Abstract
An analytical solution to the Navier–Stokes momentum equations for a compressible flow with volume and small shear viscosities as well as external friction is presented while the dynamic viscosity is set to zero. The demonstrated methodology holds in d dimensions. However, in this study, the three-dimensional case is considered in detail. The analytical solution blows up at finite times T, which is determined by a cubic relation if the initial flow velocity is not divergence-free. The existence of T is a necessary and sufficient condition for implementing a singularity. Nonetheless, for external friction μe>T−1, all analytical expressions are smooth while the averaged expressions are smooth for all times.
Highlights
Flows and vortices are of major interest in the aerospace industry
Note that turbulence theory is another application of fluid mechanics, where an exact solution to the closure problem that determines the average flow velocity is only known for the one-dimensional case
A d-dimensional analytical solution to the compressible Navier–Stokes momentum equations with volume and small shear viscosities as well as external friction was obtained while the dynamic viscosity was set to zero
Summary
Flows and vortices are of major interest in the aerospace industry. The flow around the wings is essential for an aircraft with regard to its ability to fly. As a main result of the study, smooth and globally defined ddimensional analytical expressions for the flow velocity and scalar pressure field, which solve the compressible Navier–Stokes momentum equations including small shear viscosity and external friction while the dynamic viscosity is set to zero, are presented. Which is used for closing the system of equations to be solved This linear relation between pressure and velocity divergence was used in the literature for the same purpose while small dynamic viscosity μ and large volume viscosity ζ was assumed. For vanishing dynamic viscosity μ = 0 and A = A(⃗x, t) = I3 + t∇u⃗T0 (⃗x), detA > 0, and the three-dimensional Hopf equation is solved by u⃗ = ∫ d3x′ u⃗0(⃗x′) det A(⃗x′, t) ⋅ δ(⃗x′ − ⃗x + ⃗v(t) + tu⃗0(⃗x′)), (10) To validate this main result, the corresponding equation with given u⃗ is considered.
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