__Derivation of flow rate equation for
Bingham-fluids__

Bingham-fluids are characterized by a yield stress. Contrary to
Newtonian fluids they can transmit a shear stress also without a velocity
gradient. But in order to make the Bingham-fluid flowing, the driving shear
stress has to be larger than the yield stress. Below this yield stress the fluid
will behave almost like a solid body and above as a liquid. Examples of
Bingham-fluids are tooth paste or paint. Before determining the velocity profile, the following consideration has to
be made:

The velocity profile of channel flows of Newtonian fluids features a
velocity gradient which decreases towards the centre of the channel. Hence
the shear stress transmitted by a fluid layer also decreases toward the
channel centre. For the reason that Bingham-fluids become solid when the
applied shear stress falls below the yield stress it will be clear that the
Bingham-fluid will become solid in the centre layers of the channel. There a
solid 'plug' will be moving within the flow. In the process of deriving the
velocity profile also the height of this solid area has to be determined.

The following figures show the coordinates used and give an idea of the
developing velocity profile:

*Fig.1: Used coordinate system*

*Fig 2: fully developed velocity profile*
Forces acting on a infinite small fluid element:

*Fig. 3: Forces acting on the fluid element in x-direction*In
putting up the forces acting on the fluid element the following conditions have
already been considered:
mass flow is constant along channel length (conservation of mass)
(=> no acceleration forces in x-direction)
stationary case

From necessity for the forces to be balanced it results from
Fig. 3:
(1)
The shear stress for Bingham-fluids is:
(2)
Setting eq. (2) equal to eq. (1) leads to:

(3)
Equation (3) can be rearranged:

Using the usual assumption:

Integrating twice results in:
(4)

(5)
In order to determine the constants C1 and C2
knowledge of the boundary condition is necessary:

- At y=0: No slip condition, e.g. u(y=0)=0.

Thus follows with eq.(5):
- The friction force at the walls has to be balanced by the driving
pressure force.

Thereby the friction force is given by the product of the wall surface area
with shear stress at the wall. The pressure force results from multiplying
the cross-sectional area of the channel with the pressure difference between
channel entrance an channel end. The pressure difference is here also
labeled with p..

The wall surface area is: W=2bl.

The cross-sectional area is: S=bh.

Leading to:

Rearranging results in: (6)

Because the pressure loss is constant and because the pressure decreases with
increasing x it follows: (7)

Setting eq. (7) equal to eq.(6) yields: (8)

Introducing this relation into eq.(4) gives the constant C1:
Using the constants C1 and C2
as well as eq. (5) results in u(y):
(9)

and:

__Determination of y-grenz__

After deriving the velocity profile the magnitude of y-grenz
has to be determined.

Above y-grenz the Bingham-fluid is like a solid body,
resulting in:

Introducing this condition into eq.(4) using C1
leads to:
From this equation, using the pressure loss (eq.7), y-grenz
can be determined:
(10)
__Derivation of u-max__

For the maximum velocity u-max it is valid:

This means simply eq.(10) has to be introduced into eq.
(9):
Rearranging results in the expression for u-max:
(11)/font>
__Volume flow rate__

The total volume flow rate within the channel can be written as the sum
of two partial flow rates. One part comes the moving plug in the centre of the
channel and the other part results from the areas h ³
y ³ ((h - y-grenz ) and y-grenz
³ y ³ 0 where the
velocity profile eq.(9) has to be applied. In this range the
Bingham-fluid moves as a liquid.

Using the symmetry between upper and lower half channel the total volume flow
rate can be written as: center>
(12)
The solid part is to determined easily from eq.(12) because
u-max and y-grenz are already
known. It is:

(13)
For the liquid part in eq.(12) the following integral has to
be solved (y-grenz is known):

(14) hr width="100%" >

Summing of the solid part eq.(13) and liquid part eq.(14)
yields the total volume flow rate:

Source:
http://www.tu-dresden.de/mw/ilr/lampe/bingham/bingeng.htm#(3) |