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Synovial fluid is a polymeric liquid which generally behaves as a viscoelastic fluid due to the presence of hyaluronan molecules. We restrict ourselves to the regime in which the fluid responds as a viscous fluid. A novel generalized power-law fluid model is developed wherein the power-law exponent depends on the concentration of the hyaluronan. Such a model will be adequate to describe the flows of such fluids as long as they are not subjected to instantaneous stimuli. Assuming two different structures for the form of the power law exponent, both in keeping with physical expectations, we numerically solve for the flow of the synovial fluid (described by the constraint of incompressibility, the balance of linear momentum, and a convection-diffusion equation for the concentration of hyaluronan) in a rectangular cavity. The solutions obtained with our models are compared with the predictions of those based on a model that has been used in the past to describe synovial fluids. While all the three models seem to agree well with available experimental results, one of the models proposed by us seems to fit the data the best; it would, however, be hasty to pass judgment based on this one particular experimental correlation.

Synovial fluid (SF) is a polymeric solution that is viscoelastic. That this is indeed so was documented by Ogston and Stanier [

Most biological fluids undergo a plethora of chemical reactions that leads to either degrading or enhancing their mechanical properties. To ignore such biochemical reactions and to merely model the fluid as an inert body essentially makes the studies irrelevant to most problems of physiological interest as it is the pathological situations that need to be understood and modeled appropriately. For instance, if one were to think of a biological material such as blood, it is the biochemical reactions that lead to a variety of blood disorders that present challenges with regard to the modeling; not that the fact that the flow takes place in a complex geometry is unimportant, merely that it concerns itself more with the solution methodology than the development of an appropriate mathematical model. Interestingly, in the case of SF, the response is non-Newtonian when it is healthy and normal and close to the classical Newtonian fluid when it is inflamed (see page 4). Our interest here is with regard to the mathematical modeling rather than the study of a specific equation in a complex geometry. The approach that we use, namely, that of coupling the balance of linear momentum for the body of interest with a convection-diffusion equation for a constituent which affects the properties of the body, has been adopted to study the degradation and enhancement of the flow characteristics of fluids (see Bridges and Rajagopal [

From the biochemical point of view the synovial fluid is the ultrafiltrate of blood plasma (namely, plasma free from large proteins) enriched with the locally synthesized polysaccharide molecules, called hyaluronan (HA). For closer biochemical description of SF and HA see Chapter 11 in Voet and Voet [

Since the response of the fluid, whether elastic effects or viscous effects are predominant, depends on the nature of the flow, the model for SF must depend on the “dynamics” of the flow. Higher shear rates imply higher alignment of the chains and thus a decrease in the viscosity. On the other hand, the influence of concentration works contrariwise because higher concentration of HA implies higher enlacement of the chains, which increases the viscosity. In previous studies, the viscous behavior of SF has been mathematically modeled by a shear-thinning fluid with constant concentration (Rudraiah et al. [

We consider models for the viscosity

(Model 1)

with our new model that takes into account the varying concentration influencing the shear-thinning effects in the following form:

(Model 2)

In the both models the parameters

To find a suitable form for

(Model 2a)

and a simple rational function with two free parameters

(Model 2b)

which satisfy these conditions and mainly, the fitting procedures, which are the topic of the next section, and lead to better results than for any other simple function with only one or two free parameters.

Each of the models introduced contains some unknown parameters. For model 1 they are

Fitted values of the parameters for the models;

Parameter | Value | ||

Model 1 | Model 2a | Model 2b | |

— | |||

— | — | ||

— | — |

Relative viscosity against shear rate for different concentrations. Graphs of the relative viscosity for all the models show the fitted curves and the experimental data (points) which were taken for the fitting procedure. Here we use the notation from Ogston and Stanier [

Model 1

Model 2a

Model 2b

Even though the models of class 2 fit the experimental data more accurately, the model 1 can fit the data reasonably well for some specific applications in the range of the concentrations that is being used, this means in the range of 0.14–0.25. To compare all models for higher concentration we use the results portrayed in Figure

Relative viscosity against shear rate for higher concentrations, plots (a), (c), (e), and the same graphs in the logarithmic scale, plots (b), (d), (f). Here we use notation from Ogston and Stanier [

Model 1

Model 1; logarithmic scale

Model 2a

Model 2a; logarithmic scale

Model 2b

Model 2b; logarithmic scale

For comparison, we plotted the shear-thinning index behavior of all the models that have been introduced, see Figure

Shear-thinning index for all models. The highlighted area represents the concentration range of the experimental data, thus the mean concentrations in SF under normal physiological conditions.

Even though SF is complex biological material, under normal conditions, it can be approximated as an incompressible homogeneous single constituent fluid. This homogenization can be done because the physiological mass concentration of HA is very low (usually less than

It will be useful to recast (

From now, for simplicity of the notation, we use instead of capitals the small letters for the nondimensional variables. Thus, the system of governing equations is transformed into

The mathematical analysis of such kinds of system can be found in Bulíček et al. [

For the needs of numerical simulations we consider a simple two-dimensional rectangular domain denoted by

Geometry of the domain

We decided to choose such geometry and boundary and initial conditions to demonstrate and compare the basic flow behavior of the models that are being considered. The boundary conditions are made to be continuous along the whole boundary and to exhibit continuous evolution in time from the rest state, which is advantageous in the numerical implementation.

In order to use the finite element method we introduce the governing equations system (

The discretization of the problem then follows standard finite element procedure. First we use the quadrilateral mesh

Since the reduced Reynolds number is of order

For the time discretization we use the Crank-Nicolson scheme and implicit treatment of the incompressibility constraint and pressure with time step

This nonlinear algebraic system is solved by the damped Newton method and linear subproblems by using direct linear solver.

We computed numerical solutions corresponding to all the models for the boundary conditions (

First, we present the distribution of the velocity magnitude in the whole domain (Figure

Velocity magnitude for all models.

N-S:

Model 1:

Model 2a:

Model 2b:

Comparison of velocity profiles on

Velocity profiles on line

Velocity profiles on line

Concentration distribution.

N-S:

Model 1:

Model 2a:

Model 2b:

Comparison of concentration profiles on

Concentration profiles on line

Concentration profiles on line

Viscosity distribution.

Model 1:

Model 2a:

Model 2b:

Comparison of viscosity profiles on

Viscosity profiles on line

Viscosity profiles on line

Viscosity distribution in logarithmic scale.

Model 1:

Model 2a:

Model 2b:

Comparison of viscosity profiles on

Viscosity profiles on line

Viscosity profiles on line

Shear-thinning index.

Model 2a:

Model 2b:

Comparing the flow fields, one can see significant difference between the solution of the N-S model, model 1, and models 2a, 2b. The computation for N-S model gives nearly symmetric solution along

As for the velocity profiles, the concentration profiles for model 2a and 2b are similar. The transport of concentration is fastest for the N-S model due to the higher speed of the flow compared to the other models. In Figure

Inspecting Figure

The last set of graphs in Figure

We have modeled the synovial fluid as a generalized viscous fluid as such a model is a good approximation for a class of flows of the synovial fluid provided the processes do not involve instantaneous response. We have compared three different models, with that of the Newtonian model. Model 1 combines in an artificial way two characteristics introduced in the literature, namely, exponential dependence of viscosity on the concentration and the shear-thinning ability, and the newly introduced models 2a, 2b which describe the SF as a power-law type fluid with concentration dependent exponent. While all three models fit the experimental data reasonably well it is quite clear that the model 2b provides the best fit. Of course, this might just be happenstance for this particular boundary value problem. Unfortunately additional experimental data are not available in order to corroborate the model further. In light of the results that we have established, it would be reasonable to suppose that the model 2b predicts the response of the synovial fluid the best amongst models that have been used thus far.

We numerically solved the flow in a simple configuration for all three models. There are certain differences between solutions for model 2a, 2b even though the fitted results were very close to each other. On the other hand, the solution for model 1 exhibits distinct patterns with regard to the flow field and viscosity distribution. This means that even though the viscosities for the models are close, the small difference is sufficient to produce significant differences in the flow pattern.

Since in the system of governing equations there is a convection-diffusion equation with very small diffusivity, on using standard finite element method the solutions exhibit large spurious oscillations, primarily in the solution for the concentration and consequently in the viscosity. These oscillations were reduced by the streamline diffusion method, however, it will be useful to implement better numerical stabilization methods (internal penalty method of Turek and Ouazzi [

We have solved the problem in a geometry that does not reflect the complexity of the actual physiological problem. Our aim was to merely obtain an understanding of the flow characteristics of the fluid. In the future, we plan to study the flow in a geometry which is more physiologically relevant.

J. Hron was supported by the project LC06052 (Jindřich Nečas Center for Mathematical Modeling) financed by MŠMT. J. Málek’s contribution is a part of the research project MSM 0021620839 financed by MŠMT; the support of GAČR 201/09/0917 is also acknowledged. P. Pustějovská was supported by the project LC06052 (Jindřich Nečas Center for Mathematical Modeling), financed by MŠMT and by GAUK Grant no. 2509/2007. K. R. Rajagopal thanks the National Science Foundation for its support.