Question

A fluid flow is given by : V = 10x^3 i - 8x^3 yj. Find the shear strain rate and state whether the flow is rotational or irrotationsl.

Answer 1

$$V=10x^3 \hat{i}-8x^3y\hat{j}$$

Gradient of velocity which is a second rank tensor is

$$L^T=\vec\nabla \vec V=\begin{bmatrix} \cfrac{dV_x}{dx} & \cfrac{dV_x}{dy} \\ \cfrac{dV_y}{dx} & \cfrac{dV_y}{dy} \end{bmatrix}$$

$$L^T=\begin{bmatrix} 30x^2 & 0 \\ -24x^2y & -8x^3 \end{bmatrix}$$

Taking Its transpose

$$L=\begin{bmatrix} 30x^2 & -24x^2y \\ 0 & -8x^3 \end{bmatrix}$$

Now strain rate tensor:

$$E=\cfrac{L+L^T}{2}\\ E=\cfrac{1}{2}\begin{bmatrix} 60x^2 & -24x^2y \\ -24x^2y & -16x^3 \end{bmatrix}$$

where, off-diagonal terms represent shear strain rate.

Now Spin Tensor:

$$W=\cfrac{L-L^T}{2}\\ W=\cfrac{1}{2}\begin{bmatrix} 0 & -24x^2y \\ 24x^2y & 0 \end{bmatrix}$$

Since off-diagonal term in spin tensor are non zero hence flow is rotational.