Effect of couple stresses on the unsteady convective diffusion in fluid flow through a channel

A generalized dispersion model is used to obtain exact solution for unsteady convective diffusion in the presence of couple stresses. The effect of the couple stress parameter 'a' on the most dominant dispersion coefficient is clearly depicted. The dimensionless mean concentration distribution is obtained as a function of dimensionless axial distance, time and 'a'. The results for 'pure convection' are also reported. It is shown that the effect of couple stress is predominant only for small values of 'a' and when a----infinity the flow characteristics tend to their equivalents in Newtonian theory. The results of Taylor's dispersion model are recovered as a particular case in the limit tau----infinity.     

Intrinsic time scaling in survival analysis: Application to biological populations

A method of dimensionless time-scaling based on extrinsic expectation of life at birth but intrinsic to a system generating a survival distribution is introduced. Such scaling allows the survival fraction function and its associated mortality function to serve as Green's functions for their generalized equivalents. i.e. a “population” function and a “death” function. The analytical mechanics of utilizing these concepts are formulated, applied to the classical Gompertz and Weibull survival models, and discussed with respect to biological relevance.     

The variance of the index (R) of aggregation of Clark and Evans

Summary The exact formula for the variance of the Index (R) of aggregation of Clark and Evans is derived. Due to the fact that R is a dimensionless number, its variance is independent of population density, being only a function of the sample size and so can be manipulated.     

Estimate of the sticking probability for cells in uniform shear flow with adhesion caused by specific bonds

A theory is developed for the aggregation rate of cells in uniform shear flow when the cell-cell adhesion is mediated by bonds between specific molecules on the cell surfaces such as antigen and antibody or lectin and carbohydrate. The theory is based on estimates of the frequency and duration of cell-cell collisions and of the number of bonds formed and required to hold the cells together. For high shear rates, the sticking probability is a function of a single dimensionless parameter, A, that is proportional to G-2, with G the shear rate. For low shear rates, the sticking probability is a function of a second dimensionless parameter, A' proportional to G-1. In either case the rate of cell-cell sticking is a maximum when A (or A') congruent to 1.0. For small values of A (or A') the cells collide frequently, but do not stick, whereas for large values of A (or A') the cells collide infrequently, but stick with larger probability. Studies in Couette viscometer or other flow having approximately uniform shear can test these models.     

Dimensional analysis in mathematical biology. II

The application of dimensional analysis in biology is further illustrated by functional equations composed of dimensionless numbers and dealing with renal physiology, lung physiology and plant leaf shape. Dimensional variables and dimensionless numbers are examined from the viewpoint of numerical invariant properties of a certain physical system. Utilization of the method for problems such as design of an artificial kidney is considered briefly. A tabulation of variables useful in biology is given, with suggestions for a number of new dimensional entities. A continuation of the list of dimensionless invariants from Part I (Bull. Math. Biophysics,23, 355–376, 1961) is provided and includes terms pertaining to general physiology, geometric growth, metabolism, ecological interactions, muscle kinetics and other areas. It is pointed out that use of dimensionless ratios (similarity criteria) makes possible a direct comparison of form or shape factors and relative growth ratios with a variety of physical ratios, through the use of functional equations containing only dimensionless entities. Organismal similarity during growth and development, and between genetically related species, may be analyzed in terms of “automodel” or “self-similar” systems governed by certain dimensionless invariants. Tables of biological variables and dimensionless groupings are included.     

Critical conditions for phytoplankton blooms

We motivate and analyse a reaction—advection—diffusion model for the dynamics of a phytoplankton species. The reproductive rate of the phytoplankton is determined by the local light intensity. The light intensity decreases with depth due to absorption by water and phytoplankton. Phytoplankton is transported by turbulent diffusion in a water column of given depth. Furthermore, it might be sinking or buoyant depending on its specific density. Dimensional analysis allows the reduction of the full problem to a problem with four dimensionless parameters that is fully explored. We prove that the critical parameter regime for which a stationary phytoplankton bloom ceases to exist, can be analysed by a reduced linearized equation with particular boundary conditions. This problem is mapped exactly to a Bessel function problem, which is evaluated both numerically and by asymptotic expansions. A final transformation from dimensionless parameters back to laboratory parameters results in a complete set of predictions for the conditions that allow phytoplankton bloom development. Our results show that the conditions for phytoplankton bloom development can be captured by a critical depth, a compensation depth, and zero, one or two critical values of the vertical turbulent diffusion coefficient. These experimentally testable predictions take the form of similarity laws: every plankton—water—light-system characterized by the same dimensionless parameters will show the same dynamics.     

Intensity of microcarrier collisions in turbulent flow

A model is developed, allowing estimation of the share of inelastic interparticle collisions in total energy dissipation for stirred suspensions. The model is restricted to equal-sized, rigid, spherical particles of the same density as the surrounding Newtonian fluid. A number of simplifying assumptions had to be made in developing the model. According to the developed model, the share of collisions in energy dissipation is small.List of Symbols b parameter in velocity distribution function (Eq. (28)) - c _K factor in Kolmogoroff spectrum law (Eq. (20)) - D _t(r _p ) m²/s characteristic dispersivity at particle radius scale (Eq. (13)) - E(k, t) m³/s² energy spectrum as function of k and t (Eq. (16)) - E _K (k) m³/s² energy spectrum as function of k in Kolmogoroff-region (Eq. (20)) - E _p dimensionless mean kinetic energy of a colliding particle (Eq. (36)) - E _cp dimensionless kinetic energy exchange in a collision (Eq. (37)) - G(x, s) dimensionless energy spectrum as function of x and s (Eq. (16)) - G _B(x) dimensionless energy spectrum as function of x for boundary region (Eq. (29)) - G _K(x) dimensionless energy spectrum as function of x for Kolmogoroff-region (Eq. (21)) - g m/s² gravitational acceleration - I _cp dimensionless collision intensity per particle (Eq. (38)) - I _cv dimensionless volumetric collision intensity (Eq. (39)) - k l/m reciprocal of length scale of velocity fluctuations (Eq. (17)) - K dimensionless viscosity (Eq. (13)) - n(2) dimensionless particle collision rate (Eq. (12)) - n(r) l/s particle exchange rate as function of distance from observatory particle center (Eq. (7)) - r m vector describing position relative to observatory particle center (Eq. (2)) - r m scalar distance to observatory particle center (Eq. (3)) - r _pm particle radius (Eq. (1)) - s dimensionless time (Eq. (10)) - SC kg/ms³ Severity of collision (Eq. (1)) - t s time (Eq. (2)) - u(r, t) m/s velocity vector as function of position vector and time (Eq. (2)) - u(r, t) m/s magnitude of velocity vector as function of position vector and time (Eq. (3)) - u _r(r, t) m/s radial component of velocity vector as function of position vector and time (Eq. (3)) - u _r (r, t) m/s magnitude of radial component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s latitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s magnitude of latitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s longitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s magnitude of longitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u _gsm/s superficial gas velocity - u(r) m/s root mean square velocity as function of distance from observatory particle center (Eq. (3)) - u_r(r) m/s root mean square radial velocity component as function of distance from observatory particle center (Eq. (4)) - u (r) m/s root mean square latitudinal velocity component as function of distance from observatory particle center (Eq. (4)) - u (r) m/s Root mean square longitudinal velocity component as function of distance from observatory particle center (Eq. (4)) - w(x) dimensionless root mean square velocity as function of dimensionless distance from observatory particle center (Eq. (11)) - V _pm³ particle volume (Eq. (36)) - w(2) dimensionless root mean square collision velocity (Eq. (34)) - w ^* parameter in boundary layer velocity equation (Eq. (24)) - x dimensionless distance to particle center (Eq. (9)) - x ^* value of x where G _Band G _K-curves touch (Eq. (32)) - x _K dimensionless micro-scale (Kolmogoroff-scale) of turbulence (Eq. (15)) - volumetric particle hold-up - m²/s³ energy dissipation per unit of mass - m²/s kinematic viscosity - kg/m³ density - (r) m³/s fluid-exchange rate as function of distance to observatory particle center - Latitudinal co-ordinate (Eq. (5)) - Longitudinal co-ordinate (Eq. (5))     

Nonisothermal immobilized enzyme reaction in a packed-bed reactor

Summary This work investigates the reaction behavior of immobilized enzymes in a packed-bed reactor. The effect of heat generation due to exothermic enzyme reaction is considered. Conservations of substrate and energy constitute two coupled nonlinear partial differential equations which are simultaneously solved by a numerical method. It is found that substrate conversion is generally increased at higher temperature. However, the extent of temperature heavily depends on the magnitude of the dimensionless Michaelis constant which is defined as the ratio of Michaelis constant to inlet substrate concentration. At low dimensionless Michaelis constant, substrate conversion is considerably affected by temperature, but at high dimensionless Michaelis constant, the temperature effect is negligibly small. It is also found that maximum bulk temperature of reaction mixtures occurs around a dimensionless reactor length of 1.3 for the case with high substrate conversion.     

A dimensionless number analysis of the hybridization process in diffusion- and convection-driven DNA microarray systems

The present theoretical analysis aims at providing a general understanding of the combined effect the many different process variables have on the hybridization rate in diffusion- and convection-driven DNA microarray systems. It is shown that all process variables can be grouped into only four different dimensionless numbers (the Damkohler number Da, the dimensionless association constant kappa(A), the dimensionless initial concentration C'(0) and a geometrical ratio alpha). These four numbers have a straightforward physical meaning and only contain easily measurable parameters. Reducing the solution space from 7D to 4D, the dimensionless number representation greatly facilitates the insight in the conditions leading to the occurrence of diffusion-limited hybridization rates in both diffusion- and convection-driven DNA microarray systems. This in turn simplifies their design and the interpretation of the experimental results that are obtained with these systems.     

Approximation of continuous fermentation by semicontinuous cultures

Semicontinuous fermentations, in which a fraction of a culture is replaced with fresh media at regular intervals, have been previously used as a means of approximating continuous growth. In most cases deviations from continuous operation were erroneously estimated using Fencl's model, which is only valid when the specific growth rate is independent of the substrate concentration. An approach to modeling Semicontinuous growth that incorporates the same kinetics followed in batch and continuous growth was developed and tested for Monod's expression for the specific growth rate. A dimensionless form of the model was used to simulate Semicontinuous fermentations for comparison to continuous growth. Differences between Semicontinuous and continuous growth were found to depend on three dimensionless variables: feed concentration, replacement rate, and time between replacements. For given values of the dimensionless feed concentration and time between replacements, a range of dimensionless replacement rates can be determined over which semi-continuous cultures are approximately continuous.     

Relation between left ventricular cavity pressure and volume and systolic fiber stress and strain in the wall.

Pumping power as delivered by the heart is generated by the cells in the myocardial wall. In the present model study global left-ventricular pump function as expressed in terms of cavity pressure and volume is related to local wall tissue function as expressed in terms of myocardial fiber stress and strain. On the basis of earlier studies in our laboratory, it may be concluded that in the normal left ventricle muscle fiber stress and strain are homogeneously distributed. So, fiber stress and strain may be approximated by single values, being valid for the whole wall. When assuming rotational symmetry and homogeneity of mechanical load in the wall, the dimensionless ratio of muscle fiber stress (sigma f) to left-ventricular pressure (Plv) appears to depend mainly on the dimensionless ratio of cavity volume (Vlv) to wall volume (Vw) and is quite independent of other geometric parameters. A good (+/- 10%) and simple approximation of this relation is sigma f/Plv = 1 + 3 Vlv/Vw. Natural fiber strain is defined by ef = In (lf/lf,ref), where lf,ref indicates fiber length (lf) in a reference situation. Using the principle of conservation of energy for a change in ef, it holds delta ef = (1/3)delta In (1 + 3Vlv/Vw).     

Scaling for Dynamical Systems in Biology

Asymptotic methods can greatly simplify the analysis of all but the simplest mathematical models and should therefore be commonplace in such biological areas as ecology and epidemiology. One essential difficulty that limits their use is that they can only be applied to a suitably scaled dimensionless version of the original dimensional model. Many books discuss nondimensionalization, but with little attention given to the problem of choosing the right scales and dimensionless parameters. In this paper, we illustrate the value of using asymptotics on a properly scaled dimensionless model, develop a set of guidelines that can be used to make good scaling choices, and offer advice for teaching these topics in differential equations or mathematical biology courses.     

A new method to determine optimum temperature and activation energies for enzymatic reactions

A new method for determination of the optimum temperature and activation energies based on an idea of the average rate of enzymatic reaction has been developed. A mathematical model describing the effect of temperature on a dimensionless activity for enzyme deactivation following the first-order kinetics has been derived. The necessary condition for existence of the function extreme of the optimal temperature has been applied in the model. The developed method has been verified using the experimental data for inulinase from Kluyveromyces marxianus.     

Electrodynamic properties of a non-degenerate low-temperature plasma at ħ → 0

Electrodynamics of a low-temperature plasma, including a quantized longitudinal electromagnetic field and containing a dimensionless quantum parameter, are presented. Such a parameter is the dimensionless charge inversely proportional to the root-mean-square velocity of electrons in plasma and larger than unity in magnitude. Thus, the results of numerical calculations based on the perturbation theory are now in doubt.     

An Application of Dimensional Analysis in Cultural Anthropology

Lawlike generalizations can be developed only for variables exhibiting strict constancy in concept formulation. By considering anthropology a space and its basic concepts as dimensions and borrowing from ecology, cultural materialism, and social, political, and cultural anthropology, we develop a 14-dimensional space defined by 42 scaled variables. Dimensional analysis performed yields 14 equations and 28 dimensionless expressions that satisfy them. Additionally, we derive a set of 28 ethnological expressions. Entering scaled data from five societies, we solve for each. Twenty-one expressions yield significant order-ings. Examination of their performance confirms that they function holistically.     

A method of graphically analyzing substrate-inhibition kinetics

A model of substrate inhibition for enzyme catalysis was extended to describe the kinetics of photosynthetic production of ethylene by a recombinant cyanobacterium, which exhibits light-inhibition behavior similar to the substrate-inhibition behavior in enzyme reactions. To check the validity of the model against the experimental data, the model equation, which contains three kinetic parameters, was transformed so that a linear plot of the data could be made. The plot yielded reasonable linearity, and the parameter values could be estimated from the plot. The linear-plot approach was then applied to other inhibition kinetics including substrate inhibition of enzyme reactions and inhibitory growth of bacteria, whose analyses would otherwise require nonlinear least-squares fits or data measured in constrained ranges. Plots for three totally different systems all showed reasonable linearity, which enabled visual validation of the assumed kinetics. Parameter values evaluated from the plots were compared with results of nonlinear least-squares fits. A normalized linear plot for all the results discussed in this work is also presented, where dimensionless rates as a function of dimensionless concentration lie in a straight line. The linear-plot approach is expected to be complementary to nonlinear least-squares fits and other currently used methods in analyses of substrate-inhibition kinetics. Copyright 1999 John Wiley & Sons, Inc.     

External and internal diffusion in heterogeneous enzymes systems

The intrusion of diffusion in heterogeneous enzyme reactions, which follow. Michaelis-Menten kinetics, is quantitatively characterized by dimensionless parameters that are independent of the substrate concentration. The effects of these parameters on the overall rate of reaction is illustrated on plots commonly employed in enzyme kinetics. The departure from Michaelis-Menten kinetics due to diffusion limitations can be best assessed by using Hofstee plots which are also suitable to distinguish between internal and external transport effects. A graphical method is described for the evaluation of the reaction rate as a function of the surface concentration of the substrate from measured data.     

Modela-r as a Froude and Strouhal dimensionless numbers combination for dynamic similarity in running

The aim of this study was to test the hypothesis that running at fixed fractions of Froude (Nfr) and Strouhal (Str) dimensionless numbers combinations induce dynamic similarity between humans of different sizes. Nineteen subjects ran in three experimental conditions, (i) constant speed, (ii) similar speed (Nfr) and (iii) similar speed and similar step frequency (Nfr and Str combination). In addition to anthropometric data, temporal, kinematic and kinetic parameters were assessed at each stage to measure dynamic similarity informed by dimensional scale factors and by the decrease of dimensionless mechanical parameter variability. Over a total of 54 dynamic parameters, dynamic similarity from scale factors was met for 16 (mean r=0.51), 32 (mean r=0.49) and 52 (mean r=0.60) parameters in the first, the second and the third experimental conditions, respectively. The variability of the dimensionless preceding parameters was lower in the third condition than in the others. This study shows that the combination of Nfr and Str, computed from the dimensionless energy ratio at the center of gravity (Modela-r) ensures dynamic similarity between different-sized subjects. The relevance of using similar experimental conditions to compare mechanical dimensionless parameters is also proved and will highlight the study of running techniques, or equipment, and will allow the identification of abnormal and pathogenic running patterns. Modela-r may be adapted to study other abilities requiring bounces in human or animal locomotion or to conduct investigations in comparative biomechanics.