Fluid Element Kinematics

We want to form a mathematic description of the motion of fluid elements moving in a flow field. A small fluid element in the shape of a cube which is initially in one position will move to another position during a short time interval $\delta t$. We expect the element not only to translate but also have its volume changed (linear deformation), rotate, and undergo change in shape (angular deformation).

Recall that the velocity field can be described by specifying the velocity $\mathbf{V}$ at all points, and we can write velocity as

$$\mathbf{V}=u\mathbf{i}+v\mathbf{j}+w\mathbf{k}$$

and the acceleration can be written as

$$\mathbf{a}=\frac{D\mathbf{V}}{Dt}=\frac{\partial \mathbf{V}}{\partial t}+u\frac{\partial \mathbf{V}}{\partial x}+v\frac{\partial \mathbf{V}}{\partial y}+w\frac{\partial \mathbf{V}}{\partial z}$$

Linear Motion and Deformation

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