4.1.6. Kinematics

4.1.6.1. What is the main difference between kinematics and dynamics?

  • Kinematics:

    • Quantities involving space and time only, e.g.
      • Position
      • Velocity
      • Acceleration
      • Deformation
      • Rotation
    • i.e. the geometry of fluid motion

  • Dynamics:

    • Refers to the stresses and forces that cause fluid motion
    • i.e. the conservation laws

4.1.6.2. What are the different types of motion exhibited by fluid elements?

  • Translation \(\vec{u}\)
  • Rotation \({1 \over 2} (\nabla \times \vec{u}) \times d\vec{r}\)
  • Angular deformation or linear deformation \(\vec{s} \cdot d\vec{r}\)
\[\vec{u} + {1 \over 2} (\nabla \times \vec{u}) \times d\vec{r} + \vec{s} \cdot d \vec{r}\]

4.1.6.3. How can fluid flow be decomposed into translation, deformation and rotation?

\[\vec{u} = \vec{u} + \overset{\underset{\mathrm{\rightrightarrows}}{}}{e} \cdot d\vec{c} + \overset{\underset{\mathrm{\rightrightarrows}}{}}{\omega} \times d\vec{c} = translation + deformation + rotation\]

where:

  • \(\vec{u}\) = velocity vector
  • \(\overset{\underset{\mathrm{\rightrightarrows}}{}}{e}\) = strain rate tensor
  • \(\vec{c}\) = line element vector
  • \(\overset{\underset{\mathrm{\rightrightarrows}}{}}{\omega}\) = vorticity tensor

4.1.6.4. How to decompose the velocity gradient tensor into symmetric and anti-symmetric parts?

\[{{\partial u_j} \over {\partial x_i}} = {1 \over 2} \left( {{\partial u_j} \over {\partial x_i}} + {{\partial u_i} \over {\partial x_j}} \right) + {1 \over 2} \left( {{\partial u_j} \over {\partial x_i}} - {{\partial u_i} \over {\partial x_j}} \right) = e_{ij} + \omega_{ij} = symmetric \ tensor + antisymmetric \ tensor\]

\({{\partial u_j} \over {\partial x_i}}\) appears in the Navier-Stokes equations (shear strain rate tensor).

\[e_{ij} = {1 \over 2} \left( {{\partial u_j} \over {\partial x_i}} + {{\partial u_i} \over {\partial x_j}} \right)\]

(eventually this becomes the shear stress)

4.1.6.5. What is the rate of strain tensor or deformation tensor?

\[\begin{split}\overset{\underset{\mathrm{\rightrightarrows}}{}}{e} = {1 \over 2} \left( \nabla \otimes \vec{u} + (\nabla \otimes \vec{u})^T \right) = {1 \over 2} \left( \nabla_i u_j + \nabla_j u_i \right) = \begin{bmatrix} {\partial u \over \partial x} & {1 \over 2}\left( {\partial v \over \partial x} + {\partial u \over \partial y} \right) & {1 \over 2}\left( {\partial w \over \partial x} + {\partial u \over \partial z} \right) \\ {1 \over 2}\left( {\partial u \over \partial y} + {\partial v \over \partial x} \right) & {\partial v \over \partial y} & {1 \over 2}\left( {\partial w \over \partial y} + {\partial v \over \partial z} \right) \\ {1 \over 2}\left( {\partial u \over \partial z} + {\partial w \over \partial x} \right) & {1 \over 2}\left( {\partial v \over \partial z} + {\partial w \over \partial y} \right) & {\partial w \over \partial z} \end{bmatrix}\end{split}\]

4.1.6.6. What is the vorticity tensor?

\[\begin{split}\overset{\underset{\mathrm{\rightrightarrows}}{}}{\omega} = {1 \over 2} \left( \nabla \otimes \vec{u} - (\nabla \otimes \vec{u})^T \right) = {1 \over 2} \left( \nabla_i u_j - \nabla_j u_i \right) = \begin{bmatrix} 0 & {1 \over 2}\left( {\partial v \over \partial x} - {\partial u \over \partial y} \right) & {1 \over 2}\left( {\partial w \over \partial x} - {\partial u \over \partial z} \right) \\ {1 \over 2}\left( {\partial u \over \partial y} - {\partial v \over \partial x} \right) & 0 & {1 \over 2}\left( {\partial w \over \partial y} - {\partial v \over \partial z} \right) \\ {1 \over 2}\left( {\partial u \over \partial z} - {\partial w \over \partial x} \right) & {1 \over 2}\left( {\partial v \over \partial z} - {\partial w \over \partial y} \right) & 0 \end{bmatrix}\end{split}\]

4.1.6.7. What is represented by the symmetric and anti-symmetric parts of the velocity gradient tensor?

  • symmetric part \(\longrightarrow\) angular deformation \(\longrightarrow\) shear
  • anti-symmetric part \(\longrightarrow\) rotation \(\longrightarrow\) vorticity