It is the  momentum diffusion.  The information about the wall with u=0 and a  wall shear that takes place, and they are transported  by means viscosity and the viscous stress tensor, that is, the symmetric part of velocity gradient.  It is a dissipative process converting kinetics energy into heat trough viscous dissipation. 

As the domain over the plate is by definition infinity, you can choose any high to define your domain, and apply the boundary condition: u=Uoo. As you  extend you domain in X direction, the boundary will increase, because the information from wall is continuously transported by viscous stress to the top without physical restrictions. 

But in the scale analysis you can notice that:  (delta(x) / x )~(Rex)1/2, that is, the boundary layer increases with x. This result emerges naturally from  boundary layer equation proposed by Prandtl. 

Hello Shashank,

As Ricardo Said, Because of diffusing zero momentum of the wall(u=0) in nonzero momentum of fluid, boundary layer thickness continuously increases.

The above answers given are very good. But I will try to answer in a more bascic level. The momentum of the flat plate is zero and the momentum of the uniform flow has a finite value. When the incoming uniform flow flows over a flat plate, the fluid particles near the plate will stick to the plate (no-slip condition). That means that the momentum of the flat plate is diffused to the fluid. The reason for this no-slip condition is adhesion between the flat plate and the fluid particles. The fluid particles have a force between them that holds them together, which is cohesion. When the fluid flows this cohesive force along with adhesion shows itself in the macroscopic scale as viscosity. As the flow proceeds downstream of the flat plate the viscosity is able to slow down more and more fluid layers above the flat plate. This is what is called momentum transfer. And hence the boundary layer thickness increases as the fluid moves downstream.

It is due to decrease in momentum of the fluid as the distance covered increases. This occurs due to friction between the fluid layer in contact and the plate. As the plate is fixed, it tries to drag the fluid down. The definition of boundary layer is the thickness at which the velocity of the fluid is 99% of that of free stream fluid. Since friction tries to drag the fluid layers, it is imminent that magnitude of velocity that achieves 99% of free stream along the perpendicular direction of the motion decreases as we go in direction of motion. Hence boundary layer thickness increases.  

I agree with all above comments plus entrainment of more external flow into boundary layer as you move downstream by viscous adhesion action of fluid parti

The answers are OK in the limit where the thickness is not too big: as you go downstream along the plate, the Reynolds number of the boundary layer increases and the flow becomes turbulent. Farther downstream, but nobody as far I know has a good understanding of it, the boundary layer will reach a finite thickness given by Logarithmic Prandtl theory. This is realized in atmospheric flows for instance where one observes Prandtl logarithmic layer.  

Thanks everyone for detailed answer. I agree that boundary layer develops because of diffusion of zero momentum at the wall to the top and as Ricardo said the reason for continuous increase in the boundary layer thickness is infinite domain on the top.

What if I assume a river flow over a completely flat surface and zero turbulence, what will be the boundary layer thickness. Now the domain is not infinite in the vertical direction but only in x direction.

The question, in my opinion, can simply be explained in terms of diffusion working against convection.

The easier example to start with would be the classic problem of an impulsively started infinite flat plate, where the boundary layer thickness keeps increasing with time, and not distance, as shear takes time to diffuse into the fluid away from the wall.

The same can be extended to a flat plat in flow where now the shear begins at the leading edge but is convected downstream by the flow before it can diffusively propagate outwards. Since diffusion away from the wall will always happen downstream, due to convection in flow direction, the boundary layer thickness will increase only downstream.

To add, for the follow up question, I do not believe increasing the free stream domain beyond a point affects the boundary layer thickness anyhow. Yes, if the parallel wall separation is small enough that boundary layer thickness can be of the same order, then somewhere downstream the boundary layer from the top and bottom wall will merge, assuming that transition to turbulence doesn't kick in at all. After the merge point there will be no free stream flow undisturbed by viscous diffusion of the shear caused by the walls.

In the other words, the  boundary layer is a macrosopic consequence of a microscopic chaotic molecular motion in transversal direction (which in this macroscopic, i.e. continuous world is called "momentum diffusion" or "viscosity").  In this special case of a rtiver the boundary layer (laminar and also turbulent) will grow to this cross section of a stream, where its thickness is equal to the river depth. From this point we have to do with so called "fully developed velocity profile" (and by the way - only from this point we can formally apply these popular 1D models of low, like Chezy formula).

This is just my opinion, somehow not common...

The boundary layer thickness rigorously does not exist, it is just an arbitrary construction we use for practical purposes. The Prandtl equations shows that:

dp/dy -> zero (but is not zero as it depends on the finite viscosity of the fluid)

u -> Ue (but only asymptotically for y-> +Inf).

This framework shows that you can extend the boundary layer definition in all the flow region from the Prandtl solution. That is just a mathematical consequence of the solution.

Furthermore, in practical flow conditions, the Pradtl solution is valid only for a very small lenght of the flat plate, indeed transition rapidly changes the flow behaviour

The laminar boundary layer is formed momentum diffusion at the boundary, as given by Newton's law of friction, shear stress at the boundary surface = (mu)*du/dy.  The velocity profile is found by Blasius, which shows du/dy (at the wall) at any point x will increase downstream, so more fluid particles will be retained as has been shown by Prandtl's experiment with fine aluminuum dust; pl see slide 26 of my presentation.  The 99% free stream velocity (for defining the thickness of the BL) line for all the velocity profile developed down stream will show an increase in thickness.

At a critical point instability will set in, the onset of transition flow will finally lead to turbulence; pl see my paper for determining the critical thickness.

https://www.researchgate.net/publication/265596533_THEORY_OF_BOUNDARY_LAYER_INSTABILITY_PARTICLE_OR_WAVE

Conference Paper THEORY OF BOUNDARY LAYER INSTABILITY: PARTICLE OR WAVE?

There are good answers. I want to supplement these by some points that have not yet been addressed:

1. If the plate is infinite, i.e. has no beginning, then d/dx is zero and the boundary layer grows in time.

2. If the plate is semi-infinite, with a beginning like for Blasius flow, this growth in time translates into a spatial growth (convection). Note that an isoline U=0.99U_inf (=definition of b.l. thickness=delta) is not a streamline; streamlines permanently enter the boundary layer, i.e. they cross the 0.99U_inf isoline, increasing delta successively. Moreover, the wall-normal velocity component stays constant outside the layer, and the streamlines are deflected at a constant  angle away from the wall, i.e. they are not parallel to it, even for infinite wall-normal distance. This is a consequence of the semi-infinite plate.

3. If the plate has a beginning and an end, then also the wall-normal velocity component decays to zero at some distance form the finite plate, and you have free-stream conditions sufficiently away from the plate.

If the length of the plate in infinite in both direction of x, the boundary layer becomes a logarithmic boundary layer which is not continuously getting thicker and thicker but which extends all the way to infinity, becoming a small perturbation far from the plate. At least this is the common wisdom. Actually it is very hard to make experimentally a fair distinction between a logarithmic velocity profile and a plain uniform velocity.  

To the answer of Yves Pomeau

"Common wisdom" in this case is what the reader can find, for example, in "Viscous Fluid Flow", by F.M. White, 2nd edition, for laminar flow, case of suddenly stopped plate, section 3-5: delta grows approximately with 3.64 times sqrt(nue times t).

Shahank,

The derivation of the boundary layer thickness shows the momentum diffusion thickness del = 6*SQRT(nu*t), where nu the kinematic viscosity = mu/rho, and the time of travel t = x/U.  Therefore del is proportional to SQRT(x), so that the BL thickness del will grow with distance.

kk

To Markus Kloker

Your answer is correct in the laminar case, but as I wrote before, as this boundary layer thicken it becomes turbulent. So the long time regime is  turbulent and the question then is how does this change the thickness of the boundary layer, particularly is there something like an asymptotic turbulent regime ? This question was investigated and, I think, rightly answered by Prandtl. The flow  cannot become uniform far from the plate because of the constraint that the vertical flux of horizontal momentum must be constant. This led Prandtl to the idea of Logarithmic layer by a very clever scaling argument based precisely on this flux of momentum. This is well explained  in Fluid Mechanics by Landau and Lifshitz. 

to Shashank Mishra

"What if I assume a river flow over a completely flat surface and zero turbulence, what will be the boundary layer thickness. Now the domain is not infinite in the vertical direction but only in x direction"

I think this is a quite different flow problem, you have no asymptotic external conditions...and, further, I cannot imagine a river flowing in absence of turbulence.

My answer is that you cannot define a boundary layer in the sense of the Prandtl theory.

Prandtl is a genius and master of boundary layer, for which he deserves a Nobel prize.  Upon the onset of instability of flow on the flat plate, the flow undergoes a transition as plume like structures formed and leave the wall, while the sub-layer on the surface will generate more plumes which may collide with oncoming plumes to form myriads of eddies.  Prandtl's brilliant exposition and quantification of the sub layer is most crucial in our understanding of the turbulence phenomenon.  Visiting Gotingen and attending the Symposium of 100 year of research of BL theory in 2004 is the most sacred pilgrimage for me.  He discovered and unveiled the secret and theory of flights and cruising in the ocean.

I am assuming you are asking about growth in general and not what happens during transition. I will try to explain in very simple language.

Flat plate is nothing but a momentum sink. It will extract momentum from any fluid flowing over it. Extracting momentum from a fluid is equal to slowing it, and the region where the fluid is slow (compared to the free stream flow away from the plate) is boundary layer. I you consider flow at any distance, call it x, it has some momentum, call it m. When the fluid moves a little further, say x+dx, it has to lose some momentum, and the only way to do that is slow down more fluid that is not part of the boundary layer, thus increasing the boundary layer thickness. You could argue, why not just slow down the fluid already inside the boundary layer, but the flow velocity variation has to be smooth, i.e. grow smoothly from zero at the plate to free stream velocity in the outer flow. 

Now, the plate itself cannot act as momentum sink if there is no mechanism to transfer the zero momentum of plate to the fluid. Viscosity id responsible for how effectively the momentum is extracted by the plate. In laminar flows, it is just viscosity, i.e. molecular diffusion that extracts momentum from the fluid, which is not as effective as "turbulent diffusion" in turbulent flows. Diffusion can be looked upon as mixing at microscopic scale. When there is turbulence, there is much vigorous mixing, and boundary layer grows faster.

Technically, I am not saying anything different from what most answers are trying to convey. Just another way of putting it.

Because of continuous diffusion (the Laplacian term in the NS equation) into the fluid, the BL thickness increases, and due to the advection (the non-linear terms in the NS equation) of the incoming fluid along the plate, the increment takes place along the upstream velocity, which is parallel to the plate.