Stagnation point flow
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.
When two streams either of two-dimensional or axisymmetric nature impinge on each other, a stagnation plane is created, where the incoming streams are diverted tangentially outwards; thus on the stagnation plane, the velocity component normal to that 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.
are constants (or time-dependent functions) referred as the strain rates; the three strain rates are not completely arbitrary since the continuity equation requires
The flow field can be simply described in cylindrical coordinate system
The radial stagnation flow is described using the cylindrical coordinate system
The flow due to the presence of a solid surface at
in planar stagnation-point flow was described first by Karl Hiemenz in 1911,[6] whose numerical computations for the solutions were improved later by Leslie Howarth.
According to potential flow theory, the fluid motion described in terms of the stream function
is non-zero on the solid surface indicating that the above velocity field do not satisfy no-slip boundary condition on the wall.
To find the velocity components that satisfy the no-slip boundary condition, one assumes the following form where
The existence of constant value for the viscous effects thickness is due to the competing balance between the fluid convection that is directed towards the solid surface and viscous diffusion that is directed away from the surface.
Thus the vorticity produced at the solid surface is able to diffuse only to distances of order
; analogous situations that resembles this behavior occurs in asymptotic suction profile and von Kármán swirling flow.
The problem formulated here is a special case of Falkner-Skan boundary layer.
The solution can be obtained from numerical integrations and is shown in the figure.
Hiemenz flow when the solid wall translates with a constant velocity
If the incoming stream is perpendicular to the stagnation line, but approaches obliquely, the outer flow is not potential, but has a constant vorticity
The appropriate stream function for oblique stagnation point flow is given by Viscous effects due to the presence of a solid wall was studied by Stuart (1959),[12] Tamada (1979)[13] and Dorrepaal (1986).
[14] In their approach, the streamfunction takes the form where the function
The solution for axisymmetric stagnation point flow in the presence of a solid wall was first obtained by Homann (1936).
Paul A. Libby (1974)[16](1976)[17] extended Homann's work by allowing the solid wall to translate along its own plane with a constant speed and allowing constant suction or injection at the solid surface.
The solution for this problem is obtained in the cylindrical coordinate system
Jets emerging from a slot-jets creates stagnation point in between according to potential theory.
The flow near the stagnation point can by studied using self-similar solution.
The initial study of impinging stagnation flows are due to C.Y.
flowing from opposite direction impinge, and assume the two fluids are immiscible and the interface (located at
At the interface, velocities, tangential stress and pressure must be continuous.
So there are only two parameters, which governs the flow, which are then the boundary conditions become
