When are my fluid approximations wrong? I did some classical approximations of the Navier Stokes equations, fluid is:

*

*non-viscous

*incompressible

*irrotational

When are these approximations wrong? and particularly is there a "general method" to evaluate in a theoretical way "the error" of an approximation?
For example, for a given fluid with a given velocity flow, what will be the order of the terms that I neglect?
I see some methods using dimensionnal analysis, but it wasn't very clear for me...
 A: 
When are these approximations wrong ?

Almost all real fluids will have some viscosity (an exception may be superfluid helium-4) and some degree of compressibility. And there are simple situations where flow is not necessarily irrotational e.g. flow between two concentric cylinders. So one or more of the assumptions of your model are almost always going to be incorrect. The model may still give useful results in some circumstances.
I am not sure of the meaning of the second part of your question, about the "cost" of an approximation.
A: Fluid dynamics has developed a systematic method to easily identify the correct approximations pertaining to different regimes. It is based on a set of dimensionless numbers expressing the typical ratio between different terms in Navier-Stokes, and related equations for the dynamics of fluids. The basic idea is that using typical lengths, velocities, times, etc., as units for the physical quantities appearing in the equations, it is possible to understand which terms can be neglected as a first approximation and possibly be re-introduced in a perturbative way, if necessary.
I suggest you refer to this Wikipedia page for starting information and quite an extensive list of possible dimensionless numbers.
Here I'll briefly illustrate the technique with an example.
Let's assume that we want to understand when finite compressibility plays a role in fluid dynamics. We can start with an equation containing density ($\rho$) variation and the velocity field (${\bf u}$), the continuity equation:
$$
\frac{\partial \rho}{\partial t}+\nabla\cdot \left( {\rho \bf u} \right)=0.
$$
By introducing the material derivative ($\frac{D}{Dt}$) and the equation of state to use pressure as a variable, it may be rewritten as
$$
\frac{1}{\rho c^2}\frac{Dp}{Dt}+\nabla \cdot {\bf u}=0, \tag{1}
$$
where $c$ is the speed of sound. At this point, we can introduce a typical length ($L$), a typical speed of the fluid ($U$), and a typical density ($\bar\rho$), and we can use them as new units. Equation ($1$) becomes:
$$
\frac{U^2}{\rho^* c^2}\frac{Dp^*}{Dt^*}+\nabla \cdot {\bf u}^*=0, \tag{2}
$$
where
$$
\begin{align}
t^* &= \frac{Ut}{L}\\
{\bf u}^*&=\frac{{\bf u}}{U}\\
p^*&=\frac{p}{\bar\rho U^2}\\
\rho^*&=\frac{\rho}{\bar \rho}
\end{align}
$$
and, by introducing the dimensionless Mach number $M=\frac{U}{c}$, we get
$$
\frac{M^2}{\rho^*}\frac{Dp^*}{Dt^*}+\nabla \cdot {\bf u}^*=0, \tag{2}
$$
Therefore, the importance of finite compressibility is encoded in the value of the dimensionless Mach's number. When it is negligible, the flow behaves as incompressible ($\nabla \cdot {\bf u}^* = 0$). If it is large, spatio-temporal variations of density cannot be neglected. Moreover, one could systematically introduce their effect perturbatively. However, we have to take into account that one requires some care from the mathematical point of view since the limit $M \rightarrow 0$ is non-trivial, changing the character of the resulting differential equations (see, for instance, the topic singular perturbation on Wikipedia).
A: I want to focus on the part of the question that deals with viscous effects: The dimensionless version of the conservation of linear momentum is largely equal to its dimensional version, except the term that deals with the stress tensor: In the dimensionless version, the divergence of the stress tensor is divided by the Reynolds number. Thus, for high Reynolds number flows, the shear stresses in the fluid get negligible. (The surface stresses on the boundary do not.) See, for example, Schlichting, Gersten (2017): Boundary-Layer Theory, Section 4.1 "Similarity Laws", equation 4.4.
