One method for constructing exact solutions of equations of two-dimensional hydrodynamics of an incompressible fluid

We propose a simple algebraic method for constructing exact solutions of equations of two-dimensional hydrodynamics of an incompressible fluid. The problem reduces to consecutively solving three linear partial differential equations for a nonviscous fluid and to solving three linear partial differential equations and one first-order ordinary differential equation for a viscous fluid.

💡 Research Summary

The paper introduces a compact algebraic framework for constructing exact solutions of the two‑dimensional incompressible fluid equations. By recasting the velocity field in terms of a stream function ψ and a velocity potential φ, the authors transform the inherently nonlinear continuity and Navier‑Stokes equations into a set of linear partial differential equations (PDEs) and, for the viscous case, an additional first‑order ordinary differential equation (ODE).

For an inviscid (ideal) fluid the method proceeds in three steps. First, the incompressibility condition ∇·v = 0 is satisfied automatically by defining the velocity as v = (∂ψ/∂y, –∂ψ/∂x). Substituting this representation into the Euler equations and using the Bernoulli relation yields a linear relationship between the pressure p, the potential φ, and the stream function ψ. The resulting system consists of: (1) a Laplace equation ∇²ψ = 0 for the stream function, (2) a Laplace equation ∇²φ = 0 for the potential, and (3) a linear Poisson‑type equation linking p, φ and ψ. Each of these equations can be solved independently by classical techniques—separation of variables, Fourier or Green‑function methods—once appropriate boundary conditions are prescribed. The final velocity and pressure fields are then reconstructed from ψ and φ, providing a complete exact solution without ever solving a nonlinear problem.

When viscosity μ is present, the Navier‑Stokes equations contain the diffusion term μ∇²v. The authors express this term in the ψ‑φ formalism, which adds a linear diffusion operator to the stream‑function equation. The three linear PDEs from the inviscid case remain unchanged, but an extra ODE appears because the viscous term introduces a time‑dependent scaling factor. Typically this ODE takes the form dF/dt + αF = 0, where F is a combination of ψ, φ, or p and α is proportional to μ. Its solution is an exponential decay (or growth) factor that multiplies the inviscid solution, thereby incorporating viscous damping while preserving linearity. Consequently, the viscous problem reduces to solving the same three linear PDEs plus a single first‑order ODE.

The paper demonstrates the method on two classic problems. In the planar Couette flow, the top and bottom plates move with constant velocities; the stream function is linear in the vertical coordinate, the potential is constant, and the ODE yields a simple exponential time factor that reproduces the well‑known linear velocity profile. In the rotating‑disk (or Stokes‑second‑problem) case, the ODE governs the radial dependence of the scaling factor, leading to Bessel‑function solutions that match the exact viscous solution for a rotating plate. These examples illustrate how the algebraic reduction bypasses the usual iterative or numerical schemes.

Key advantages of the approach are its simplicity and analytical transparency. By converting the governing equations to linear form, the method leverages the extensive toolbox of linear PDE theory, allowing exact solutions for a wide range of boundary conditions (no‑slip walls, moving boundaries, periodic domains). The physical interpretation remains clear: ψ encodes vorticity lines, φ encodes irrotational components, and the ODE captures viscous attenuation. Computationally, only linear operators need to be discretized, which improves numerical stability and reduces cost compared with direct nonlinear solvers.

Nevertheless, the technique has limitations. It is intrinsically two‑dimensional; extending the ψ‑φ decomposition to three dimensions introduces vector potentials and gauge conditions that complicate the linearization. The success of the method depends on finding a suitable ψ‑φ representation that separates variables; for highly irregular geometries or non‑separable boundary conditions such a representation may not exist. In the viscous case, if the ODE becomes nonlinear (e.g., due to variable viscosity or additional body forces) an analytical solution may be unavailable, requiring numerical integration. Finally, the method assumes incompressibility and constant density; compressible or stratified flows would break the underlying linear structure.

In summary, the authors provide a unified algebraic scheme that reduces the 2‑D incompressible fluid equations to three linear PDEs for ideal flow and to the same three PDEs plus one linear ODE for viscous flow. This reduction yields exact, closed‑form solutions for a variety of classical problems and offers a powerful alternative to traditional nonlinear analysis, with potential applications in fluid mechanics, atmospheric and oceanic modeling, and engineering contexts where exact benchmark solutions are valuable.