In incompressible viscous flows in a confined domain, vortices are known to form at the corners and in the vicinity of separation points. The existence of a sequence of vortices (known as Moffatt vortices) at the corner with diminishing size and rapidly decreasing intensity has been indicated by physical experiments as well as mathematical asymptotics. In this work, we establish the existence of Moffatt vortices for the flow in the famous Lid-driven square cavity at moderate Reynolds numbers by using an efficient Navier-Stokes solver on non-uniform space grids. We establish that Moffatt vortices in succession follow fixed geometric ratios in size and intensities for a particular Reynolds number. In order to eliminate the possibility of spurious solutions, we confirm the physical presence of the small scales by pressure gradient computation along the walls.
The notion of infiniteness as pointed out in [1,2] comes from the solution of the biharmonic equation(1)The above is a simplified version of the Navier-Stokes equations for creeping flow. The author of [2] V. Shtern contends that although the solution of (1) leads to a sequence of infinite number of eddies, in physical reality, the sequence of eddies is finite. This is a clear admission that (1) does not accurately model the physical reality. In a self-contradictory defence, the author ([2]) comments that " "the sequence of corner eddies is finite", which is valid from the physical point of view, is incorrect for the mathematical solution."It is highly probable that tweaked statements such as "incorrect for mathematical solution" and "valid from physical point of view" could divert the scientific community further from the ground reality, which, on the contrary is not very difficult to fathom in the current context! If a partial differential equation modelling a physical system produces a mathematically correct solution having no physical relevance, it is a clear indication that the equation has failed to model the physical system correctly. Viscous effects on the wall, which is a pre-requisite for flow separation leading to the formation of vortex is completely absent in equation (1). We have also reiterated in our original work [1] that one of the sources of this non-physical result is the discontinuity of the boundary conditions at r = 0, ie., at the corner and failure of equation (1) in adhering to the continuum hypothesis on which the foundation of Navier-Stokes equation is built upon. R. L. Panton, in his highly acclaimed book Incompressible flows [4] concurs with our views: citing a specific case of Stokes flow in corners, he states that the remedy of removing this discontinuity lies in solving another problem that allows for a small gap at the corner (the junction of the walls). If the gap is actu-
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