Demonstration #4: Pressure and Force

    At the outset, we noted that these vector fields are the results of the flow of an inviscid, incompressible fluid. Such fluids satisfy Euler's equations; these equations have two variables- the fluid velocity Overscript[v, ⇀] and the fluid pressure p. If we write Overscript[v, ⇀] = (v_1, v_2), then the equations are
        v_1∂Overscript[v, ⇀]/∂x + v_2∂Overscript[v, ⇀]/∂y + ∇p = 0,
        ∂v_1/∂x + ∂v_2/∂y = 0.
In addition to these auditions, the velocity field must (and does) satisfy appropriate boundary conditions. If C is any boundary of the flow, then Overscript[v, ⇀] · Overscript[n, ⇀] = 0 on C, where Overscript[n, ⇀] is the outward normal to C.
    If the velocity field is also conservative, meaning ∂v_1/∂y = ∂v_2/∂x, then the pressure can be determined directly from the velocity. Indeed, taking the first component of the first equation, we find
        v_1∂v_1/∂x + v_2∂v_1/∂y + ∂p/∂x = v_1∂v_1/∂x + v_2∂v_2/∂x + ∂p/∂x = 0
so that
        ∂/∂x (1/2v_1^2 + 1/2v_2^2 + p) = 0.
A similar argument for the second component shows that
        ∂/∂y (1/2v_1^2 + 1/2v_2^2 + p) = 0,
and hence we have
        p = p_0 - 1/2 | Overscript[v, ⇀] |^2.
This fact is called Bernoulli's Equation. Because only the gradient of the pressure appears in Euler's equations, the pressure is only determined up to an additive constant p_0.
    All of the flows we have considered are conservative, as they all can (locally) be written as the gradient of some potential.    
    Because we are using fluid flows that neglect the effects of friction and viscosity, the only way that the fluid exerts a force on an object is through the pressure. In particular, if Ω is an obstacle in the path of a fluid with boundary curve C, then the force that the fluid exerts on the obstacle is
        Overscript[F, ⇀] = -∲_C p Overscript[n, ⇀] ds
where Overscript[n, ⇀] is the outward normal to the obstacle.

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