Recall our discussion on internal fluid forces in . Here and in the next two sections we consider the explicit form of the shear stresses and in particular the deviatoric stress matrix. This is necessary if we want to consider/model any real fluid, i.e. non-ideal fluid. We explain shear stresses as follows – see , p. 31).

The well-posedness of smooth solutions to the three-dimensional incompressible Navier–Stokes equations globally in time is a major open mathematical problem. Our goal in this chapter is to provide a succinct though comprehensive introduction to the main well-posedness results that are known. In the three-dimensional case, we indicate how smooth solutions may develop singularities in finite time. At the same time we establish classical regularity assumptions/conditions that guarantee well-posedness globally in time, i.e. if we could prove three-dimensional incompressible Navier–Stokes solutions satisfied those assumptions/conditions, then we would establish global regularity.

Herein we focus on two-dimensional irrotational flows which have a rich structure in the sense that they are intimately connected to complex variable theory and analysis.

Let us consider the forces that act on a small parcel of fluid in a fluid flow. There are two types:

1. 1. External or body forces, these may be due to gravity or external electromagnetic fields. They exert a force per unit volume on the continuum.

2. 2. Surface or stress forces, these are forces, molecular in origin, that are applied by the neighbouring fluid across the surface of the fluid parcel.

The Reynolds numbers associated with flows past aircraft or ships are typically large, indeed of the order of ${10}^8$ of ${10}^9$; recall Remark 4.36 and . For individual wings or fins, the Reynolds number may be an order of magnitude or two smaller. However, such Reynolds numbers are still large and the flow around a wing for example is well approximated by Euler flow. We can imagine the flow over the top of the wing of an aircraft has a high relative velocity tangential to the surface wing directed towards the rear edge of the wing.

A material exhibits flow if shear forces, however small, lead to a deformation which is unbounded – we could use this as a definition of a fluid. A solid has a fixed shape, or at least a strong limitation on its deformation when force is applied to it. Within the category of ‘fluids’, we include liquids and gases. The main distinguishing feature between these two fluids is the notion of compressibility. Gases are usually compressible – as we know from everyday aerosols. Liquids are generally incompressible – a feature essential to all modern car brakes. However, some gas flows can also be incompressible, particularly at low speeds.

Consider the homogeneous incompressible Navier–Stokes equations with no body force in dimensionless form: