Semester 1: Equations Flashcards
How can Cartesian coordinates be written in terms of plane polar coordinates?
What are the unit vectors for plane polar coordinates in Cartesian coordinates?
How can cylindrical coordinates be written in terms of Cartesian coordinates?
How can Cartesian coordinates be written in terms of cylindrical coordinates?
What are the unit vectors for cylindrical coordinates in Cartesian coordinates?
How can spherical polar coordinates be written in terms of Cartesian coordinates?
How can Cartesian coordinates be written in terms of spherical polar coordinates?
What are the unit vectors for spherical polar coordinates in Cartesian coordinates?
What is the equation for the dot product?
What is the equation for the cross product?
What is the equation for the del operator?
What is the equation for the del operator in cylindrical coordinates?
What is the equation for the del operator in spherical polar coordinates?
What is the equation for the Laplacian operator?
What are the three types of line/path integral?
What is the equation for flux?
What is the equation for the divergence theorem?
What is the equation for Stoke’s theorem?
What is the equation for vorticity?
ζ = vorticity
v = fluid velocity
What is the equation for circulation?
Γ = circulation
v = fluid velocity
What is the equation that relates vorticity and circulation?
What is the equation for the velocity of a forced vortex?
v = velocity
ρ = distance from origin
ω = angular speed
What is the equation for the angular momentum of a free vortex?
L = angular momentum (= Iω)
I = moment of inertia
ω = angular speed
m = mass
ρ = distance from origin
v = velocity
What is the continuity equation for an incompressible fluid flowing between two tubes?
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?
