diff --git a/diffusion-equation.html b/diffusion-equation.html index 821caddd87013d6bcd743465963c154759ba924f..4dfeab8155080746dfe545b36b7dfcb468c9b5c4 100644 --- a/diffusion-equation.html +++ b/diffusion-equation.html @@ -46,23 +46,25 @@ </mfrac> </mrow> <mo>=</mo> - <mi mathvariant="normal">∇<!-- ∇ --></mi> + <mi mathvariant="normal" data-dref="nabla">∇<!-- ∇ --></mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> - <mi>D</mi> - <mo stretchy="false">(</mo> - <mi>ϕ<!-- ϕ --></mi> - <mo>,</mo> - <mrow class="MJX-TeXAtom-ORD"> - <mi mathvariant="bold">r</mi> - </mrow> - <mo stretchy="false">)</mo> + <mrow data-dref="Dphir"> + <mi data-dref="D">D</mi> + <mo stretchy="false">(</mo> + <mi>ϕ<!-- ϕ --></mi> + <mo>,</mo> + <mrow class="MJX-TeXAtom-ORD"> + <mi mathvariant="bold">r</mi> + </mrow> + <mo stretchy="false">)</mo> + </mrow> <mtext> </mtext> - <mi mathvariant="normal">∇<!-- ∇ --></mi> + <mi mathvariant="normal" data-dref="nabla">∇<!-- ∇ --></mi> <mrow data-dref="phirt"> <mi data-dref="phi">ϕ<!-- ϕ --></mi> <mo stretchy="false">(</mo> @@ -92,8 +94,11 @@ <p>where <span data-decl="phi"><i>ϕ</i>(<b>r</b>, <i>t</i>) is the <a href="/wiki/Density" title="Density">density</a> of the diffusing material <span data-decl="r"> at location <b>r</b></span> and <span data-decl="t"> time <i>t</i></span></span> and -<span data-decl="Dphir"><i>D</i>(<i>ϕ</i>, <b>r</b>) is the collective <a href="/wiki/Diffusion_coefficient" class="mw-redirect" title="Diffusion coefficient">diffusion coefficient</a> for density <i>ϕ</i> at location <b>r</b></span>; and ∇ represents the vector <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> <a href="/wiki/Del" title="Del">del</a>. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.</p> -<p>More generally, when <i>D</i> is a symmetric <a href="/wiki/Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">positive definite matrix</a>, the equation describes <a href="/wiki/Anisotropy" title="Anisotropy">anisotropic</a> diffusion, which is written (for three dimensional diffusion) as:</p> +<span data-decl="Dphir"><i>D</i>(<i>ϕ</i>, <b>r</b>) is the collective <a +href="/wiki/Diffusion_coefficient" class="mw-redirect" title="Diffusion +coefficient">diffusion coefficient</a> for density <i>ϕ</i> at location <b>r</b></span>; +and <span data-decl="nabla">∇ represents the vector <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> <a href="/wiki/Del" title="Del">del</a></span>. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.</p> +<p>More generally, when <span data-decl="D"><i>D</i> is a symmetric <a href="/wiki/Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">positive definite matrix</a></span>, the equation describes <a href="/wiki/Anisotropy" title="Anisotropy">anisotropic</a> diffusion, which is written (for three dimensional diffusion) as:</p> <dl> <dd> <table cellpadding="5" style="border:2px solid #50C878;background: #ECFCF4; text-align: center;"> @@ -108,18 +113,20 @@ and <span data-decl="t"> time <i>t</i></span></span> and <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> - <mi>ϕ<!-- ϕ --></mi> - <mo stretchy="false">(</mo> - <mrow class="MJX-TeXAtom-ORD"> - <mi mathvariant="bold">r</mi> + <mrow data-dref="phirt"> + <mi data-dref="phi">ϕ<!-- ϕ --></mi> + <mo stretchy="false">(</mo> + <mrow class="MJX-TeXAtom-ORD"> + <mi mathvariant="bold" data-dref="r">r</mi> + </mrow> + <mo>,</mo> + <mi data-dref="t">t</mi> + <mo stretchy="false">)</mo> </mrow> - <mo>,</mo> - <mi>t</mi> - <mo stretchy="false">)</mo> - </mrow> + </mrow> <mrow> - <mi mathvariant="normal">∂<!-- ∂ --></mi> - <mi>t</mi> + <mi mathvariant="normal">∂<!-- ∂ --></mi> + <mi>t</mi> </mrow> </mfrac> </mrow> @@ -163,7 +170,7 @@ and <span data-decl="t"> time <i>t</i></span></span> and <mrow> <mo>[</mo> <mrow> - <msub> + <msub data-dref="D"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> @@ -181,14 +188,16 @@ and <span data-decl="t"> time <i>t</i></span></span> and <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> - <mi>ϕ<!-- ϕ --></mi> - <mo stretchy="false">(</mo> - <mrow class="MJX-TeXAtom-ORD"> - <mi mathvariant="bold">r</mi> - </mrow> - <mo>,</mo> - <mi>t</mi> - <mo stretchy="false">)</mo> + <mrow data-dref="phirt"> + <mi data-dref="phi">ϕ<!-- ϕ --></mi> + <mo stretchy="false">(</mo> + <mrow class="MJX-TeXAtom-ORD"> + <mi mathvariant="bold" data-dref="r">r</mi> + </mrow> + <mo>,</mo> + <mi data-dref="t">t</mi> + <mo stretchy="false">)</mo> + </mrow> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi>