Consider the following mathematical object constructed out of tensors:
$$\mathbf{Re} = \frac{\rho \vec{v} \cdot\nabla \vec{v}}{\mu \nabla \cdot \nabla \vec{v}}$$
$\rho$ and $\mu$ are $(0,0)$ tensors and $\vec{v}$ is a $(1,0)$ tensor.
Is this object a tensor, i.e. does it transform like a tensor?
For clarity, if I fixed an orthonormal coordinate system to ignore raising/lowering indexes, my expression for the components of $\mathbf{Re}$ would be the following in index notation, following the Einstein summation convention:
$$Re_{ij} = \frac{\rho v_k \frac{\partial v_k}{\partial x_i}}{\mu \frac{\partial^2 v_j}{\partial x_l \partial x_l}}$$