I am searching the precise and mathematical definition of a fluid for a long time but I did not find it anywhere. What I mean by precise and mathematical can be understood by the following:
There is a subject "Topology". One can define what one means by a topological space or a topology. There is Group Theory. One can define what one means by a group. But there is Fluid Dynamics or Fluid Mechanics. So, what does one mean by fluid precisely and mathematically?
On
What defines the mechanics of a material mathematically is the constitutive equation describing the relationship between stress and motion of the material. The most common type of fluids are Newtonian. For the case of an in-compressible Newtonian fluid (water, oil etc.) the constitutive relationship is as follows: $\mathbf{\sigma}_{ij}= - p \delta_{ij} + \mu\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i} \right)$
Anything that satisfies the axioms of fluid mechanics is, for that purpose, a fluid. The modern approach is not to define what something is in terms of simpler things, but rather to say what properties, i.e., axioms, does something satisfy. After all, that is all we care about. It's the properties of something that make it what it is. Its internal composition is irrelevant.