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