Semester 1: Equations Flashcards by Hannah Emery

Q

How can Cartesian coordinates be written in terms of plane polar coordinates?

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Q

What are the unit vectors for plane polar coordinates in Cartesian coordinates?

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Q

How can cylindrical coordinates be written in terms of Cartesian coordinates?

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Q

How can Cartesian coordinates be written in terms of cylindrical coordinates?

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Q

What are the unit vectors for cylindrical coordinates in Cartesian coordinates?

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Q

How can spherical polar coordinates be written in terms of Cartesian coordinates?

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Q

How can Cartesian coordinates be written in terms of spherical polar coordinates?

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Q

What are the unit vectors for spherical polar coordinates in Cartesian coordinates?

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Q

What is the equation for the dot product?

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Q

What is the equation for the cross product?

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Q

What is the equation for the del operator?

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Q

What is the equation for the del operator in cylindrical coordinates?

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Q

What is the equation for the del operator in spherical polar coordinates?

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Q

What is the equation for the Laplacian operator?

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Q

What are the three types of line/path integral?

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Q

What is the equation for flux?

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Q

What is the equation for the divergence theorem?

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Q

What is the equation for Stoke’s theorem?

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Q

What is the equation for vorticity?

A

ζ = vorticity
v = fluid velocity

Q

What is the equation for circulation?

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Γ = circulation
v = fluid velocity

Q

What is the equation that relates vorticity and circulation?

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Q

What is the equation for the velocity of a forced vortex?

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v = velocity
ρ = distance from origin
ω = angular speed

Q

What is the equation for the angular momentum of a free vortex?

A

L = angular momentum (= Iω)
I = moment of inertia
ω = angular speed
m = mass
ρ = distance from origin
v = velocity

Q

What is the continuity equation for an incompressible fluid flowing between two tubes?

A

ρ = density
v = fluid velocity
A = area of flow tube

What is the continuity equation for a steady flow?

What is the equation for the steady flow of an incompressible fluid in two dimensions (the x-y plane)?

u = velocity in x direction v = velocity in y direction

What are the equations for the two velocity components of a stream function?

What is the equation for the slope of a streamline?

What are the three ways that the calculus version of the continuity equation can be written?

What is the calculus version of the continuity equation?

ρ = density t = time v = fluid velocity

What is the continuity equation for a steady, incompressible flow?

What is the continuity equation for a steady, incompressible flow?

What is the Laplace equation?

v = fluid velocity F = scalar function

What is the equation for the fluid velocity of a line source in 2D?

v = fluid velocity m = mass

What is the equation for the fluid velocity of a 3D source?

v = fluid velocity m = mass

What is the equation for the velocity of a free vortex?

v = velocity k = constant ρ = distance from origin

What is the equation for the acceleration of a particle?

a = acceleration v = velocity t = time

What is the equation for acceleration in terms of the del operator?

a = acceleration v = velocity t = time

What is the equation for a small change in the force acting on a small mass of fluid due to gravity?

f = force M = mass g = acceleration due to gravity

What is the equation for a small change in the force acting on a small mass of fluid due to pressure?

f = force p = pressure S = area

What is the formula for the Euler equation?

ρ = density v = velocity t = time g = acceleration due to gravity p = pressure

What is the formula for the Bernoulli equation?

ρ = density v = velocity g = acceleration due to gravity p = pressure

What is the equation for dynamical viscosity

µ = dynamical viscosity F = force A = area v = velocity d = distance

What is the equation for the Reynolds number?

Re = Reynolds number ρ = density of fluid v = mean velocity of fluid relative to surface L = characteristic linear dimension (e.g. pipe radius) µ = dynamic viscosity

What is the formula for the Navier-Stokes equation in vector form?

ρ = density of fluid v = velocity t = time µ = dynamic viscosity f = body forces per unit volume

How can plane polar coordinates be written in terms of Cartesian coordinates?