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Figure 1. Bingham Plastic flow as described by Bingham. Figure 1 shows a graph of the behaviour of an ordinary viscous (or Newtonian) fluid in red, for example in a pipe. If the pressure at one end of a pipe is increased this produces a stress on the fluid tending to make it move (called the shear stress) and the volumetric flow rate increases proportionally.
For an incompressible and isotropic Newtonian fluid in laminar flow only in the direction x (i.e. where viscosity is isotropic in the fluid), the shear stress is related to the strain rate by the simple constitutive equation = where is the shear stress ("skin drag") in the fluid,
Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow .
The viscosity can easily be calculated from shear stress (from the torque) and shear rate (from the angular velocity). If a test with any geometries runs through a table of several shear rates or stresses, the data can be used to plot a flow curve, that is a graph of viscosity vs shear rate.
This is, however, specific to idealised flow; in the case of a cross-slot geometry the extensional flow is not ideal, so the stress, although singular, remains integrable, although the stress is infinite in a correspondingly infinitely small region. [15] If the solvent viscosity is zero, the Oldroyd-B becomes the upper convected Maxwell model.
Dimensionless numbers (or characteristic numbers) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena. [1] They include the Reynolds and the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, and flow speed.
A Newtonian fluid is a power-law fluid with a behaviour index of 1, where the shear stress is directly proportional to the shear rate: = These fluids have a constant viscosity, μ, across all shear rates and include many of the most common fluids, such as water, most aqueous solutions, oils, corn syrup, glycerine, air and other gases.
The viscosity of a non-Newtonian fluid is defined by a power law: [5] = ˙ where η is the viscosity after shear is applied, η 0 is the initial viscosity, γ is the shear rate, and if <, the fluid is shear thinning, >, the fluid is shear thickening,