In fluid dynamics, a stagnation point flow refers to a fluid flow in the neighbourhood of a stagnation point (in two-dimensional flows) or a stagnation line (in three-dimensional flows) with which the stagnation point/line refers to a point/line where the velocity is zero in the inviscid approximation. The flow specifically considers a class of stagnation points known as saddle points wherein incoming streamlines gets deflected and directed outwards in a different direction; the streamline deflections are guided by separatrices. The flow in the neighborhood of the stagnation point or line can generally be described using potential flow theory, although viscous effects cannot be neglected if the stagnation point lies on a solid surface.
Stagnation point flow without solid surfaces When two streams of two-dimensional or axisymmetric nature impinge on each other, a stagnation plane is created and the incoming streams are diverted tangentially outwards from the plane. Thus, the velocity component normal to the stagnation plane is zero; whereas, the tangential component is non-zero. In the neighborhood of the stagnation point, a local description for the velocity field can be described.
General three-dimensional velocity field The stagnation point flow corresponds to a linear dependence on the coordinates, that can be described in the Cartesian coordinates ( x , y , z ) {\displaystyle (x,y,z)} with velocity components ( v x , v y , v z ) {\displaystyle (v_{x},v_{y},v_{z})} as follows
v x = α x , v y = β y , v z = γ z {\displaystyle v_{x}=\alpha x,\quad v_{y}=\beta y,\quad v_{z}=\gamma z}
where ( α , β , γ ) {\displaystyle (\alpha ,\beta ,\gamma )} are constants (or time-dependent functions) referred to as the strain rates. The three strain rates are not completely arbitrary since the continuity equation requires α + β + γ = 0 {\displaystyle \alpha +\beta +\gamma =0} ; therefore, only two of the three constants are independent. We shall assume γ < 0 ≤ α {\displaystyle \gamma <0\leq \alpha } , meaning the flow is towards the stagnation point in the z {\displaystyle z} direction and away from the stagnation point in the x {\displaystyle x} direction. Without loss of generality, one can assume that β ≥ α {\displaystyle \beta \geq \alpha } . The flow field can be categorized into different types based on a single parameter
λ = α − β α + β {\displaystyle \lambda ={\frac {\alpha -\beta }{\alpha +\beta }}}
Planar stagnation-point flow The two-dimensional stagnation-point flow belongs to the case β = 0 ( λ = 1 ) {\displaystyle \beta =0\,(\lambda =1)} . The flow field is described as follows
v x = k x , v z = − k z {\displaystyle v_{x}=kx,\quad v_{z}=-kz}
where we let k = α = − γ > 0 {\displaystyle k=\alpha =-\gamma >0} . This flow field is investigated as early as 1934 by G. I. Taylor. In the laboratory, this flow field is created using a four-mill apparatus, although these flow fields are ubiquitous in turbulent flows. This type of flow also finds application in electrochemical reactors, where perpendicular feed streams can generate a uniform mass transfer boundary layer along the electrode surface, enhancing reaction uniformity and efficiency throughout the equipment.
Axisymmetric stagnation-point flow The axisymmetric stagnation point flow corresponds to α = β ( λ = 0 ) {\displaystyle \alpha =\beta \,(\lambda =0)} . The flow field can be simply described in cylindrical coordinate system ( r , θ , z ) {\displaystyle (r,\theta ,z)} with velocity components ( v r , 0 , v z ) {\displaystyle (v_{r},0,v_{z})} as follows
v r = k r , v z = − 2 k z {\displaystyle v_{r}=kr,\quad v_{z}=-2kz}
where we let k = α = β = − γ / 2 > 0 {\displaystyle k=\alpha =\beta =-\gamma /2>0} .
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