potential flow Note that the velocity potential is undefined to an arbitrary

Potential flow note that the velocity potential is

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potential flow . Note that the velocity potential is undefined to an arbitrary additive constant. We have demonstrated that a velocity potential necessarily exists in a fluid whose velocity field is irrotational. Conversely, when a velocity potential exists the flow is necessarily irrotational. This follows because [see Equa- tion (A.176)] ω = ∇ × v = −∇ × ∇ φ = 0 . (5.30) Incidentally, the fluid velocity at any given point in an irrotational fluid is normal to the constant- φ surface that passes through that point. If a flow pattern is both irrotational and incompressible then we have v = −∇ φ (5.31) and ∇ · v = 0 . (5.32) These two expressions can be combined to give (see Section A.21) 2 φ = 0 . (5.33) In other words, the velocity potential in an incompressible irrotational fluid satisfies Laplace’s equation. According to Equation (5.24), if the flow pattern in an incompressible inviscid fluid is also irrotational, so that ω = 0 and v = −∇ φ , then we can write parenleftBigg p ρ + 1 2 v 2 + Ψ ∂φ t parenrightBigg = 0 , (5.34) which implies that p ρ + 1 2 v 2 + Ψ ∂φ t = C ( t ) , (5.35)
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Incompressible Inviscid Fluid Dynamics 83 P C B A Figure 5.4: Two-dimensional flow. where C ( t ) is uniform in space, but can vary in time. In fact, the time variation of C ( t ) can be eliminated by adding the appropriate function of time (but not of space) to the velocity potential, φ . Note that such a procedure does not modify the instantaneous velocity field v derived from φ . Thus, the above equation can be rewritten p ρ + 1 2 v 2 + Ψ ∂φ t = C , (5.36) where C is constant in both space and time. Expression (5.36) is a generalization of Bernoulli’s theorem (see Sec- tion 5.3) that takes non-steady flow into account. However, this generalization is only valid for irrotational flow. For the special case of steady flow, we get p ρ + 1 2 v 2 + Ψ = C , (5.37) which demonstrates that for steady irrotational flow the constant in Bernoulli’s theorem is the same on all streamlines. (See Section 5.3.) 5.8 Two-Dimensional Flow Fluid motion is said to be two-dimensional when the velocity at every point is parallel to a fixed plane, and is the same everywhere on a given normal to that plane. Thus, in Cartesian coordinates, if the fixed plane is the x - y plane then we can express a general two-dimensional flow pattern in the form v = v x ( x ,y, t ) e x + v y ( x ,y, t ) e y . (5.38) Let A be a fixed point in the x - y plane, and let ABP and ACP be two curves, also in the x - y plane, that join A to an arbitrary point P . See Figure 5.4. Suppose that fluid is neither created nor destroyed in the region, R (say), bounded by these curves. Since the fluid is incompressible, which essentially means that its density is uniform and constant, fluid continuity requires that the rate at which the fluid flows into the region R , from right to left across the curve ABP , is equal to the rate at which it flows out the of the region, from right to left across the curve ACP . Now, the rate of fluid flow across a surface is generally termed the flux . Thus, the flux (per unit length parallel to the z -axis) from right to left across ABP
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