Fluids Flashcards by Hannah Fryer

Which forces are supported in fluids?

tension is not supported
compression is supported and results in a small elastic deformation
shear is supported but results in flow

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du/dx + dv/dy + dw/dz = 0

all flux in all three directions must equal to zero
mass must be concerved

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In which direction of pressure change is fluid driven?

From high to low pressure

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τ = F/A

Shear stress= force/area

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τ= μ δu/δy

Shear stress = dynamic viscosity * (velocity/ depth)

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What is the shear stress in viscous flow?

No longer zero

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Kinematic viscosity?

ν=μ/ρ (m^2/ s)

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Dynamic viscosity?

μ (kg/ms)

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What navier stokes simplification can you make by assuming the flow to be steady?

All time derivatives will be zero (d/dt)=0

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What navier stokes simplification can you make by assuming the flow to be fully developed?

The velocity component in the x-direction will remain constant (du/dx = 0)

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What navier stokes simplifications can you make when the sides are incredibly long. (Flow in one direction, boundaries in another)?

No flow in the the third direction and no effect by the boundaries.
W=0
d/dz=0

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What simplification to navier stokes equations by assuming its laminar flow?

The gradient of v is zero and v is zero at the boundary, v must be zero everywhere. (V is in the y direction)

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Re= ρUL/μ

Inertial forces/ viscous forces
Reynolds number= densityvelocity scale length scale/ dynamic viscosity

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What are Reynolds numbers for laminar flow?

Re < Recrit

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What Reynold numbers are turbulent flow?

Re> Recrit

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Q= π/4 D^2 U

Volume flux= π/4 * diameter^2 * velocity

(Only for uniform flow)

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What is Q along a pipe?

Constant

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What is M along a pipe

Momentum is constant along a pipe

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M= π/4 D^2 U^2 = QU

Momentum = π/4 * diameter ^2 * velocity ^2

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Hydraulic grade line?

Represents piezometric head (h)

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Energy grade line?

Represents total head line (H)

Piezometric head?

Pressure head + gravitational head combined

p1/ρg + z1 = p2/ρg + z2 + τ0PL/ ρgA

p= pressure
ρ= density
g= gravitational acceleration
z= height
τ0= shear stress
P= wetted perimeter
L= length
A= area

τ0= f/8 ρ U^2
where
f=f(Re,ks/D)

Shear stress = function of Reynolds and relative roughness * density * velocity^2/8

In what conditions is f= 64/Re

For laminar flow/ Re< 2300

In what condition is 1/f^(1/2) = -2log10[k/D/3.71 + 2.51/Re*f^(1/2)]

For turbulent flow/ for Re> 4000

When is 1/f^(1/2)= 2log10[Ref^(1/2)] -0.8

For turbulent flow in smooth pipes/ for Re>4000

hL = εU^2/2*g

minor head loss= loss coefficient* velocity^2/2*gravitational acceleration

ε= (1-Α1/A2)^2

Loss coefficient for sudden pipe enlargement = (1-small area before/ small area after)^2

RH= A/P

Hydraulic radius= flow area/ wetted perimeter

hf= f* L/4RH * U^2/2*g

head loss= function * length/4*hydraulic radius * velocity^2/2*gravitational acceleration

Entry length?

Region where boundary layer develops

Does laminar or turbulent flow boundary layers take longer to develop?

It takes longer for laminar pipe flow boundary layers to develop with longer entry lengths

Fully developed flow?

When boundary layers converges to pipe centreline

How does shear stress vary across a pipe?

Linearly with being 0 at the centreline

What effect of shear force have a long a plate?

The effect of shear force increases along a plate, increasing the thickness of fluid layer that is affected. Reynolds number also increases along a plate leading to turbulent flow conditions. There’s a laminar region, transition region and then turbulent region

u/u* = u*y/ ν

u= velocity u* = wall shear velocity ν= velocity viscosity y= depth/ distance from sublayer This works when yu*/ν is between 0 and 3~5

Sublayer thickness = (3-5) ν/ u*

ν= velocity viscosity u* = wall shear velocity

Irrationality equation?

dv/ dx - du/ dy = 0

What is the velocity normal to the cylinder on the surface of the cylinder?

Zero

Φ = Uo (r * D^2/4*r) cos θ

Φ = potential flow around a cylinder U0 = velocity r = distance from centre of cylinder D= diameter θ = angle to the point

ur = dΦ/ dr uθ = 1/r * dΦ/dθ

Velocity components in polar coordiantes

What are velocity on the surface of the cylinder?

ur = 0 uθ = -2U0 sinθ

P = 1/2 ρ U0^2 (1-4sinθ^2)

P = inviscid pressure distribution

What are the velocities on the surface of the cylinder in a real fluid?

ur = 0 (no flow in/ out of cylinder) uθ = 0 (friction between surface and fluid) No slip conditions

τ = μ duθ/dr

Shear stress = velocity gradient * dynamic viscosity

Drag force = Σ (τ ) ds

Drag force is the sum of all shear stresses taken round the surface of the cylinder

Explain the difference between upstream and downstream normal pressures for larger U0

For larger U0 the difference between upstream and downstream normal pressures contributes more significantly to the drag force. This pressure difference is caused by flow separation from the surface of the cylinder. In real solution pressure is a lot lower downstream than potential.

What causes flow seperation

What does flow separation depend on?