Pustam | पुस्तम | পুস্তম🇳🇵<p>Convection–diffusion equation<br />The convection-diffusion equation is a more general version of the scalar transport equation. It is a combination of the diffusion and convection (advection) equations. It describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection.<br />\[\dfrac{\partial c}{\partial t} = \mathbf{\nabla} \cdot (D \mathbf{\nabla} c - \mathbf{v} c) + R\]</p><p>\[\dfrac{\partial c}{\partial t} = \underbrace{\mathbf{\nabla} \cdot (D \mathbf{\nabla} c)}_{\text{diffusion}}-\overbrace{\underbrace{\mathbf{\nabla}\cdot (\mathbf{v} c)}_{\text{advection}}}^\text{convection} + \overbrace{\underbrace{R}_\text{destruction}}^\text{creation}\] </p><p>\(\mathbf{\nabla} \cdot (D \mathbf{\nabla} c)\) is the contribution of diffusion.<br />\(- \mathbf{\nabla}\cdot (\mathbf{v} c)\) is the contribution of convection or advection.<br />\(R\) describes the creation or destruction of the quantity.</p><p>where<br />\(c\) is the variable of interest.<br />\(D\) is the diffusivity.<br />\(\mathbf{v}\) is the velocity field, and<br />\(R\) is the sources or sinks of the quantity \(c\).</p><p><a href="https://mathstodon.xyz/tags/Convection" class="mention hashtag" rel="tag">#<span>Convection</span></a> <a href="https://mathstodon.xyz/tags/Diffusion" class="mention hashtag" rel="tag">#<span>Diffusion</span></a> <a href="https://mathstodon.xyz/tags/Transport" class="mention hashtag" rel="tag">#<span>Transport</span></a> <a href="https://mathstodon.xyz/tags/Advection" class="mention hashtag" rel="tag">#<span>Advection</span></a> <a href="https://mathstodon.xyz/tags/Equation" class="mention hashtag" rel="tag">#<span>Equation</span></a> <a href="https://mathstodon.xyz/tags/ConvectionDiffusionEquation" class="mention hashtag" rel="tag">#<span>ConvectionDiffusionEquation</span></a> <a href="https://mathstodon.xyz/tags/DifferentialEquations" class="mention hashtag" rel="tag">#<span>DifferentialEquations</span></a> <a href="https://mathstodon.xyz/tags/AdvectionEquation" class="mention hashtag" rel="tag">#<span>AdvectionEquation</span></a> <a href="https://mathstodon.xyz/tags/DiffusionEquation" class="mention hashtag" rel="tag">#<span>DiffusionEquation</span></a> <a href="https://mathstodon.xyz/tags/TransportEquation" class="mention hashtag" rel="tag">#<span>TransportEquation</span></a> <a href="https://mathstodon.xyz/tags/ConvectionEquation" class="mention hashtag" rel="tag">#<span>ConvectionEquation</span></a></p>
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Pustam | पुस्तम | পুস্তম🇳🇵<p>LINEAR TRANSPORT EQUATION<br />The linear transport equation (LTE) models the variation of the concentration of a substance flowing at constant speed and direction. It's one of the simplest partial differential equations (PDEs) and one of the few that admits an analytic solution.</p><p>Given \(\mathbf{c}\in\mathbb{R}^n\) and \(g:\mathbb{R}^n\to\mathbb{R}\), the following Cauchy problem models a substance flowing at constant speed in the direction \(\mathbf{c}\).<br />\[\begin{cases}<br />u_t+\mathbf{c}\cdot\nabla u=0,\ \mathbf{x}\in\mathbb{R}^n,\ t\in\mathbb{R}\\ <br />u(\mathbf{x},0)=g(\mathbf{x}),\ \mathbf{x}\in\mathbb{R}^n<br />\end{cases}\]<br />If \(g\) is continuously differentiable, then \(\exists u:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}\) solution of the Cauchy problem, and it is given by<br />\[u(\mathbf{x},t)=g(\mathbf{x}-\mathbf{c}t)\]</p><p><a href="https://mathstodon.xyz/tags/LinearTransportEquation" class="mention hashtag" rel="tag">#<span>LinearTransportEquation</span></a> <a href="https://mathstodon.xyz/tags/LinearTransport" class="mention hashtag" rel="tag">#<span>LinearTransport</span></a> <a href="https://mathstodon.xyz/tags/Cauchy" class="mention hashtag" rel="tag">#<span>Cauchy</span></a> <a href="https://mathstodon.xyz/tags/CauchyProblem" class="mention hashtag" rel="tag">#<span>CauchyProblem</span></a> <a href="https://mathstodon.xyz/tags/PDE" class="mention hashtag" rel="tag">#<span>PDE</span></a> <a href="https://mathstodon.xyz/tags/PDEs" class="mention hashtag" rel="tag">#<span>PDEs</span></a> <a href="https://mathstodon.xyz/tags/CauchyModel" class="mention hashtag" rel="tag">#<span>CauchyModel</span></a> <a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="tag">#<span>Math</span></a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="tag">#<span>Maths</span></a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="tag">#<span>Mathematics</span></a> <a href="https://mathstodon.xyz/tags/Linear" class="mention hashtag" rel="tag">#<span>Linear</span></a> <a href="https://mathstodon.xyz/tags/LinearPDE" class="mention hashtag" rel="tag">#<span>LinearPDE</span></a> <a href="https://mathstodon.xyz/tags/TransportEquation" class="mention hashtag" rel="tag">#<span>TransportEquation</span></a> <a href="https://mathstodon.xyz/tags/DifferentialEquations" class="mention hashtag" rel="tag">#<span>DifferentialEquations</span></a></p>
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