scholarly journals A Mathematical Model for Transport and Growth of Microbes in Unsaturated Porous Soil

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Ramesh Chandra Timsina ◽  
Harihar Khanal ◽  
Andrei Ludu ◽  
Kedar Nath Uprety
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Vertical Direction ◽  
Reaction Diffusion Equations ◽  
Water Infiltration ◽  
Diffusion Equations ◽  
Richards Equation ◽  
Auxiliary Equation ◽  
Highly Nonlinear ◽  
Porous Soil

In this work, we develop a mathematical model for transport and growth of microbes by natural (rain) water infiltration and flow through unsaturated porous soil along the vertical direction under gravity and capillarity by coupling a system of advection diffusion equations (for concentration of microbes and their growth-limiting substrate) with the Richards equation. The model takes into consideration several major physical, chemical, and biological mechanisms. The resulting coupled system of PDEs together with their boundary conditions is highly nonlinear and complicated to solve analytically. We present both a partial analytic approach towards solving the nonlinear system and finding the main type of dynamics of microbes, and a full-scale numerical simulation. Following the auxiliary equation method for nonlinear reaction-diffusion equations, we obtain a closed form traveling wave solution for the Richards equation. Using the propagating front solution for the pressure head, we reduce the transport equation to an ODE along the moving frame and obtain an analytic solution for the history of bacteria concentration for a specific test case. To solve the system numerically, we employ upwind finite volume method for the transport equations and stabilized explicit Runge–Kutta–Legendre super-time-stepping scheme for the Richards equation. Finally, some numerical simulation results of an infiltration experiment are presented, providing a validation and backup to the analytic partial solutions for the transport and growth of bacteria in the soil, stressing the occurrence of front moving solitons in the nonlinear dynamics.

The Influence of the Diffusion Space Geometry on Behavior of some Processes in Biochemistry and Electrochemistry

2000 ◽  
Vol 5 ◽  
pp. 3-38 ◽  
Author(s):  
R. Baronas ◽  
F. Ivanauskas ◽  
J. Kulys ◽  
M. Sapagovas ◽  
A. Survila
Keyword(s):  
Mathematical Model ◽  
Reaction Diffusion ◽  
Electrode Reaction ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Carbon Paste ◽  
Diffusion Type ◽  
Function Modelling ◽  
Partially Blocked Electrodes ◽  
The Mathematical Model

The reaction-diffusion and diffusion equations were applied for modelling of some processes in biochemistry and electrochemistry. Modelling of the amperometric biosensors based on carbon paste electrodes encrusted with a single nonhomogeneous microreactor is analyzed. The mathematical model of the biosensor operation is based on nonstationary reaction-diffusion equations containing a non-linear term given by Michaelis-Menten function. Modelling of a simple redox-electrode reaction, involving two soluble species, is also considered. The model of the electrode behavior, taking into account the resist layer of the partially blocked electrodes, was expressed as a system of differential equations of the diffusion type with initial and boundary conditions. The mathematical model generalizing both processes: biochemical and electrochemical is presented in this paper. The generalized problem was solved numerically. The finite-difference technique was used for discretisation of the model. Using the numerical solution of the generalized problem, the influence of the size, shape and position of a microreactor as well as the thickness of the resist layer on the current dynamics was investigated.

scholarly journals Calculation of the apparent Michaelis constant for biosensors using reaction-diffusion equations

2014 ◽  
Vol 55 ◽  
Author(s):  
Pranas Katauskis ◽  
Feliksas Ivanauskas ◽  
Sigitas Laukevičius
Keyword(s):  
Mathematical Model ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Michaelis Constant ◽  
Amperometric Biosensors ◽  
The Relationship

The relationship between the apparent Michaelis constant and the diffusion module for one and two-layer biosensors is studied using a mathematical model describing action of the amperometric biosensors.

Fore-Aft Forces in Tire-Wheel Assemblies Generated by Unbalances and the Influence of Balancing

1991 ◽  
Vol 19 (3) ◽  
pp. 142-162 ◽  
Author(s):  
D. S. Stutts ◽  
W. Soedel ◽  
S. K. Jha
Keyword(s):  
Mathematical Model ◽  
Elastic Foundation ◽  
Torsional Oscillation ◽  
Vertical Direction ◽  
Torsional Stiffness ◽  
Rolling Motion ◽  
Vertical Force ◽  
Horizontal Force ◽  
Wheel Rim ◽  
Open Question

Abstract When measuring bearing forces of the tire-wheel assembly during drum tests, it was found that beyond certain speeds, the horizontal force variations or so-called fore-aft forces were larger than the force variations in the vertical direction. The explanation of this phenomenon is still somewhat an open question. One of the hypothetical models argues in favor of torsional oscillations caused by a changing rolling radius. But it appears that there is a simpler answer. In this paper, a mathematical model of a tire consisting of a rigid tread ring connected to a freely rotating wheel or hub through an elastic foundation which has radial and torsional stiffness was developed. This model shows that an unbalanced mass on the tread ring will cause an oscillatory rolling motion of the tread ring on the drum which is superimposed on the nominal rolling. This will indeed result in larger fore-aft than vertical force variations beyond certain speeds, which are a function of run-out. The rolling motion is in a certain sense a torsional oscillation, but postulation of a changing rolling radius is not necessary for its creation. The model also shows the limitation on balancing the tire-wheel assembly at the wheel rim if the unbalance occurs at the tread band.

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