(1)
K. H. M. Ali, S. Pan and
B. A. O¡¯Connor
Department of Civil Engineering, University of Liverpool
Brownlow Street, Liverpool, L69 3GQ, U.K.
Telephone: +44 151 794 5242
Fax: +44 151 794 5218
E-mail: S.Pan@liv.ac.uk
Abstract: The paper gives theoretical study of some hydrodynamic and rheological behaviours of fluid mud by using an assumed fluid mud velocity distribution and sediment transport equation. The analysis was verified by data obtained from the field for different bed slopes and initial concentrations. The results demonstrated that the fluid mud can be better described as a Newtonian fluid rather than a non-Newtonian fluid. The results also revealed the enhanced kinematic viscosity for the fluid mud.
Keywords: fluid mud, hydrodynamics, rheology, sediment concentration, viscosity, newtonian fluid
Studies of sediment transport in estuaries have usually been performed through field studies and physical and numerical modelling. Such studies have generally been carried out using an appropriate combination of the above methods.
Estuarine cohesive sediment, commonly known as mud, is composed primarily of silt and clay. Mud contains a large proportion of very small particles which have a large specific area such that the effect of the surface physio-chemical forces becomes as important as the effect of gravity (Odd and Rodger, 1986).
The prediction of the movement of cohesive sediments has an important economical and ecological significance in the development of new engineering works and the maintenance of existing installations (Ali and Georgladis, 1991). Also, this ability is crucial in the understanding of the distribution of certain pollutants, in particular heavy metals which are absorbed on to the clay and silt particles.
Fluid mud is a dense suspension containing a concentration of mud flocs which is high enough to change significantly the physical properties of the mud/water mixture compared with that of clear water with the same salinity and temperature. Fluid mud is formed under a variety of conditions. Once formed it may flow under the influence of gravity and hydrostatic forces and then settle and dewater in, say, navigation channel causing significant rates of siltation over and above that due to settlement directly from suspension.
The paper gives an analysis for the variation of fluid mud velocity with the depth and the rheological behaviours of the fluid mud by using an assumption of fluid mud to be a power law fluid and sediment concentration distribution through the water column. The analysis was verified using field data obtained by H R Wallingford (Hydraulic Research Wallingford Ltd, 1992), who conducted extensive experiments to measure fluid mud velocity and concentrations over a four-day period (the measured results were subsequently identified with the profile numbers). This analysis also yielded a relationship between the viscosity of fluid mud and the mud concentration.
As given by Wilkinson (1960) if assuming fluid mud to be a non-Newtonian power law fluid, the shear stress can be expressed in the following form:
(1)
where k and n are parameters dependent on concentration and du/dy is the velocity gradient. Figures 1 and 2 show variations of n and k in Eq.(1) with concentrations for various suspensions obtained from the experiments of Jones and Golden (H R Wallingford, 1992). In spite of the considerable scatter, the trends are well established. This also indicates that as the concentration of fluid mud varies in the water column, k and n varies through the water depth.
Based on the above-mentioned evidence, this paper proposes a new approach to examine the hydrodynamic and rheological behaviours for the fluid mud. Not to loss the generality, the approach assumes a power-law velocity distribution as illustrated in Figure 3 for the fluid mud. Ali et al. (1997) showed the velocity gradient expressed in the following expression, which was also similar to that proposed by Ippen and Harleman (1952) and Harleman (1961):
(2)
where
is the maximum velocity and Z1
is the distance from the bed to the point where the maximum velocity occurred.
Due to variation of the concentration of the fluid mud in the water column, it seems to be natural to make use of the sediment transport equation for the vertical concentration distribution. Assuming in a steady state uniform flow, the sediment distribution in a water column is governed by:
(3)
where C is the concentration at elevation y; ws is the
fall velocity of fluid mud and
is the sediment diffusion
coefficient. In order to obtain
, we assume that there exists the following relation between
and velocity gradient:
(4)
Also for fluid mud, the density and fall velocity are assumed to be given by:
(5)
and
(6)
where
and k2 are constants.
Substituting for
,
and w
in Eq.(3) and integrating, we obtain:
+ constant
(7)
where
(8)
and
(9)
The field results obtained by H R Wallingford
(1992) from River Parrett were used to verify the theoretical relationships.
According to Eq. (7), for a Newtonian fluid
(n = 1) we should get a straight line for the variation of
with y. This trend is well
established in Figure 4.
However, if fluid mud behaves as a power law
fluid the according to Eq. (7), we should get a straight line for the variation
of
with
. Figure 5 shows the correlations between F(f)
and F(y) if the fluid mud is
considered as a non-Newtonian fluid (n=0.5).
The results indicate that only in some of the experiments this trend is verified
and in the majority of the cases fluid mud behaves as a Newtonian fluid with
viscosity much higher than that of clear water.
Using the experimental results obtained by H R Wallingford (1992) from the River Parrett, as well as these obtained from the narrow channel (Ali and Georgidis, 1991) and the Race Track Flume at Liverpool University (Ali et al, 1997) according to Eq. (7), the kinematic viscosity n (= k/r) of fluid mud was also calculated. The calculations revealed a strong dependence of the kinematic viscosity on fluid mud concentration as:
(10)
where
is the kinematic viscosity of clear
water.
Particularly in case of River Parrett, it was found that the relative dynamic viscosity of fluid mud is 9 ~ 550 time that of clear water.
(1) The analysis for the distribution of fluid mud concentration was verified using field data. Better agreement was obtained on assuming fluid mud to be a Newtonian fluid then assuming it to be a power law fluid. However, the values of viscosity obtained from either assumption were comparable.
(2) Viscosity of fluid mud increases with the increase in concentration.
(3) The relative dynamic viscosity of fluid mud obtained from the River Parrett was 9 ~550 times that of clear water.
References
Ali, K H M and Georgiadis, K (1991), ¡°Laminar motion of fluid mud¡±, Proc. Instn. Civil Engrs., Part 2, 91, pp 795-821.
Ali, K H M, Crapper, M and O¡¯Connor, B A (1997), ¡°Fluid mud transport¡±, Proc. Instn. Civil Engrs., Wat., Marit. & Energy, 1997 124, pp 64-78.
Harleman, D R F (1961), ¡°Stratified flow¡± in ¡°Handbook of fluid dynamics¡± (Street, V L Ed.), McGraw-Hill, New York, Chapter 26.
Hydraulic Research Wallingford Ltd. (1992), ¡°Fluid mud in estuaries¡±, ETSUTID 4084, Energy Technology Support Unit, UK Department of Energy.
Ippen, A T and Harleman, D R F (1952), ¡°Steady-state characteristics of sub-surface flow¡±, Circular No. 521, US National Bureau of Standards, pp 79-93.
Odd, N V M and Rodge, J G (1986), ¡°Analysis of the behaviour of fluid mud in estuaries¡±, Report No SR 84, Hydraulic Research Wallingford Ltd.
Wilkinson, W L (1960), ¡°New-Newtonian Fluids¡±, Pergamon Press, London.
Fig. 1 Variation of n with C
Fig. 2 Variation of k with C
Fig. 3 Definition sketch for the assumed velocity
profile
Fig. 4 Variation of F(f) with y (Newtonian fluid: n=1)
Fig. 5 Variation of F(f) with F(y) (Non-Newtonian fluid: n=0.5)