Well, I don't think vorticity in the walls is the reason for the no-slip boundary condition, but the two go together. Basically, for flows with small viscosity, the velocity goes from some finite velocity, the free stream velocity, to zero in a thin region near the wall. That means a large transversal gradiente in the tangential components of velocity near the boundary, which means lots of shear near the boundary, which is another word for vorticity. The reason for no-slip is to avoid non-physical infinite stress at the boundary.

Hello Sir,

 Thanks for answering. The velocity gradient at the wall will result in vorticity (as you mentioned). Vorticity means rotation of fluid particle, how can a rotating fluid particle at wall still satisfy the no slip condition which  essentially indicates that the fluid particle near to the wall is at rest ?

Hello Senthil,

If you think in the fluid as formed by particles with a size d such  spheres this doubt will arise. But the fluid  is a continuum at macro scale and the continuum hypothesis means that at each point (x,t) is associated only  one particle of fluid. But, mathematically, points  do not have dimension. 

If imagine little particles with a size not equal to zero, if this particle has 10 Angstroms or 10 micrometers, doesn't matter once the continuum hypothesis to be valid in the flow conditions. But the fluid is a material, formed by molecules, and the surface is other material with imperfections, cavities and not a perfect flat plate.

hence, what is the "particle of fluid"? the molecule? A very little drop? At this scale Navier Stokes  Equations  does not have meaning. It is a mathematical concept because the matter has no continuities. After a determined scale this fluctuations disappear and the medium can be see as a continuum. So, there is no particle rotation over a wall. There are  a lot of collisions and interactions at molecular level. And at macroscopic level, the statistical behavior is like a rested fluid over a perfect wall. And you can define a derivative over a continuum field of velocities, density and viscosity

If you identify a vertical material line  inside of a boundary layer, for example, this material line will rotate due the velocity gradient. - higher velocity at the top. But  is the fluid rotating? No! Vorticity does not imply in rotation in the common sense, but in gradient velocities. A vortex has rotation and gradient velocity, and consequently, vorticity.  A laminar boundary layer over a flat wall have parallel streamlines, with shear between fluid layers, but without a vortex, and due the velocity gradient it has vorticity as you has pointed. 

And as Milton said, the stress is not infinity at the wall. It is a useful and valid mathematical concept at scales where the momentum equation are valid. 

This may probably help. It explains the vorticity of flow in the boundary layer, and in other scenarios as well.

https://www.youtube.com/watch?v=loCLkcYEWD4 - Vorticity 1 of 2

https://www.youtube.com/watch?v=h6bmrRFYFbc - Vorticity 2 of 2

Definition: Vorticity is matrix of i,j,k (unit vectors) ,d/dx,d/dy,d/dz, and u,v,w (velocity components). Therefore , there is no conflict ;for viscous flow velocity at the wall itself is zero( if wall is stagnant, if it moves e.g. moving belt it is equal to the wall velocity) while vorticity exists infinitesimaly close to the wall (not on the wall) because of the existence of velocity gradients

Vorticity is the key to explain the no-slip condition. First of all vorticity does not mean vortex. the two terms are not synonyms. The two concepts are different. The vorticity it is a measure of the "surfacial density" of vortices (Stokes theorem). Imagine the motion of a fluid element close to a wall in a pipe. With the no-sleep boundary condition the velocity at the wall is zero. You can expand the velocity in Taylor series, since the velocity is a vector, you will have a matrix as first order derivative which you can split in 3 terms, an isotropic one (expansion-contraction), an an-isotropic symmetric, (distorsion) and an an-isotropic anti-symmetric (rotation/vorticity) . The only possibility for the fluid element to move is to be distorted and rotated (locally). The two mechanism are almost always joint, because when a fluid element changes its shape (distortion) its moment of inertia changes, and therefore it is subject to a rotation. There is also an "energetic" explanation. A real fluid when it moves dissipates energy. Therefore its acceleration cannot have a scalar potential and so the curl of the velocity must be non-zero.

hello ! Ricardo Vicente de Paula Rezende,Milton Lopes Filho,N.C. Markatos and Andrea Boghi.

After read this discussion there are several doubts comes in my mind which i want to share it over here and hope i ill get reply from your side soon.

what is the difference between the terms vortex and vorticity ? whether they both are dependent on each other presents or present of vorticity leads to the result where vortex present ?

thanks

Hi,

well, I do not think that vorticity in the walls is the reason for maintaining the no-slip boundary condition, but instead, no-slip condition is source for vorticity generation at the wall. Concerning the vorticity equation for a divergence free Navier-Stokes vector field, there is no source term explicit in the governing equation. Vorticity may be amplified or attenuated when undergoing stretching or squeezing, but it can only be generated at the wall via some Kelvin–Helmholtz like instability.

Shrish,

Vortices are topological structures of the velocity field, where you see some spiral-like behaviours of velocity "arrows", you may get a general idea by thinking Karman vortex street. Vorticity, however, is another vector field which is defined as the curl of the velocity vector field. So, what is the linkage between vortices and vorticity? Well, vorticity dominated regions, equivalently, the region with low pressure (please referring to the potential theory), are generally be reinterpreted as vortex cores... 

That is, vortices are subregions in the velocity vector field, which are located to the extremes of the corresponding vorticity vector field. 

Hope it clarifies.

Thanks.

Wang Zhe