Generalised Bernoulli Equation Flashcards Preview

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Flashcards in Generalised Bernoulli Equation Deck (29)
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1
Q

What does dv/dt equal for irrotational flow?

A

dv/dt = -∇(P/ρ + gz + v^2/2)

2
Q

What is the generalised Bernoulli equation?

A

dФ/dt + v^2/2 + P/ρ + gz = const

3
Q

In cylindrical polars, what does r(hat) equal?

A

r(hat) = cosθ i + sinθ j

4
Q

In cylindrical polars, what does θ(hat) equal?

A

θ(hat) = -sinθ i + cosθ j

5
Q

In cylindrical polars, what does z(hat) equal?

A

z(hat) = k

6
Q

What is the potential vortex?

A

A flow with circular paths around an axis such that ∇v = 0

7
Q

What does ∇v = 0 equal in cylindrical coordinates?

A

1/r d/dr(rv(θ)) - 1/r *d/dθ(v(r)) = 0

8
Q

For the potential vortex, what does the cylindrical coordinates version of ∇v = 0 equal and why?

A

Potential vortex only has v(θ)(r), so ∇v = 0 -> d/dr(r*v(θ)) = 0, so v(θ) = const/r = K/2πr

9
Q

What is the equation for v?

A

v = v(θ) θ(hat) = v(θ)*(-sinθ i + cosθ j)

10
Q

What is the equation for l, the vector to the path?

A

l = rcosθ i + rsinθ j

11
Q

What is the closed integral of v.dl equal to?

A

integral from 0 to 2π of v.dl/dθ dθ, then sub in the differentiated equation for l and the equation for v, and simplifies to equal K

12
Q

What does the closed integral of v.dl show us?

A

That circulation around loop enclosing axis is not zero, and circulation around loop not enclosing axis is zero as ∇v = 0.

13
Q

What is the equation for P far from the centre?

A

P = P0 - 1/2ρ(k^2/4π^2r^2)

14
Q

What is Ф equal to for the potential vortex v = k/2πr θ(hat)?

A

v = ∇Ф, so Ф = K/2π θ (because ∇Ф = dФ/dr + 1/r *dФ/dθ + dФ/dx in cylindrical)

15
Q

What is Ф equal to for uniform flow?

A

v = v i(hat), so Ф = v(x)

16
Q

What is Ф equal to for a point source/sink?

A

v = 2/2πr r(hat), so Ф = 2/2π *ln(r)

17
Q

For potential flow over a cylinder, what is the first stage of the 3 stage process to build the flow?

A

Have a source-sink pair (flowing out of one point on left and into other point on right)

18
Q

For potential flow over a cylinder, what is the second stage of the 3 stage process to build the flow?

A

Add a uniform flow from left to right to the diagram. Will be a stagnation point at each end where the flows flow into eachother. Closed streamline joins stagnation points and acts like a solid object.

19
Q

For potential flow over a cylinder, what is the third stage of the 3 stage process to build the flow?

A

As source/sink brought together, the closed streamline approximates a circle.

20
Q

What does the diagram of the source sink pair look like after the third stage?

A

Triangle split in 2 with θ1 on left, θ in middle and θ2 on right (outside of triangle). Same with r1, r and r2, and 2a across top. Point P joining r1 and r2.

21
Q

What is the equation for the added potentials of the source and sink at point P?

A

Ф = q/2π *ln(r1) - q/2π *ln(r2) = q/2π *ln(r1/r2) = q/4π *ln(r1^2/r2^2)

22
Q

What is the equation for r1, the distance from the source to the point P?

A

r1^2 = r^2*sin^2(θ) + (rcos(θ)+a)^2

23
Q

What can we approximate the equations for r1 and r2 to?

A

With r&raquo_space; a, can approximate r1^2 = r^2 +2racosθ, r2^2 = r^2 - 2racosθ

24
Q

What is the equation for the potential Ф after simplifying r1 and r2?

A

Ф = v(x) + qa/π *cosθ/r, as we add the potential for the uniform flow.

25
Q

What is the final equation for the potential Ф?

A

Ф = vr(1+R^2/r^2)*cosθ, since x = rcosθ, with R^2 = qa/πv

26
Q

How do we find v(r) and v(θ)?

A

v(r) = dФ/dr, v(θ) = 1/r dФ/dθ ,so v(r) = v(1-R^2/r^2)cosθ and v(θ) = -v(1+R^2/r^2)*sinθ

27
Q

How can we approximate v(r) and v(θ) for a cylinder surface?

A

r = R, so v(r) = 0 and v(θ) = -2v*sinθ

28
Q

What is the equation for P(s), the pressure on the surface of a cylinder?

A

P(s) = P0 + 1/2ρv^2(1-4sin^2(θ))

29
Q

What is the equation for the net force on the cylinder?

A

F = -double integral of P(s) ds, so F(x) = double integral from 0 to L and 0 to 2π of P(s)(cosθR)dθ dz = 0