In fluid dynamics, inviscid flow is the flow of fluid that is not viscous. The principles of inviscid flow can also be applied to the flow of fluids of low viscosity in regions of the flow field where it is known there is little viscous activity. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. Where viscous forces are non-existent the Navier–Stokes equations can be simplified to a form known as the Euler equations. This simplified equation is derived by considering an inviscid fluid. Using the Euler equation, many fluid dynamics problems involving low viscosity are easily solved; however, the assumption of negligible viscosity is not valid in the region of the flow field near a solid boundary (the boundary layer) or, more generally in regions with large velocity gradients which are evidently accompanied by viscous forces. The flow of a superfluid is inviscid.
Euler equations
Euler equations describe the dynamics of inviscid flow, originally published by Leonhard Euler in 1757. This equations are the following
D ρ D t + ρ ∇ ⋅ v = 0 ; {\displaystyle {D\rho \over Dt}+\rho \nabla \cdot \mathbf {v} =0;}
ρ D v D t = − ∇ p + ρ g ; {\displaystyle \rho {D\mathbf {v} \over Dt}=-\nabla p+\rho \mathbf {g} \,;}
ρ = h ( p ) , {\displaystyle \rho =h(p),} where
The first equation is the continuity equation related to the conservation of mass. The second equation is the equivalent of Newton's second law of motion for fluids. The third equation is an equation of state and must be introduced to obtain a self-consisted system of equations, fluids following this relations are called homoentropic and applies to most fluids with uniform composition. If the fluid is incompressible, the first equation reduces to the condition that the fluid is solenoidal, ∇ ⋅ v = 0 {\displaystyle \nabla \cdot \mathbf {v} =0} . Assuming inviscid flow allows the Euler equations to be applied to flows in which viscous forces are insignificant. Some examples include flow around an airplane wing, upstream flow around bridge supports in a river, and ocean currents.
Derivation from Navier–Stokes equations In 1845, George Gabriel Stokes published another important set of equations, today known as the Navier–Stokes equations. Claude-Louis Navier developed the equations first using molecular theory, which was further confirmed by Stokes using continuum theory. The Navier–Stokes equations describe the motion of fluids:
ρ D v D t = − ∇ p + μ ∇ 2 v + ρ g . {\displaystyle \rho {D\mathbf {v} \over Dt}=-\nabla p+\mu \nabla ^{2}\mathbf {v} +\rho \mathbf {g} .}
When the fluid is inviscid, or the viscosity can be assumed to be negligible, the Navier–Stokes equations simplifies to the Euler equations: This simplification is much easier to solve, and can apply to many types of flow in which viscosity is negligible. Some examples include flow around an airplane wing, upstream flow around bridge supports in a river, and ocean currents. The Navier–Stokes equation reduces to the Euler equations when μ = 0 {\displaystyle \mu =0} . Another condition that leads to the elimination of viscous force is ∇ 2 v = 0 {\displaystyle \nabla ^{2}\mathbf {v} =0} , and this results in an "inviscid flow arrangement". Such flows are found to be vortex-like.
Applicability
Reynolds number The Reynolds number (Re) is a dimensionless quantity that is commonly used in fluid dynamics and engineering. Originally described by George Gabriel Stokes in 1850, it became popularized by Osborne Reynolds after whom the concept was named by Arnold Sommerfeld in 1908. The Reynolds number is calculated as:
R e = l c v ρ μ {\displaystyle \mathrm {Re} ={l_{c}v\rho \over \mu }}
The value represents the ratio of inertial forces to viscous forces in a fluid, and is useful in determining the relative importance of viscosity. In inviscid flow, since the viscous forces are zero, the Reynolds number approaches infinity. When viscous forces are negligible, the Reynolds number is much greater than one. In such cases (Re>>1), assuming inviscid flow can be useful in simplifying many fluid dynamics problems.
Solid boundaries Negligible viscosity can no longer be assumed near solid boundaries, such as the case of the airplane wing. In turbulent flow regimes (Re >> 1), viscosity can typically be neglected, however this is only valid at distances far from solid interfaces. When considering flow in the vicinity of a solid surface, such as flow through a pipe or around a wing, it is convenient to categorize four distinct regions of flow near the surface:
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