0.6 Solution of the partial differential equations  (Page 2/13)

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Second order PDE. The classification of second order PDEs as elliptic, parabolic, and hyperbolic arise from a transformation of the independent variables. The classification apply to quasilinear (i.e., linear in the highest order derivatives) but we will only discuss linear equations with constant coefficients here. Numerical solutions are needed for quasilinear systems. Again let $u$ denote the dependent variables and $t$ , $x$ , $y$ , $z$ as the independent variables. Examples of the different classes of equations are

$\begin{array}{c}0=\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}+\frac{{\partial }^{2}u}{\partial {z}^{2}}+\rho ={\nabla }^{2}u+\rho ,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathrm{elliptic}\phantom{\rule{0.277778em}{0ex}}\mathrm{equation}\hfill \\ \frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}+\frac{{\partial }^{2}u}{\partial {z}^{2}}+\rho ={\nabla }^{2}u+\rho ,\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{parabolic}\phantom{\rule{0.277778em}{0ex}}\mathrm{equation}\hfill \\ \frac{{\partial }^{2}u}{\partial {t}^{2}}=\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}+\frac{{\partial }^{2}u}{\partial {z}^{2}}+\rho ={\nabla }^{2}u+\rho ,\phantom{\rule{1.em}{0ex}}\mathrm{hyperbolic}\phantom{\rule{0.277778em}{0ex}}\mathrm{equatio}n\hfill \end{array}$

The $\rho$ term represents sources. When the cgs system of units is used in electrostatics and $\rho$ is the charge density, the source is expressed as $4\pi \phantom{\rule{0.166667em}{0ex}}\rho$ . The factor $4\pi$ is absent with the mks or SI system of units. The parabolic PDEs are sometimes called the diffusion equation or heat equation. In the limit of steady-state conditions, the parabolic equations reduce to elliptic equations. The hyperbolic PDEs are sometimes called the wave equation. A pair of first order conservation equations can be transformed into a second order hyperbolic equation.

Convective-diffusion equation. The above equations represented convection without diffusion or diffusion without convection. When both the first and second spatial derivatives are present, the equation is called the convection-diffusion equation.

$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=\frac{1}{{N}_{Pe}}\frac{{\partial }^{2}u}{\partial {x}^{2}}$

Usually a dimensionless group such as the Reynolds number, or Peclet number appears as a factor to quantify the relative contribution of convection and diffusion.

Systems described by the poisson and laplace equation

We saw earlier that an irrotational vector field can be expressed as the gradient of a scalar and if in addition the vector field is solenoidal, then the scalar potential is the solution of the Laplace equation.

$\begin{array}{cc}\mathbf{v}=-\nabla \varphi ,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{irrotational}\phantom{\rule{0.277778em}{0ex}}\mathrm{flow}\hfill \\ \nabla •\mathbf{v}=\Theta =-{\nabla }^{2}\varphi \hfill \\ \nabla •\mathbf{v}=0=-{\nabla }^{2}\varphi ,\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{incompressible},\mathrm{irrotational}\phantom{\rule{0.277778em}{0ex}}\mathrm{flow}\hfill \end{array}$

Also, if the velocity field is solenoidal then the velocity can be expressed as the curl of the vector potential and the Laplacian of the vector potential is equal to the negative of the vorticity. If the flow is irrotational, then the vorticity is zero and the vector potential is a solution of the Laplace equation.

$\begin{array}{cc}\mathbf{v}=\nabla ×\mathbf{A},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{incompressible}\phantom{\rule{0.277778em}{0ex}}\mathrm{flow}\hfill \\ \nabla ×\mathbf{v}=\mathbf{w}=-{\nabla }^{2}\mathbf{A},\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{for}\phantom{\rule{0.277778em}{0ex}}\nabla •\mathbf{A}=0\hfill \\ {\nabla }^{2}\psi =-w,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{in}\phantom{\rule{0.277778em}{0ex}}\mathrm{two}\phantom{\rule{0.277778em}{0ex}}\mathrm{dimensions}\hfill \\ \nabla ×\mathbf{v}=0=-{\nabla }^{2}\mathbf{A},\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{irrotational}\phantom{\rule{0.277778em}{0ex}}\mathrm{flow}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\nabla •\mathbf{A}=0\hfill \\ {\nabla }^{2}\psi =0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{for}\phantom{\rule{0.277778em}{0ex}}\mathrm{two}\phantom{\rule{0.277778em}{0ex}}\mathrm{dimensional},\mathrm{irrotational},\mathrm{incompressible}\mathrm{flow}\hfill \end{array}$

Other systems, which are solution of the Laplace equation, are steady state heat conduction in a homogenous medium without sources and in electrostatics and static magnetic fields. Also, the flow of a single-phase, incompressible fluid in a homogenous porous media has a pressure field that is a solution of the Laplace equation.

Systems described by the diffusion equation

Diffusion phenomena occur with viscous flow, thermal conduction, and molecular diffusion. Heat conduction and diffusion without convection are described by the diffusion equation. Convection is always present in fluid flow but its contribution to the momentum balance is neglected for creeping (low Reynolds number) flow or cases where the velocity is perpendicular to the velocity gradient. In this case

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