0.6 Solution of the partial differential equations

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Topics covered in the chapter

• Classes of partial differential equations
• Systems described by the Poisson and Laplace equation
• Systems described by the diffusion equation
• Greens function, convolution, and superposition
• Green's function for the diffusion equation
• Similarity transformation
• Complex potential for irrotational flow
• Solution of hyperbolic systems

Classes of partial differential equations

The partial differential equations that arise in transport phenomena are usually the first order conservation equations or second order PDEs that are classified as elliptic, parabolic, and hyperbolic. A system of first order conservation equations is sometimes combined as a second order hyperbolic PDE. The student is encouraged to read R. Courant, Methods of Mathematical Physics, Volume II Partial Differential Equations , 1962 for a complete discussion.

System of conservation laws. Denote the set of dependent variables (e.g., velocity, density, pressure, entropy, phase saturation, concentration) with the variable $u$ and the set of independent variables as $t$ and $x$ , where $x$ denotes the spatial coordinates. In the absence of body forces, viscosity, thermal conduction, diffusion, and dispersion, the conservation laws (accumulation plus divergence of the flux and gradient of a scalar) are of the form

$\begin{array}{c}\frac{\partial g\left(u\right)}{\partial t}+\nabla ·f\left(u\right)=\frac{\partial g\left(u\right)}{\partial t}+\frac{\partial f\left(u\right)}{\partial x}=\frac{\partial u}{\partial t}+\frac{\partial f}{\partial g}\frac{\partial u}{\partial x}\hfill \\ \mathrm{where}\hfill \\ \frac{\partial f}{\partial g}={\left[\frac{\partial \left({g}_{1},\phantom{\rule{0.166667em}{0ex}}{g}_{2},\cdots {g}_{n}\right)}{\partial \left({u}_{1},\phantom{\rule{0.166667em}{0ex}}{u}_{2},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{u}_{n}\right)}\right]}^{-1}\frac{\partial \left({f}_{1},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{f}_{2},\phantom{\rule{0.277778em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{f}_{n}\right)}{\partial \left({u}_{1},\phantom{\rule{0.166667em}{0ex}}{u}_{2},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{u}_{n}\right)},\phantom{\rule{1.em}{0ex}}\mathrm{Jacobian}\phantom{\rule{0.277778em}{0ex}}\mathrm{matrix}\hfill \end{array}$

This is a system of first order quasilinear hyperbolic PDEs. They can be solved by the method of characteristics. These equations arise when transport of material or energy occurs as a result of convection without diffusion.

The derivation of the equations of motion and energy using convective coordinates (Reynolds transport theorem) resulted in equations that did not have the accumulation and convective terms in the form of the conservation laws. However, by derivation of the equations with fixed coordinates (as in Bird, Stewart, and Lightfoot) or by application of the continuity equation, the momentum and energy equations can be transformed so that the accumulation and convective terms are of the form of conservation laws. Viscosity and thermal conductivity introduce second derivative terms that make the system non-conservative. This transformation is illustrated by the following relations.

$\begin{array}{ccc}\hfill \frac{D\rho }{Dt}+\rho \phantom{\rule{0.166667em}{0ex}}\nabla •\mathbf{v}& =& \frac{\partial \rho }{\partial t}+\nabla •\left(\rho \phantom{\rule{0.166667em}{0ex}}\mathbf{v}\right)=0,\phantom{\rule{1.em}{0ex}}\mathrm{continuity}\phantom{\rule{0.277778em}{0ex}}\mathrm{equation}\hfill \\ \hfill \rho \frac{DF}{Dt}& =& \rho \frac{\partial F}{\partial t}+\rho \phantom{\rule{0.166667em}{0ex}}\mathbf{v}•\nabla F,\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}F=\mathbf{v},\phantom{\rule{0.277778em}{0ex}}S,\phantom{\rule{0.277778em}{0ex}}\mathrm{or}\phantom{\rule{0.277778em}{0ex}}E\hfill \\ \hfill \frac{\partial \left(\rho \phantom{\rule{0.166667em}{0ex}}F\right)}{\partial t}& =& \rho \frac{\partial F}{\partial t}+F\frac{\partial \rho }{\partial t}\hfill \\ \hfill \nabla •\left(\rho \phantom{\rule{0.166667em}{0ex}}F\phantom{\rule{0.166667em}{0ex}}\mathbf{v}\right)& =& \rho \phantom{\rule{0.166667em}{0ex}}F\phantom{\rule{0.166667em}{0ex}}\nabla •\mathbf{v}+\mathbf{v}•\nabla \left(\rho \phantom{\rule{0.166667em}{0ex}}F\right)=\rho \phantom{\rule{0.166667em}{0ex}}F\phantom{\rule{0.166667em}{0ex}}\nabla •\mathbf{v}+F\phantom{\rule{0.166667em}{0ex}}\mathbf{v}•\nabla \rho +\rho \phantom{\rule{0.166667em}{0ex}}\mathbf{v}•\nabla F\hfill \\ \hfill \rho \frac{DF}{Dt}& =& \frac{\partial \left(\rho \phantom{\rule{0.166667em}{0ex}}F\right)}{\partial t}-F\frac{\partial \rho }{\partial t}+\nabla •\left(\rho \phantom{\rule{0.166667em}{0ex}}F\phantom{\rule{0.166667em}{0ex}}\mathbf{v}\right)-\rho \phantom{\rule{0.166667em}{0ex}}F\phantom{\rule{0.166667em}{0ex}}\nabla •\mathbf{v}-F\phantom{\rule{0.166667em}{0ex}}\mathbf{v}•\nabla \rho \hfill \\ & =& \frac{\partial \left(\rho \phantom{\rule{0.166667em}{0ex}}F\right)}{\partial t}+\nabla •\left(\rho \phantom{\rule{0.166667em}{0ex}}F\phantom{\rule{0.166667em}{0ex}}\mathbf{v}\right)-F\left[\frac{\partial \rho }{\partial t}+\rho \phantom{\rule{0.166667em}{0ex}}\nabla •\mathbf{v}+\mathbf{v}•\nabla \rho \right]\hfill \\ & \therefore & \\ \hfill \rho \frac{DF}{Dt}& =& \frac{\partial \left(\rho \phantom{\rule{0.166667em}{0ex}}F\right)}{\partial t}+\nabla •\left(\rho \phantom{\rule{0.166667em}{0ex}}F\phantom{\rule{0.166667em}{0ex}}\mathbf{v}\right),\phantom{\rule{1.em}{0ex}}\mathrm{Q}.\mathrm{E}.\mathrm{D}.\hfill \end{array}$

Assignment 7.1

1. For the case of inviscid, nonconducting fluid, in the absence of body forces, derive the steps to express the continuity equation, equations of motion, and energy equations as conservation law equations for mass, momentum, the sum of kinetic plus internal energy, and entropy.
2. For the case of isentropic, compressible flow, express continuity equation and equations of motions in terms of pressure and velocity. Transform it to a second order hyperbolic equation in the case of small perturbations.

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