Non-Newtonian fluids don't follow the rules of regular fluids. They can get thicker or thinner when you push on them, or need a certain force to start moving. This makes them act weird in pipes and channels.
To understand these fluids, we use special equations that describe how they flow. These help us figure out things like how much pressure we need to move them through pipes. Dealing with non-Newtonian fluids can be tricky, but they're used in lots of cool stuff like making plastic toys and food products.
Flow Behavior of Non-Newtonian Fluids
Flow of non-Newtonian fluids
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Non-Newtonian fluids exhibit complex flow behavior that deviates from Newtonian fluids
Shear-thinning (pseudoplastic) fluids experience decreasing viscosity with increasing shear rate (ketchup, paint)
Shear-thickening (dilatant) fluids experience increasing viscosity with increasing shear rate (cornstarch suspension, some colloids)
Bingham plastic fluids require a yield stress to initiate flow and then behave as a Newtonian fluid (toothpaste, mayonnaise)
Flow in pipes differs from Newtonian fluids
Velocity profile deviates from the parabolic profile observed in Newtonian fluids
Shear-thinning fluids have flatter velocity profiles due to reduced viscosity near the pipe wall
Shear-thickening fluids have more pronounced velocity gradients near the pipe wall due to increased viscosity
Flow in channels exhibits similar behavior to pipe flow but with a rectangular cross-section
Velocity profile and flow characteristics depend on the specific rheology of the non-Newtonian fluid
Calculations for non-Newtonian fluids
Rheological models describe the relationship between shear stress and shear rate for non-Newtonian fluids
Power-law (Ostwald-de Waele) model: τ = K γ ˙ n \tau = K \dot{\gamma}^n τ = K γ ˙ n
τ \tau τ represents shear stress
K K K represents consistency index
γ ˙ \dot{\gamma} γ ˙ represents shear rate
n n n represents flow behavior index, where n < 1 n < 1 n < 1 for shear-thinning, n > 1 n > 1 n > 1 for shear-thickening, and n = 1 n = 1 n = 1 for Newtonian fluids
Bingham plastic model: τ = τ 0 + μ p γ ˙ \tau = \tau_0 + \mu_p \dot{\gamma} τ = τ 0 + μ p γ ˙
τ 0 \tau_0 τ 0 represents yield stress
μ p \mu_p μ p represents plastic viscosity
Pressure drop calculations for non-Newtonian fluids
Modified Hagen-Poiseuille equation for power-law fluids: Δ P = 2 K L R 1 + 1 n ( 3 n + 1 4 n ) n ( 8 V D ) n \Delta P = \frac{2KL}{R^{1+\frac{1}{n}}}\left(\frac{3n+1}{4n}\right)^n\left(\frac{8V}{D}\right)^n Δ P = R 1 + n 1 2 K L ( 4 n 3 n + 1 ) n ( D 8 V ) n
Δ P \Delta P Δ P represents pressure drop
L L L represents pipe length
R R R represents pipe radius
V V V represents average velocity
D D D represents pipe diameter
Buckingham-Reiner equation for Bingham plastic fluids: Δ P L = 4 τ 0 3 R ( 1 − 4 3 λ + 1 3 λ 4 ) + 8 μ p V R 2 ( 1 − λ 2 ) \frac{\Delta P}{L} = \frac{4\tau_0}{3R}\left(1-\frac{4}{3}\lambda+\frac{1}{3}\lambda^4\right) + \frac{8\mu_pV}{R^2}\left(1-\frac{\lambda}{2}\right) L Δ P = 3 R 4 τ 0 ( 1 − 3 4 λ + 3 1 λ 4 ) + R 2 8 μ p V ( 1 − 2 λ )
λ = R τ 0 2 μ p V \lambda = \frac{R\tau_0}{2\mu_pV} λ = 2 μ p V R τ 0 represents the dimensionless yield stress parameter
Velocity profile calculations for non-Newtonian fluids
Power-law fluids: v ( r ) = ( n n + 1 ) ( Δ P R 2 K L ) 1 n [ R n + 1 n − r n + 1 n ] v(r) = \left(\frac{n}{n+1}\right)\left(\frac{\Delta PR}{2KL}\right)^{\frac{1}{n}}\left[R^{\frac{n+1}{n}}-r^{\frac{n+1}{n}}\right] v ( r ) = ( n + 1 n ) ( 2 K L Δ PR ) n 1 [ R n n + 1 − r n n + 1 ]
v ( r ) v(r) v ( r ) represents velocity at radial position r r r
Bingham plastic fluids: v ( r ) = Δ P R 2 4 μ p L ( 1 − r 2 R 2 ) − τ 0 R 2 μ p ( 1 − r R ) v(r) = \frac{\Delta PR^2}{4\mu_pL}\left(1-\frac{r^2}{R^2}\right) - \frac{\tau_0R}{2\mu_p}\left(1-\frac{r}{R}\right) v ( r ) = 4 μ p L Δ P R 2 ( 1 − R 2 r 2 ) − 2 μ p τ 0 R ( 1 − R r ) for r R ≥ 2 μ p V τ 0 R \frac{r}{R} \geq \frac{2\mu_pV}{\tau_0R} R r ≥ τ 0 R 2 μ p V
Challenges with non-Newtonian fluids
Pumping and transport of non-Newtonian fluids
Higher pressure drop compared to Newtonian fluids due to complex rheological behavior
Potential for flow instabilities and non-uniform flow, leading to processing issues
Selection of appropriate pump types and materials to handle the specific fluid properties
Heat transfer in non-Newtonian fluids
Modified heat transfer correlations are needed to account for the fluid's rheological behavior
Potential for reduced heat transfer efficiency due to viscosity effects and non-uniform flow
Mixing and agitation of non-Newtonian fluids
Increased power requirements for mixing due to higher viscosities and yield stresses
Potential for dead zones and poor mixing efficiency, leading to product quality issues
Selection of appropriate impeller types and configurations to ensure adequate mixing
Rheological characterization of non-Newtonian fluids
Accurate measurement of fluid properties is critical for proper design and operation of systems
Consideration of shear rate range, temperature, and time-dependent effects is necessary
Selection of appropriate rheological models and parameters to describe the fluid behavior
Applications of non-Newtonian fluids
Polymer processing involves non-Newtonian fluid behavior
Extrusion processes, where molten polymers exhibit shear-thinning behavior (plastic sheets, pipes)
Injection molding, where flow behavior affects mold filling and product quality (plastic parts, toys)
Fiber spinning, where elongational viscosity is critical for process stability and fiber properties (synthetic fibers, textiles)
Food engineering relies on understanding non-Newtonian fluid behavior
Pumping and transport of food products, such as ketchup and mayonnaise
Extrusion of food products, like pasta and snack foods
Mixing and blending of food ingredients, such as dough and sauces
Other industrial applications involving non-Newtonian fluids
Drilling fluids in the oil and gas industry, which exhibit shear-thinning and yield stress behavior
Blood flow in biomedical engineering, as blood exhibits shear-thinning behavior
Slurry transport in mining and construction, where suspensions can display non-Newtonian characteristics
Printing and coating processes, where ink and coating formulations may exhibit non-Newtonian behavior