# Category:CDm 1.Introduction

(Difference between revisions)
 Revision as of 17:08, 2 September 2013 (view source)Antonia (Talk | contribs) (Created page with "Category: Convection Diffusion module") Latest revision as of 13:23, 3 October 2013 (view source)Antonia (Talk | contribs) Line 1: Line 1: + The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. This paper describes the Convection Diffusion Applications for solving this equation. + + ====Examples==== + =====Example1===== + Below two examples of the application are shown. + + The first one deals with a differentially heated cavity. A square domain filled with air contains a "cold" and "hot" vertical walls, where temperatures of 295 and 305 C are prescribed (respectively). + The horizontal walls are adiabatic. + [[Image:c_model.jpg|thumb|left|300px|Heated cavity: model]] + Due to buoyancy convection occurs, Temperature propagates within the domain due to both the convection and the diffusion. Stationary solution is obtained. + [[Image:cavity.jpg|thumb|left|300px|Heated cavity: solution]] + =====Example2===== + To be filled + + ====Theory==== + + Under the assumption of incompressibility, the governing equations are given by + + [[Image:placa.jpg|thumb|right|300px|Problem description]] + $+ \rho C \frac{\partial T}{\partial t} + \rho C v \cdot \nabla T + \nabla + \cdot q = 0 +$ + (1) + + + + $+ q = -\kappa \nabla \cdot T +$ + (2) + + In the context of mass difussion within a fluid, (1) is is the mass + conservation equation and (2) is a contitutive law proposed by Fourier. The + notation is standard: $\rho$ is the density, $C$ the heat capacity, $\kappa$ + the thermal conductivity, $T$ is the temperature, $v$ is the velocity field + and $q$ is the diffusive flux per unit fluid density. + + + Remark: systen can be decoupled since we can plug (2) into (1) and solve the + scalar equation + + $+ \rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T - + \nabla \cdot (\kappa \nabla \cdot T) = 0 +$ + (3) + + + ====Problem statement==== + + Let us consider the transport by convection and diffusion in an open set + $\Omega$ $\subset \Re^d$(d=2 or 3) \ with piecewise smooth boundary $\Gamma$, + such that $\Gamma = \Gamma_d \cup \Gamma_N$. The unit outward normal vector to + $\Gamma$ is denoted $n$. The convection-diffusion initial-boundary value + problem can be stated as follows: given a divergence-free velocity field a, + the diffusion tensor $\kappa$ and adequate initial and boundary conditions, find $T$ : + $\bar{\Omega} \times \left[ 0, T \left] \rightarrow \Re \right. \right.$ such + that + + $+ \rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T - + \nabla \cdot (\kappa \nabla T) = 0 in \Omega \times (0, T) +$ + (4) + + + $+ T(x, 0) = T_0 (x) on \Omega +$ + + $+ T = T_D on \Gamma_D \times (0, T) +$ + + $+ k (\nabla T) \cdot n = q on \Gamma_N \times (0, T) +$ + + + + + + + + + + + + + + + + + + + + + + + [[Category: Convection Diffusion module]] [[Category: Convection Diffusion module]]

## Latest revision as of 13:23, 3 October 2013

The numerical solution of convection-diffusion transport problems arises in many important applications in science and engineering. These problems occur in many applications such as in the transport of air and ground water pollutants, oil reservoir flow, in the modeling of semiconductors, and so forth. This paper describes the Convection Diffusion Applications for solving this equation.

## Contents

#### Examples

##### Example1

Below two examples of the application are shown.

The first one deals with a differentially heated cavity. A square domain filled with air contains a "cold" and "hot" vertical walls, where temperatures of 295 and 305 C are prescribed (respectively). The horizontal walls are adiabatic.

Heated cavity: model

Due to buoyancy convection occurs, Temperature propagates within the domain due to both the convection and the diffusion. Stationary solution is obtained.

Heated cavity: solution

To be filled

#### Theory

Under the assumption of incompressibility, the governing equations are given by

Problem description

$\rho C \frac{\partial T}{\partial t} + \rho C v \cdot \nabla T + \nabla \cdot q = 0$ (1)

$q = -\kappa \nabla \cdot T$ (2)

In the context of mass difussion within a fluid, (1) is is the mass conservation equation and (2) is a contitutive law proposed by Fourier. The notation is standard: ρ is the density, C the heat capacity, κ the thermal conductivity, T is the temperature, v is the velocity field and q is the diffusive flux per unit fluid density.

Remark: systen can be decoupled since we can plug (2) into (1) and solve the scalar equation

$\rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T - \nabla \cdot (\kappa \nabla \cdot T) = 0$ (3)

#### Problem statement

Let us consider the transport by convection and diffusion in an open set Ω $\subset \Re^d$(d=2 or 3) \ with piecewise smooth boundary Γ, such that $\Gamma = \Gamma_d \cup \Gamma_N$. The unit outward normal vector to Γ is denoted n. The convection-diffusion initial-boundary value problem can be stated as follows: given a divergence-free velocity field a, the diffusion tensor κ and adequate initial and boundary conditions, find T : $\bar{\Omega} \times \left[ 0, T \left] \rightarrow \Re \right. \right.$ such that

$\rho C \frac{\partial T}{\partial t} + \rho Cv \cdot \nabla T - \nabla \cdot (\kappa \nabla T) = 0 in \Omega \times (0, T)$ (4)

T(x,0) = T0(x)onΩ

$T = T_D on \Gamma_D \times (0, T)$

$k (\nabla T) \cdot n = q on \Gamma_N \times (0, T)$

## Pages in category "CDm 1.Introduction"

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