## Explore chapters and articles related to this topic

### Introduction to Diffusive Processes

*Published in Ranjit Kumar Upadhyay, Satteluri R. K. Iyengar, Spatial Dynamics and Pattern Formation in Biological Populations, 2021*

*Ranjit Kumar Upadhyay, Satteluri R. K. Iyengar*

Advection: Advection is a transport mechanism of a fluid due to the fluid’s bulk motion. A simple example of advection is the transport of pollutants or silt in a river by bulk water flow downstream. Another example of advection is energy or enthalpy. The fluid’s motion is described mathematically as a vector field, and the transported material is described by a scalar field showing its distribution over space. Since advection requires currents in a fluid, it cannot happen in rigid solids. It does not include transport of substances by molecular diffusion. In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean such as heat, humidity (moisture), or salinity. Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle.

### Application of Numerical Methods to Selected Model Equations

*Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020*

*Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar*

Commonly used models for heat and mass transfer are analogous to each other. A large variety of physical and numerical issues are encountered in modeling the transport of chemical species of considerable importance in chemical and metallurgical engineering. In this section, models of 1-D transport are examined and a few unique features that must be captured by numerical schemes are described. The transport of a species (as measured by its concentration in a given control volume) is modeled by describing the pathways for its convection and diffusion. In the literature, one often sees the words advection and convection being used synonymously. This does not lead to any fundamental difficulties, and we note that formally convection refers to the movement of a carrier fluid, while advection is the movement of the species carried by the fluid. Denote the concentration of a species s present in a fluid mixture by Cs, the species molar concentration in moles per unit volume of a mixture. Assume that this quantity is transported by the local fluid velocity (convection) and satisfies simple conservation law: ∂Cs∂t+∇⋅Js=0

### Governing Equations and Classification of PDE

*Published in D. G. Roychowdhury, Computational Fluid Dynamics for Incompressible Flows, 2020*

As we already know, fluid dynamics is governed by the conservation of mass, momentum, energy and any additional equations describing the scalar transport of species (e.g. transport of particles or the concentration of dissolved solids). In a fluid flow problem, variables can be written in terms of a single generic equation: the scalar-transport or advection-diffusion equation. The advection is associated with the transport with flow (i.e. a “bulk” phenomenon), whereas the diffusion is a molecular phenomenon and arises due to the concentration gradient. This single scalar-transport equation represents all the physical quantities such as mass, momentum, energy, and we can deal with the discretization of a single equation instead of dealing with all the equations one by one. All these physical principles are expressed mathematically either in differential or integral form.

### A temperature-dependent numerical study of a moving boundary problem with variable thermal conductivity and convection

*Published in Waves in Random and Complex Media, 2023*

*Vikas Chaurasiya, Subrahamanyam Upadhyay, K.N. Rai, Jitendra Singh*

In this case, it is assumed that when solid material heated then the fluid flows exponentially within the melt region. In this case, we discuss the impact of exponential form of temperature dependent convection in term of Péclet number. Convection plays very important role during heat flow in a phase change material. It describes the heat transfer from one end to another due to fluid flow. Convection heat transfer involves combined process of conduction as well as advection. In the current case, we describe the convection effect on both temperature configuration as well as on moving melting front by varying Péclet number Pe. Figures 10(a) and 10(b) plotted at and . From Figure 10(a), it is observed that with increasing the value of Pe, temperature within the medium rises. Further, for higher convection rate the solid/liquid interface get accelerated, see Figure 10(b). Consequently, the melting process becomes fast and thus, less time is required to complete the process. Figures 10(a) and 10(b) also shows that the process is faster with convection while it deterred without convection.

### Unsteady three dimensional radiative-convective flow and heat transfer of dusty nanofluid within porous cubic enclosures

*Published in Journal of Dispersion Science and Technology, 2023*

The SIMPLE scheme[52,53] in 3 D dimensional case is used to treat the pressure distributions for the nanofluid and dusty particles in the previous systems. Also, the FVM (Finite Volume Method) approaches are applied to treat the advection, the diffusion, and the source terms. Equations (13)–(22) are taken the following integral form: where is a parameter that is equal 0,0,1 in equations, respectively, is a unit vector on in the outside direction of

### Large-Scale Numerical Investigation of Coherent Structure Dynamics in Film Cooling

*Published in Heat Transfer Engineering, 2022*

Solving advection-diffusion equation is the most commonly used method for simulating advection-diffusion problems. To discretize the advection-diffusion equation, a finite difference (FD) scheme is applied in this work: where is the temperature and is the time step used in the finite difference operation. Position and time are the same as those used in LBM. is thermal diffusivity, which is defined as and are the thermal conductivity of fluid and specific heat capacity at constant pressure, respectively. To discretize convection and diffusion terms shown in the right side of Eq. (10), second-order upwind scheme and second-order central scheme are adopted respectively, whose accuracies are ensured by the high resolution of grids. By assuming that the fluid is ideal air, the value of Prandtl number is 0.71.