley reduces to Bingham plastic model
Herschel-Bulkley model more versatile, describes behavior of wider range of non-Newtonian fluids (yield-stress fluids with shear-thinning or shear-thickening properties)