List of Symbols/Notation#
This page contains a list of symbols used in all the following chapters
Notation#
a or \(a\) - scalar
\(\mathbf{v}\) or \(\bar{\mathbf{v}}\) - vector
\(\mathbf{T}\) - tensor of rank > 1
\(\bar{\bar{\mathbf{T}}}\) or \(\underline{\underline{\mathbf{T}}}\) - rank 2 tensor
\(\mathbf{\hat{e}}_v\) - unit vectors along direction of \(\mathbf{v}\)
\(\mathbf{\hat{n}}\) - unit vector in direction of outward facing normal for a plane
General#
\(\rho\) - Density (\(kg/m^3\))
\(M\) - Mass (\(kg\))
\(L\) - Length (\(m\))
\(t\) - Time (\(s\))
\(T\) - Temperature (\(K\) or \(^o C\))
\(\mathbf{v}\) - Velocity (\(m/s\))
Intro to Vector/Tensor Calculus#
\(\nabla\) - Del Operator
\(\nabla a\), \(\nabla \mathbf{v}\) - Gradient of scalar \(a\), gradient tensor of vector \(\mathbf{v}\)
\(\nabla \cdot \mathbf{v}\) - Divergence of vector \(\mathbf{v}\)
\(\nabla \times \mathbf{v}\) - Curl of vector \(\mathbf{v}\)
\(\nabla \cdot \nabla\)= \(\Delta\) - Laplacian
\(\delta_{ij}\) - Kronecker delta
\(\epsilon_{ijk}\) - Permutation symbol, Levi-Civita tensor
\(d/d t\) - derivative to time (when no dependence on other parameters)
\(\partial/\partial t\) - partial derivative to time
Stress and Tensors#
\(\bar{\bar{\mathbf{\sigma}}}\) - Stress tensor (\(N/m^2\))
\(\mathbf{t}\) - Traction, stress vector (\(N/m^2\))
\(\mathrm{tr}(\bar{\bar{\mathbf{\sigma}}})\) = \(p\) - trace of the stress tensor = pressure (\(N/m^2\))
\(\bar{\bar{\mathbf{\sigma'}}}\) - Deviatoric stress (\(N/m^2\))
Kinematics#
\(D/D t\) - material time derivative
\(\mathbf{a}\) - Acceleration vector (\(m/s^2\))
\(\mathbf{v}\) or \(\mathbf{v}'\) - Velocity vector (Eulerian or Material) (\(m/s\))
\(\mathbf{x}\) or \(\bf{\xi}\) - Position vector (Eulerian or Material) (\(m\))
\(\mathbf{u}\) - Displacement vector (\(m\))
\(\nabla \mathbf{u}\) - Displacement gradient tensor (\(unitless\))
\(\bar{\bar{\mathbf{\omega}}}\) - Infinitesimal strain rigid body rotation tensor (\(unitless\))
\(\bar{\bar{\mathbf{\varepsilon}}}\) - Infinitesimal strain internal deformation/strain tensor (\(unitless\))
Conservation Equations#
\(\mathbf{F}\) - Body force (\(N\))
\(\mathbf{f}\) - Body force per unit volume (\(N/m^3\))
\(\mathbf{q}\) - Heat flux (\(W/m^2\))
\(k\) - Thermal conductivity (\(W\))
\(H\) - Heat production per unit mass (\(W/kg\))
\(A\) - Heat production per unit volume (\(W/m^3\))
\(C_P\) - Heat capacity at constant pressure (\(J/kg K\))
\(\kappa\) = \(k/\rho/C_P\) - Thermal diffusivity (\(m^2/s\))
\(\mathbf{D}\) - Infinitesimal-strain strain rate tensor (\(s^{-1}\))
\(\lambda\) and \(\mu\) - elastic Lamé constants (\(N/m^2\))
\(K\) - Elastic bulk modulus (\(N/m^2\))
\(G\) = \(\mu\) - Elastic shear modulus \(N/m^2\)
\(E\) - Elastic Youngs modulus \(N/m^2\)
\(\nu\) - Elastic Poisson’s ratio
\(p\) - Hydrostatic pressure \(N/m^2\)
\(\zeta\) - Bulk viscosity \(Pa \hspace{0.1cm} s\) = \(N s/m^2\)
\(\eta\) - Shear viscosity \(Pa \hspace{0.1cm} s\)
\(\nu\)= \(\eta/\rho\) - Kinematic viscosity \(J s / kg\)
\(\mathbf{g}\) - Acceleration due to gravity \(m/s^2\)
Dimensional Analysis#
\(D\) - Diffusivity \(m^2/s\)
\(C\) - Concentration
\(H\) - Thickness \(m\)
\(h\) - External heat transfer coefficient
\(Pe\) - Peclet number
\(Re\) - Reynolds number
\(Ga\) - Galileo number
\(Fr\) - Froude number
\(\lambda\) - Wavelength of an instability \(m\)
\(\gamma\) - Surface tension \(N/m\)
\(Q\) - Rate of energy transfer \(W\)
\(V\) - Volumetric fluid flow rate \(m^3/s\)
Potential Flow#
\(\phi\) - Potential
\(k\) - Permeability
\(F\) - Flux
\(\xi\) - Residual
Turbulent and Non-Newtonian Flows#
\(f\) - Fanning Friction Factor
\(\mathbf{u}\) - Instantaneous velocity \(m/s\)
\(\mathbf{U}\) - Average velocity \(m/s\)
\(\mathbf{u^{'}}\) - Fluctuating component of velocity \(m/s\)
\(k\) - Turbulent kinetic energy \(J\)
\(\varepsilon\) - Turbulent dissipation rate
Interpolation Quadrature#
\(P_N(x)\) - Interpolated polynomial
\(\alpha_i\) - Weight for interpolation function
\(\phi_i\) - Basis functions for interpolation function
\(L(x)\) - Lagrange polynomial
\(l_i(x)\) - Lagrange basis polynomial
\(R_N(x)\) - Remainder function/Interpolation error
\(I_M\) - Integral by using midpoint rule
\(I_T\) - Integral by using trapezoidal rule
\(I_S\) - Integral by using Simpson’s rule
\(I_W\) - Integral by using Weddle’s rule
ODE Solvers#
\(t_0\) - Initial time
\(T\) - Upper limit of integral in time
\(\Delta t_n\) - Step size
\(\mathcal{O}\) - Big O approximation for higher order terms
\(\bar{f}\) - Average value of a function
\(\tau\) - Local truncation error
\(E\) - Global error
PDE Solvers#
\(c\) - Concentration
\(\kappa\) - Diffusivity coefficient
\(\xi\) - Computational coordinate
\(C\) - Courant number
\(r\) - r-parameter
\(k_m\) - Wavenumber
Finite Element Methods#
\(e\) - Single element
\(C\) - Constant
\(\phi\) - Basis function
\(\alpha\) - Weights
\(P_h\) - \(L^2\) norm
\(K\) - Discretisation matrix