Vectors, Tensors and the Basic Equations of Fluid Mechanics
Agenda
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Vectors and Tensors: The Basics
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The Kinematics of Fluid Motion
Vectors and Tensors
The mathy bits
\bold{a}=a_x \hat{i}+a_y \hat{j}+a_z \hat{k}
Vector Notation
\(\bold{x} \times \bold{y} = (x_j y_k-x_k y_j )\hat{i} + (x_k y_i-x_i y_k)\hat{j} + (x_iy_j-x_jy_i)\hat{k}\)
Cross Product
\bold{x} \times \bold{y}: \left[ {\begin{array}{cc}
\hat{i}&\hat{j}&\hat{k}\\
x_i&x_j&x_k\\
y_i&y_j&y_k
\end{array} } \right]
What is a Tensor?
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components
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n base units
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Changes with the coordinate system
3^{n-1}
\bold{n}=\left[ {\begin{array}{cc}
n_x\\
n_y\\
n_z
\end{array} } \right]
\bold{m}=\left[ {\begin{array}{cc}
n_{xx}&n_{xy}&n_{xz}\\
n_{yx}&n_{yy}&n_{yz}\\
n_{zx}&n_{zy}&n_{zz}
\end{array} } \right]
Second Rank Tensor (Vector)
Third Rank Tensor
Text
J=\frac{\delta (x,y)}{\delta (r,\theta)}=det(\left[ {\begin{array}{cc}
\frac{\delta x}{\delta r}&\frac{\delta x}{\delta \theta}\\
\frac{\delta y}{\delta r}&\frac{\delta y}{\delta \theta}
\end{array} } \right])
=\frac{\delta x}{\delta r} \cdot \frac{\delta y}{\delta \theta} - \frac{\delta x}{\delta \theta} \cdot \frac{\delta y}{\delta r}
The Jacobian
The Kinematics of Fluid Motion
Continuum Mechanics
\bold{x}=\bold{x}(\bold{\sigma},t)
where \(\sigma\) is the inital postion vector
x_i=x_i (\sigma_x,\sigma_y,\sigma_z,t)
\bold{\sigma}=\bold{\sigma}(\bold{x},t)
\sigma_i=\sigma_i (x_x,x_y,x_z,t)
Material Properties:
Spatial
Properties:
p(\bold{x},t)=p(\sigma(\bold{x},t),t)
The property of the material at position \(\bold{x}\) and time \(t\) is the value for a particle which is at \(\bold{x}\) at time \(t\) !
dV=JdV_0
Dilation
Vectors and the Basic Equations of Fluid Mechanics
By Dalton Scott
Vectors and the Basic Equations of Fluid Mechanics
A brief presentatation on Vectors and the Basic Equations of Fluid Mechanics
