# Difference between partical shear stress and boundary shear stress

Hi all,

I am currently studying civil hydraulics in my civil engineering course and we are going through estimating critical shear stresses for sediments. I am confused about the difference between boundary shear stress and particle shear stress. In terms of estimating critical shear stress, is there a difference between using particle shear stress and using boundary shear stress?

How would our results differ by choosing boundary/particle shear stress to begin with?

Thanks everyone in advance!!

## Answers and Replies

I assume you are studying open channel flow, which is generally turbulent, except in the boundary layer.
This is reflected in an additional term in the viscosity - shear relationship
The total transport in any fluid is given by the sum of the molecular transport and the turbulent trnasport.

Measuring y from the bottom up with y' the thickness of the boundary layer

For turbulent flow the general equations are

momentum transfer

$${\tau _{xy}} = \rho (\upsilon + \varepsilon )\frac{{\partial \overline v }}{{\partial y}}$$

mass transfer

$$w = - \left( {D + {E_m}} \right)\frac{{\partial \overline c }}{{\partial y}}$$

Heat transfer

$$q = - \rho {C_p}\left( {\alpha + {E_h}} \right)\frac{{\partial \overline \theta }}{{\partial y}}$$

leading to

$${\tau _{xy}} = \rho \varepsilon \frac{{\partial \overline v }}{{\partial y}}\;for\;y \ge y'$$

Where epsilon is the turbulent factor

From this you can develop the various open channel flow formulae.

See Von Karman and Prandtl in particular.

boneh3ad
Science Advisor
Gold Member
I am not a sediment transport expert but I know that in order for a sedimentary particle to be moved by a fluid, the shear stress exerted on it by the fluid must be greater than a critical shear stress determined by the particle's size and density. The boundary shear stress is the shear stress between the fluid and the particle at the boundary between the two. I am not really familiar with the particle shear stress terminology.

Thanks Studiot for your very detailed explanation, however we haven't been doing much on the quantitative side so I found it a bit hard to follow the formulas you have mentioned above as I don't have much background in it. Would there be a more qualitative way of going about this question. Would you be able to explain the particle shear stress terminology as I have not been able to find anything on it anywhere..

Boneh3ad, thanks for the explanation about the boundary shear stress. I have been researching particle shear stress and haven't been successful in finding anything.

OK, let's take a step back.

Do you understand what the boundary layer is? and importantly why there has to be one?

From my understanding the boundary layer determines the amount of bed shear stress? so it helps us with calculations. But I am not too sure why there needs to be one.

OK I am sorry I thought you were looking to develop a more theoretical approach.

The whole subject of hydraulics and in particular rough erodible channel hydraulics is semi-empirical.

It is known that water flow in such channels exerts a traction parallel to the channel sides and bed. Thus this is a shear.

The common semiempirical Shields equation is

$${\tau _{critical}} = c\left( {{\rho _{solid}} - {\rho _{water}}} \right)gd$$

Where c is an empirical constant around 0.05

This describes the critical shear to drive an exposed aprticle of diameter d along a horizontal bed.
This is reduced on sloping side walls by a function of the angle of repose.

In order to find a value for The shear imposed by a given flow we again employ semi-empirical methods and formulae associated wth Darcy, Chezy or Manning, Muller, Einstein or Meyer
These provide the tractive force or shear exerted on the boundary bed and walls by a given flow velocity or discharge rate (which amounts to the same thing knowing the cross sectional area).

Thus we can estimate the flow rate at which the tractive shear will first exceed the critical shear.

The appropriate values are built into these equations. which are deduced on dimensional arguments and then brought into line with reality by measured constants. That is what is meant by semi-empirical.

There is no one value for shear in the boundary layer. The boundary layer exists because water obeys the no slip boundary condition. That is the water touching the container boundary is at rest relative to it and the viscous shear increases rapidly from zero to the constant value in the bulk fluid.

To estimate this one has to consider momentum transport across a section of the boundary layer parallel to the flow and integrate perpendicular to the flow. I will post a derivation if you like.