A
Newtonian fluid (named after
Isaac Newton) is defined to be a
fluid whose
shear stress is linearly proportional to the
velocity gradient in the direction
perpendicular to the plane of shear. This definition means regardless of the forces acting on a fluid, it
continues to flow. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the
drag of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare
friction). Important fluids, like water as well as most gasses, behave—to good approximation—as a Newtonian fluid under normal conditions on Earth. By contrast, stirring a
non-Newtonian fluid can leave a "hole" behind. This will gradually fill up over time—this behavior is seen in materials such as pudding,
oobleck, or
sand (although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip
paints). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property—for example, most fluids with long molecular chains can react in a non-Newtonian manner.
Equations for a Newtonian fluid The constant of proportionality between the viscous stress tensor and the velocity gradient is known as the
viscosity. A simple equation to describe incompressible Newtonian fluid behavior is :\tau = -\mu\frac{\mathrm{d} u}{\mathrm{d} n} where :\tau is the shear stress exerted by the fluid ("
drag"), :\mu is the fluid viscosity—a constant of proportionality, and :\frac{\mathrm{d} u}{\mathrm{d} n} is the velocity gradient perpendicular to the direction of shear. For a Newtonian fluid, the viscosity, by definition, depends only on
temperature, not on the forces acting upon it. If the fluid is
incompressible the equation governing the viscous stress (in
Cartesian coordinates) is :\tau_{ij} = \mu\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i} \right) where :\tau_{ij} is the shear stress on the i^{th} face of a fluid element in the j^{th} direction :v_i is the velocity in the i^{th} direction :x_j is the j^{th} direction coordinate. If the fluid is not incompressible the general form for the viscous stress in a Newtonian fluid is :\tau_{ij} = \mu \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} - \frac{2}{3} \delta_{ij} \nabla \cdot \mathbf{v} \right) + \kappa \delta_{ij} \nabla \cdot \mathbf{v} where \kappa is the second viscosity coefficient (or bulk viscosity). If a fluid does not obey this relation, it is termed a
non-Newtonian fluid, of which there are several types. Non-Newtonian fluids can be either plastic, Bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic. In some applications, another rough broad division among fluids is made: ideal and non-ideal fluids. An ideal fluid is non-viscous and offers no resistance whatsoever to a shearing force. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. One example of this is the flow far from solid surfaces. In many cases, the viscous effects are concentrated near the solid boundaries (such as in boundary layers) while in regions of the flow field far away from the boundaries the viscous effects can be neglected and the fluid there is treated as it were inviscid (ideal flow). When the viscosity is neglected, the term containing the viscous stress tensor \mathbf{\tau} in the Navier–Stokes equation vanishes. The equation reduced in this form is called the
Euler equation. ==See also==