If your velocity field is correct, it must satisfy the conservation of energy and the Second Law of Thermodynamics (entropy generation).
The core challenge in advanced fluid mechanics is the , which describe the motion of viscous fluids. While a general solution is one of the unsolved Millennium Prize Problems , exact solutions exist for specific "reduced" scenarios where non-linear terms cancel out. Problem: Combined Couette-Poiseuille Flow advanced fluid mechanics problems and solutions
Force balance on cylindrical shell: ( \tau_rz (2\pi r L) = \Delta p \pi r^2 ) ⇒ ( \tau_rz = \fracG r2 ). If your velocity field is correct, it must
Imagine a fluid trapped between two infinite parallel plates. The bottom plate is stationary, while the top plate moves at a constant velocity . This is known as Couette flow . Coordinate System & Assumptions: Use Cartesian coordinates . Assume steady flow ( ), incompressible fluid ( ), and fully developed flow ( Continuity Equation: . For this geometry, this simplifies to . Given our assumptions, this confirms the velocity is only a function of the height This is known as Couette flow
Imagine fluid trapped between two cylinders, one spinning inside the other. The Problem
For steady, fully developed, 1D flow, the N-S equations reduce to: