# Fundamentals ## Governing Equations of Fluid Dynamics and Heat Transfer - Conservation of chemical species (mass) - Conservation of momentum (Newton's second law) - Conservation of energy (1st law of thermodynamics) Concept of continuous medium consisting of elementary volumes - Lagrangian approach - equations formulated wrt individual particle - Eulerian approach - equations formulated wrt to distributed properties ### Material Derivative Consider element with velocity $\boldsymbol{V}(x,y,z,t)$ in fluid with density of $\rho(x,y,z,t)$ - Differentiation of $\rho$ wrt $t$ yields:$\frac{\partial\rho}{\partial t}+\frac{\partial\rho}{\partial x}\frac{dx(t)}{dt}+\frac{\partial\rho}{\partial y}\frac{dy(t)}{dt}+\frac{\partial\rho}{\partial z}\frac{dz(t)}{dt}=\frac{\partial\rho}{\partial t}+u\frac{\partial\rho}{\partial x}+v\frac{\partial\rho}{\partial y}+w\frac{\partial\rho}{\partial z}$ - Right hand side is the material derivative$\frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+u\frac{\partial\rho}{\partial x}+v\frac{\partial\rho}{\partial y}+w\frac{\partial\rho}{\partial z}=\frac{\partial\rho}{\partial t}+\boldsymbol{V}\cdot\nabla\rho$ - Derivative has two parts: 1. Time varying part at given location 2. Part due to motion of element Volume changes as result of flow:$\frac{1}{\delta\mathcal{V}}\frac{d(\delta\mathcal{V})}{dt}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=\nabla\cdot \boldsymbol{V}$ ### Mass Conservation Consider fluid element $\delta\mathcal{V}$ - mass $\delta m=\rho\delta\mathcal{V}$ must be constant$\frac{d(\rho\delta\mathcal{V})}{dt}=\delta\mathcal{V}\frac{D\rho}{Dt}+\rho\frac{d(\delta\mathcal{V})}{dt}=0$ Divide by $\delta\mathcal{V}$ to get the *continuity equation*:$\frac{1}{\delta\mathcal{V}}\frac{d(\delta\mathcal{V})}{dt}=\frac{D\rho}{Dt}+\rho\nabla\cdot \boldsymbol{V} = 0$ Can be rewritten as:$\frac{\partial\rho}{\partial t} + V\cdot\nabla\rho+\rho\nabla\cdot \boldsymbol{V} = \frac{\partial\rho}{\partial t} + \nabla\cdot(\rho \boldsymbol{V}) = 0$ For incompressible fluid:$\nabla\cdot V=0$ ### Conservation of Chemical Species Transport via diffusion quantified by $\boldsymbol{J}_i(x,t)$ - denotes direction and rate of mass flux of species *i* - movement from high concentration to lower concentration - measured with - *mass fraction* $m_i(x,t)$, mass ratio of species *i* to mass of element - *concentration of species* $C_i=m_i\rho$. mass of species *i* per unit volume - Conservation given by:$\frac{\partial}{\partial t}(\rho m_i)+\nabla\cdot(\rho m_i\boldsymbol{V}+\boldsymbol{J}_i) = R_i$ - $R_i$ is a source term to account for production/consumption of species by chemical reactions #### Fick's Law of Diffusion Empirical relation, used when variations of concentration are not very strong$\boldsymbol{J}_i=-\Gamma_i\nabla m_i$ - Concentration becomes:$\frac{\partial}{\partial t}(\rho m_i)+\nabla\cdot(\rho m_i\boldsymbol{V}) = R_i+\nabla\cdot(\Gamma_i\nabla m_i)$ - If diffusion coefficients $\Gamma_i$ are constants:$\frac{\partial}{\partial t}(\rho m_i)+\nabla\cdot(\rho m_i\boldsymbol{V}) = R_i+\Gamma_i\nabla^2 m_i$ ### Conservation of Momentum Newton's second law:$\frac{d}{dt}(m\boldsymbol{V})=\boldsymbol{F}$ - Replace normal derivative with material derivative:$\rho\frac{D}{Dt}(\boldsymbol{V})=\rho\bigg[\frac{\partial}{\partial t}(\boldsymbol{V})+(\boldsymbol{V}\cdot\nabla)\boldsymbol{V}\bigg]$ Two types of forces that affect fluid elements: 1. Body forces - act directly on the element, come from external source - gravity, electric, magnetic, etc. - total body force is proportional to its mass, written as $\rho\boldsymbol{f}$ 2. Surface forces - pressure and friction between neighboring fluid elements and walls - vector field of surface forces can represented by 3x3 stress tensor $\tau$ - component $\tau_{ij}$ is the *i*-component of stress acting normal to the *j*-axis - diagonal elements cause extension/contraction of the fluid element, offdiagonal elements represent deformation by shear Conservation of momentum written as:$\rho\frac{Du}{Dt}=\rho f_x+\frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{\partial z}$$\rho\frac{Dv}{Dt}=\rho f_y+\frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{zy}}{\partial z}$$\rho\frac{Dw}{Dt}=\rho f_z+\frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz}}{\partial y}+\frac{\partial\tau_{zz}}{\partial z}$ Stress tensor can separated into isotropic presure (always present) and viscous friction (zero if fluid at rest):$\tau_{ij}=-p\delta_{ij}+\sigma_{ij}$$\delta_{ij}=\begin{cases} 1\quad\text{if}\quad i=j \\ 0\quad\text{if}\quad i\neq j\end{cases}$ Model for viscous stresses:$\sigma_{ij}=\lambda\delta_{ij}(\nabla\cdot\boldsymbol{V})+\mu\bigg(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\bigg)$ - $\mu$ and $\lambda$ are first and second viscosity coefficients - for incompressible fluid, $\lambda$ disappears - for compressible fluid, $\lambda=-\frac{2}{3}\mu$ is general approximation Final form of momentum conservation, Navier-Stokes equations:$\begin{align}\rho\frac{Du}{Dt}=\rho f_x-\frac{\partial p}{\partial x}+&\frac{\partial}{\partial x}\bigg[\mu\bigg(-\frac{2}{3}\nabla\cdot\boldsymbol{V}+2\frac{\partial u}{\partial x}\bigg)\bigg]\\+&\frac{\partial}{\partial y}\bigg[\mu\bigg(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\bigg)\bigg]+\frac{\partial}{\partial z}\bigg[\mu\bigg(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\bigg)\bigg] \end{align}$$\begin{align}\rho\frac{Dv}{Dt}=\rho f_y-\frac{\partial p}{\partial y}+&\frac{\partial}{\partial y}\bigg[\mu\bigg(-\frac{2}{3}\nabla\cdot\boldsymbol{V}+2\frac{\partial v}{\partial y}\bigg)\bigg]\\+&\frac{\partial}{\partial x}\bigg[\mu\bigg(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\bigg)\bigg]+\frac{\partial}{\partial z}\bigg[\mu\bigg(\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\bigg)\bigg] \end{align}$$\begin{align}\rho\frac{Dw}{Dt}=\rho f_z-\frac{\partial p}{\partial z}+&\frac{\partial}{\partial z}\bigg[\mu\bigg(-\frac{2}{3}\nabla\cdot\boldsymbol{V}+2\frac{\partial w}{\partial z}\bigg)\bigg]\\+&\frac{\partial}{\partial x}\bigg[\mu\bigg(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\bigg)\bigg]+\frac{\partial}{\partial y}\bigg[\mu\bigg(\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\bigg)\bigg] \end{align}$ Can be simplified by introducing rate of strain tensor:$S_{ij}=\frac{1}{2}\bigg(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\bigg)$ $\rho\frac{Du_i}{Dt}=\rho f_i -\frac{\partial p}{\partial x_i}+\frac{\partial}{\partial x_j}\bigg[2\mu S_{ij}-\frac{2}{3}\mu(\nabla\cdot\boldsymbol{V})\delta_{ij}\bigg]$ For incompressible fluid, constant viscosity $\mu$:$\rho\frac{D\boldsymbol{V}}{Dt}=-\nabla p+\mu\nabla^2\boldsymbol{V}+\rho\boldsymbol{f}$ For inviscid fluid, $\mu = \lambda = 0$, results in the Euler equations:$\rho\frac{D\boldsymbol{V}}{Dt}=-\nabla p+\rho\boldsymbol{f}$ ### Conservation of Energy $\rho\frac{De}{Dt}=-\nabla\cdot\boldsymbol{q}-p(\nabla\cdot\boldsymbol{V})+\dot{Q}$ - $e(x,t)\quad$internal energy per unit mass - $\boldsymbol{q}(x,t)\quad$vector field of heat flux by thermal conduction - represented by Fourier's law:$\boldsymbol{q}=-\kappa\nabla T$ - $T(x,t)\quad$temperature field - $\kappa\quad$thermal conductivity - $\dot{Q}\quad$rate of internal heat generation, caused by friction, radiation, etc. Neglect internal heat generation, assume incompressible (Boussinesq approximation):$\rho C\frac{DT}{Dt}=\rho C\bigg(\frac{\partial T}{\partial t}+\boldsymbol{V}\cdot\nabla T\bigg)=\nabla\cdot(\kappa\nabla T)$ Reduces to classical heat conduction equation if $\kappa$ constant:$\rho C\frac{\partial T}{\partial t} = \kappa\nabla^2T$ ### Equation of State Ideal gas model:$\frac{p}{\rho}=RT,\quad e=e(T)$ Incompressible fluid model:$\rho=const, \quad e=CT$ ### Equations in Integral Form Conservation of mass:$\frac{d}{dt}\int_\Omega\rho d\Omega+\int_S\rho\boldsymbol{V}\cdot\boldsymbol{n}dS =0$ - $\Omega\quad$control volume - $S\quad$boundary of control volume Conservation of momentum:$\frac{d}{dt}\int_\Omega\rho u d\Omega+\int_S\rho u\boldsymbol{V}\cdot\boldsymbol{n}dS=\int_S\boldsymbol{t}_x\cdot\boldsymbol{n}dS+\int_\Omega\rho f_xd\Omega$$\frac{d}{dt}\int_\Omega\rho v d\Omega+\int_S\rho v\boldsymbol{V}\cdot\boldsymbol{n}dS=\int_S\boldsymbol{t}_y\cdot\boldsymbol{n}dS+\int_\Omega\rho f_yd\Omega$$\frac{d}{dt}\int_\Omega\rho w d\Omega+\int_S\rho w\boldsymbol{V}\cdot\boldsymbol{n}dS=\int_S\boldsymbol{t}_z\cdot\boldsymbol{n}dS+\int_\Omega\rho f_zd\Omega$ Full Navier-Stokes for the *i*-component of momentum: $\begin{align}&\frac{d}{dt}\int_\Omega\rho u_id\Omega+\int_S\rho u_i\boldsymbol{V}\cdot\boldsymbol{n}dS\\&=\int_S\bigg[(-p+\lambda\nabla\cdot\boldsymbol{V})n_i+\sum_j\mu\bigg(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\bigg)n_j\bigg]dS+\int_\Omega\rho f_id\Omega\end{align}$ Conservation of energy: - must consider both conduction ($\boldsymbol{q}\cdot\boldsymbol{n}$) and convection ($\rho E\boldsymbol{V}$)$\begin{align}\frac{d}{dt}&\int_\Omega\rho Ed\Omega+\int_S\boldsymbol{q}\cdot\boldsymbol{n}dS+\int_S\rho E\boldsymbol{V}\cdot\boldsymbol{n}dS\\&=\int_S-p\boldsymbol{V}\cdot\boldsymbol{n}dS+\int_\Omega\dot{Q}d\Omega+\int_\Omega\rho\boldsymbol{f}\cdot\boldsymbol{V}d\Omega\end{align}$ - using Boussinesq assumption:$\frac{d}{dt}\int_\Omega\rho CTd\Omega+\int_S\rho CT\boldsymbol{V}\cdot\boldsymbol{n}dS=\int_S\kappa\nabla T\cdot\boldsymbol{n}dS$ Different integral types: - ***Convective flux***: surface integrals that don't contain derivatives of the conserved field - represent transport by velocity through the boundary of the control volume - ***Diffusive flux***: surface integrals that contain first derivatives of the conserved field - represent transport by diffusion, conduction, or viscosity - ***Volume source***: correspond to distributed sources or sinks of the conserved quantity within the control volume - ***Surface force***: represent work by normal surface forces on boundary of the control volume Illustrated in conservation of an arbitrary scalar field $\Phi$:$\underbrace{\frac{d}{dt}\int_\Omega\Phi d\Omega}_\text{Rate of change}+\underbrace{\int_S\Phi\boldsymbol{V}\cdot\boldsymbol{n}dS}_\text{Convective flux}=\underbrace{\int_S\mathcal{X}\nabla\Phi\cdot\boldsymbol{n}dS}_\text{Diffusive flux}+\underbrace{\int_\Omega Qd\Omega}_\text{Volume source}$