Oceanography 540--Marine Geological Processes--Winter Quarter 2001

Marine Benthic Boundary Layers
Shear Stress

Physical and Geological Perspective

Sediment transport in the marine environment is driven by fluid motion. Some important transport mechanisms are depicted in Figure 1 in the context of the continental shelf:


lec99-2-f1.gif Figure 34-1

Different regimes exist across the continental shelf:


lec99-2-f2.gif

Figure 34-2


Internal waves can act in any of these regimes, though their importance increases in the outer zones. The important processes in deeper water are the same as on the outer shelf, i.e., dominated by geostrophic flow and secondarily internal waves.

Context for Study of Boundary Layers

A boundary layer exhibits a velocity profile connecting a background oceanographic flow and a no-slip, zero velocity condition at a solid boundary.


lec99-2-f3.gif

Figure 34-3


Transfer of momentum toward the boundary imposes a shear stress; when it is sufficient (i.e., exceeds a critical value), it can cause transport of sediment either as bedload or suspended load.

We begin by considering how a moving fluid interacts with a solid boundary and the nature of the velocity profile.

Shear Stress

Stress, denoted t, is a force applied to a plane surface per unit area. Dimensionally:

Eq 34-1 dimensions

Consider a volume element. Each of the three surfaces can have three forces acting on it, so there are nine components in all. They are denoted where the i subscript describes the surface (e.g., z would indicate the surface perpendicular to the z-axis) and the j subscript describes the direction of the force (e.g., x would indicate the force acting in the direction of the x-axis).


lec99-2-f4.gif

Figure 34-4


The three components acting perpendicular to the surfaces represent the pressure force acting on the volume:

Eq 34-2

Of the remaining six components, we are interested in the two acting in the x and y directions on the plane perpendicular to the z-axis :

Eq 34-3

At the seabed these stresses are denoted and .

For simplicity we will consider flow in the x-direction that is steady () and uniform () so henceforth we denote as and the stress at the boundary as .

Consider the case of laminar flow (no turbulence) between a fixed surface and a plate of area A, separated by a height h, and moving at a steady velocity u (such that the applied force is opposed equally by the drag of the solid surface).


f-s2-1.gif

Figure 34-5


In this situation the fluid velocity is proportional to the plate separation and the force per unit area:

Eq 34-4: eq 22-1

For laminar flow (of a Newtonian fluid, see below), the velocity profile would be linear:

Eq 34-5: eq 22-2

The gradient of velocity which appears in this equation is called the shear. Since the stress is the force per unit area, we use Eq 34-4 and 34-5 to write:

Eq 34-6: eq 22-3

The proportionality constant µ is the absolute viscosity and is a measure of the resistance to deformation of a fluid. Dimensionally the stress has units of force per unit area (g cm/s^2)/(cm^2) and the velocity gradient units of (s^-1) so the viscosity has units of g/(cm s). One g/(cm s) is one poise. In seawater the viscosity depends on temperature and salinity:

 

Fresh Water

Seawater, S=35 o/oo

Temperature

Density

Absolute Viscosity

Kinematic Viscosity

Density

Absolute Viscosity

Kinematic Viscosity

C

g/cm3

centipoise

cm2/s

(x 10-2)

g/cm3

centipoise

cm2/s

(x 10-2)

0

1.0000

1.52

1.52

1.028

1.61

1.57

10

.9997

1.31

1.31

1.027

1.39

1.35

15

.9991

1.14

1.14

1.026

1.22

1.19

20

.9982

1.005

1.00

1.025

1.07

1.05

30

.9956

0.801

0.804

1.022

0.87

0.85

An alternative way of thinking about the stress due to shear is as a transfer of momentum in a direction perpendicular to the surface on which the stress is acting. Multiplying the numerator and denominator by r :

Eq 34-7

The stress is the product of the gradient of fluid momentum and a constant n, the kinematic viscosity.

The discussion to this point pertains to a laminar flow and is for a Newtonian fluid. We say a fluid is Newtonian when the viscosity as defined by the proportionality in Eq 34-6 is a constant.


lec99-2-f5.gif

Figure 34-6


Other materials may resist deformation until a critical stress is reached, for example a so-called Bingham plastic which does not deform until a yield strength is exceeded after which it deforms linearly with increasing stress. Pseudo-plastic materials follow a path intermediate to a Newtonian fluid and a Bingham plastic. Water and dilute suspensions exhibit Newtonian behavior, while dense clay suspensions (say > 10 g/l) exhibit pseudo-plastic behavior (flocs are broken down to the constituent units before deformation takes place).


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