Analytical Solutions of Incompressible Fluid Dynamics

Analytical solutions of incompressible fluid dynamics provide exact expressions for velocity, pressure and related fields under simplifying assumptions. Such solutions to the Navier–Stokes equations and associated continuity equation yield fundamental insight into flow structure, stability and transport mechanisms. Classical examples include Couette, Poiseuille and Hiemenz flows, Beltrami and Trkal vortical motions, and stratified or multi-layer configurations. In non-isothermal contexts, the Boussinesq approximation permits coupling of momentum and buoyancy, leading to exact thermal convection solutions. Exact formulae inform boundary-layer theory, validate numerical schemes and illuminate parameter dependencies such as Reynolds, Grashof and Marangoni numbers. They are indispensable for the design of microfluidic devices, geophysical modelling of oceanic and atmospheric currents, chemical reactors and heat-exchange equipment, and for probing nonlinear phenomena such as counterflows, cellular convection and vortex breakdown. Recent advances have broadened these solution classes to include spatially inhomogeneous shear, stratification by density or viscosity, and combined heat and mass transfer effects.

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Analytical Solutions of Incompressible Fluid Dynamics publication trend

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Technical terms

Navier–Stokes equations: Non-linear partial differential equations governing momentum conservation in viscous, incompressible fluids.

Incompressibility condition: A divergence-free constraint on the velocity field ensuring constant density.

Boussinesq approximation: Simplification that treats density variations only in buoyancy terms, coupling momentum and thermal fields.

Lin–Sidorov–Aristov class: A family of exact solutions in which flow variables depend linearly on certain spatial coordinates, with coefficients as functions of remaining variables.

Beltrami flow: A flow in which vorticity is everywhere parallel to velocity, yielding eigenfunction solutions of the curl operator.

References

1. Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows. Symmetry (2023).
2. Couette - Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid, with allowance made for heat recovery. Вестник Самарского государственного технического университета Серия Физико-математические науки (2018).
3. Exact solutions to generalized plane Beltrami--Trkal and Ballabh flows. Вестник Самарского государственного технического университета Серия Физико-математические науки (2020).
4. Exact solutions to the Navier-Stokes equations describing stratified fluid flows. Вестник Самарского государственного технического университета Серия Физико-математические науки (2021).
5. Exact Solutions to Navier–Stokes Equations Describing a Gradient Nonuniform Unidirectional Vertical Vortex Fluid Flow. Dynamics (2022).