Stagnation point flow

Stagnation point flow represents a fluid flow in the immediate neighborhood of solid surface at which fluid approaching the surface divides into different streams or a counterflowing fluid streams encountered in experiments. Although the fluid is stagnant everywhere on the solid surface due to no-slip condition, the name stagnation point refers to the stagnation points of inviscid Euler solutions. Stagnation point flow represents a fluid flow in the immediate neighborhood of solid surface at which fluid approaching the surface divides into different streams or a counterflowing fluid streams encountered in experiments. Although the fluid is stagnant everywhere on the solid surface due to no-slip condition, the name stagnation point refers to the stagnation points of inviscid Euler solutions. Hiemenz formulated the problem and calculated the solution numerically in 1911 and subsequently by Leslie Howarth(1934). The flow in the neighborhood of the stagnation point can be modeled by a flow towards an infinite flat plate, even though the whole body is a curved one(locally curvature effects are negligible). Let the plate be in the x z {displaystyle xz} plane with ( x , y ) = ( 0 , 0 ) {displaystyle (x,y)=(0,0)} representing the stagnation point. The inviscid stream function ψ {displaystyle psi } and velocity ( u , v ) {displaystyle (u,v)} from Potential flow theory are where k {displaystyle k} is an arbitrary constant (represents strain rate in the counter flow setup). For real fluid(including viscous effects), there exists a self-similar solution if one defines where ν {displaystyle u } is the Kinematic viscosity and δ = ν / k {displaystyle delta ={sqrt { u /k}}} is a boundary layer thickness but it is constant(vorticity generated at the solid surface is prevented diffusing far away by an opposing convection, similar profiles are Blasius boundary layer with suction, Von Kármán swirling flow etc.,). Then the velocity components and subsequently pressure and the equation for F ( η ) {displaystyle F(eta )} using Navier–Stokes equations are and the boundary condition due to no penetration and no-slip and the free stream condition for u {displaystyle u} (Note boundary conditions for v {displaystyle v} far away from the plate is not specified, because it is part of the solution - a typical boundary layer problem) are The asymptotic forms for large η → ∞ {displaystyle eta ightarrow infty } are where δ ∗ {displaystyle delta ^{*}} is the displacement thickness. Stagnation point flow with moving plate with constant velocity U {displaystyle U} can be considered as model for rotating solids near the stagnation points. The stream function is where G ( η ) {displaystyle G(eta )} satisfies the equation