The Navier-Stokes equation is used to describe the motion of a viscous fluid and was named after Claude-Louis Navier, a French scientist, and George Stokes, an Irish scientist. To derive the equation, let’s consider the flow of an incompressible fluid through an infinitesimal cubic control volume* depicted in the diagram below.
* the control volume here refers to the volume of an object in the fluid.
The x-component of the velocity of the fluid is a function of x, y, z and t, i.e. u_{x} (x, y, z, t) where t is time. The total differential of u_{x }is:
Dividing the equation by dt and letting dx/dt = u_{x}, dy/dt = u_{y}, dz/dt = u_{z}, we have:
Let’s analyse the hydrostatic and hydrodynamic forces acting on each face of the infinitesimal control volume in the x-direction (see diagram above where point C is the origin). They are summarised as follows:
Face |
Force per unit area |
Force |
BDFH |
σ_{xx}(x + dx) | σ_{xx}(x + dx)dydz |
ACEG |
σ_{xx}(x) | σ_{xx}(x)dydz |
ABHG |
τ_{yx}(y + dy) | τ_{yx}(y + dy)dxdz |
CDFE |
τ_{yx}(y) | τ_{yx}(y)dxdz |
ABDC |
τ_{zx}(z) | τ_{zx}(z)dxdy |
GHFE |
τ_{zx}(z + dz) | τ_{zx}(z + dz)dxdy |
σ_{xx} = F/A , i.e. the force F acting on a unit area of the x-plane in the x-direction, thereby the double-x subscript. The notation σ_{xx}(x) refers to σ_{xx} evaluated at a distance of (x) while σ_{xx}(x + dx) is evaluated at a distance of (x + dx). A different notation τ is given to the forces that act parallel to the faces of the control volume, e.g. τ_{yx}(y + dy) is the force acting on a unit area of the y-plane in the x-direction that is evaluated at a distance of (y + dy).
With reference to the defined directions of the forces, F_{H }the sum of hydrostatic and hydrodynamic forces acting on the infinitesimal control volume in the positive x-direction is:
In addition to hydrostatic and hydrodynamic forces, an inertia force F_{g} (gravitational force) also acts on the infinitesimal control volume in the x-direction:
where ρ is the density of the fluid and g_{x} is the x-component of the acceleration due to gravity.
The total force F_{T }acting on the infinitesimal control volume in the x-direction is therefore:
Next, by i) substituting F_{H} and F_{g} and eq24 in eq25; ii) dividing the substituted equation by dxdydz and iii) letting dx, dy, dz → 0 for an infinitesimal control volume, we have:
Using the same logic and repeating the above steps, we have for the y-direction and the z-direction:
From the articles on constitutive relation, . We therefore substitute , and in eq26, noting that (see eq14) to give:
Using the same logic and repeating the above step for eq27 and eq28, we have:
Eq29, eq30 and eq31 are the Cartesian form of the Navier-Stokes equations, which when multiplied throughout by the unit vectors , and respectively and summed, becomes the vector form:
where
is the convective derivative defined as
is the gradient defined as
is the vector Laplacian defined as
is the viscosity of the fluid