Consider a fluid of constant density ρ and viscousity μ. The flow is assumed to be governed by the Navier-Stokes equation:

ρ(∂v/∂t+(v·∇)v)=-∇p+μ∆v
and the continuity (incompressibility) condition:
∇·v=0
where v denotes the velocity and p is the pressure. Let S0 denotes the surface of the body which moves with constant velocity in one direction. In this case, show that the most commonly used force formula (on the body) is;
∬ pn dA + μ∬ n×ω dA
S0 S0

where ω is the vorticity vector and n denotes unit normal vector to the body surface.

To show that the most commonly used force formula on a body moving with constant velocity in one direction is given by:

∬ pn dA + μ∬ n×ω dA

where ω is the vorticity vector and n denotes the unit normal vector to the body surface, we need to analyze the forces acting on the body.

The force acting on a body immersed in a fluid can be decomposed into two components: the pressure force and the viscous force.

Let's start by considering the pressure force. The pressure force acting on an infinitesimal surface element dA can be calculated using the formula pn, where p is the pressure and n is the unit normal vector to the surface.

The total pressure force on the body can be obtained by integrating the pressure force over the entire surface S0:

∬ pn dA

Next, let's analyze the viscous force. The viscous force arises due to the friction between the fluid layers moving at different velocities. This force can be described using the stress tensor, which relates the rate of deformation of the fluid to the stress within the fluid.

In this particular case, the viscous force formula involves the vorticity vector ω. The vorticity vector describes the local rotation of fluid particles within the flow.

The viscous force acting on an infinitesimal surface element dA can be calculated as n×ω, where n is the unit normal vector to the surface and ω is the vorticity vector.

The total viscous force on the body can be obtained by integrating the viscous force over the entire surface S0:

∬ n×ω dA

Therefore, the most commonly used force formula on the body, considering both pressure and viscous forces, is:

∬ pn dA + μ∬ n×ω dA

where ω is the vorticity vector and n denotes the unit normal vector to the body surface.