According to the book written by Pope, the vorticity equations exhibit material-frame indifference when the flow is two-dimensional and the rotations of the frame are steady. But how can we judge the Navier-Stokes equations possess material-frame indifference?
When the frame rotates, the Coriolis force in the fictitious force is non-zero. It seems that the fictitious force can't be absorbed into modifier pressure, which follows the Navier Stokes equations do not have material-frame indifference.
Just consider the vanishing time derivative in the last term of (2.100) and rememember that in 2D the vorticity is constrained to have only the component normal to the x1,x2 plane (no stretching term).
The vorticity equation is nothing but the momentum equation under curl application and satisfies the divergence-free constraint. Actaully, that is only a different form of the NSE.
Material indifference means that the mathematical formulation is not altered by a change of reference frame. The Navier-Stokes equation involves scalar (time, pressure, density), vector (velocity, force fields), and tensor quantities (stresses). NS equation is a vectorial equation that is expressed using mathematical operators (vector, gradients of scalars or divergence of tensors of order 2), the formulations of which are by definition independent of the reference frame. Hence the property of material indifference of NS equation. This is valid whatever the dimension considered: 1D, 2D or 3D
The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluid substances, accounting for both inertia and viscous effects. These equations govern various fluid flow phenomena, such as the motion of air and water.
Material-frame indifference, also known as Galilean invariance or objectivity, refers to the property of a physical theory or equations that remains unchanged under coordinate transformations between different observers or reference frames. In the context of fluid dynamics, it means that the equations describing fluid flow should produce the same results regardless of the reference frame chosen to observe the flow.
For two-dimensional flow, the Navier-Stokes equations can be written as:
∂u/∂t + u ∂u/∂x + v ∂u/∂y = -1/ρ ∂p/∂x + ν (∂²u/∂x² + ∂²u/∂y²)
∂v/∂t + u ∂v/∂x + v ∂v/∂y = -1/ρ ∂p/∂y + ν (∂²v/∂x² + ∂²v/∂y²)
where u and v are the velocity components in the x and y directions, respectively, t is time, p is the pressure, ρ is the fluid density, ν is the kinematic viscosity, and ∂/∂x and ∂/∂y represent partial derivatives with respect to x and y.
In two-dimensional flow, the velocity components are functions of only two variables, say x and y, while time t remains as a parameter. This implies that the velocity field is independent of the z-coordinate. Consequently, the equations do not contain any terms that explicitly depend on z or involve the ∂/∂z derivative.
The absence of z-dependence makes the equations invariant under rotations about the z-axis, which means that the equations will yield the same results regardless of the orientation of the coordinate system in the z-direction. This property of the equations leads to material-frame indifference for two-dimensional flow, as the equations remain unchanged under coordinate transformations or rotations about the z-axis.
It is important to note that this material-frame indifference property does not hold for three-dimensional flow, as the Navier-Stokes equations for three-dimensional flow involve additional terms that explicitly depend on the z-coordinate and do not exhibit the same rotational symmetry around the z-axis.
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