Axisymmetric flow using Stokes Streamfunction
Bernoulli Equation applies for steady, irrotational fluid (where is the body force potential):
Cauchy’s equation of motion preserves momentum, applies to all fluids:
Vorticity equation expresses conservation of angular momentum (take curl of Euler):
Name Equation Applies to Stress Tensor
Euler equation
Inviscid flow
Laplace equation
Navier-Stokes equation
Useful Definitions and Tools
Streamlines are space-curves whose tangent vectors are parallel to velocity:
Particle paths are time-curves which follow the flow of a particle over time:
Lagrangian velocity
For sphere use with :
No-slip (solid boundary viscid flow):
and
Solving ODEs
Choose particular solutions of the form:
The Cauchy-Euler equation has the form:
Where are constants. To solve this equation we make the trial solution :
ODE Systems Analysis
First order ODEs can be solved graphically using vector field, or by linear stability analysis
Linear stability analysis for
can be derived by considering small perturbation
, constructing taylor series of about and solving the resulting DE for an exponential
function
All linear systems have the solution:
To find the direction of spiral curvature, plot a few velocity vectors
Stable or unstable nodes have phase portrait lines which curve inwards towards (becoming
tangent to) the axis with the lower eigenvalue (slower eigendirection)
Systems with degenerate eigenvalues have a single eigendirection, and phase portrait lines
that intersect it like a big ‘S’
For linear stability analysis, evaluate the Jacobian matrix at each of the fixed points
If the eigenvalues are all zero, linear stability analysis fails (Hartman-Grubman theorem)
Conservative systems have a non-constant function that is constant on time trajectories.
They cannot have attractive fixed points
To show a system is conservative:
Method of Characteristics
Write out full derivative and use to obtain two separate ODEs
Solve each ODE, one for the solution of the u equation and one for the characteristic
If there is a source or sink term, density will not be constant along characteristics, so first we
need to solve for density, then substitute this into the characteristic ODE and solve that
If the initial conditions of u are not constant AND there is a source/sink term, there will be
two equations for u: one given by initial conditions, and the other given by the ODE for u.
They will be equal when t=0, use this fact to solve for any unknown constants
Finally, substitute the characteristic s into ‘x’ in the solution of the main equation either
from the ODE or ICs or both) to come up with overall solution
If fans are present, re-arrange characteristic equation with s set to where the fan starts
If shocks are present, solve using
, where is the coefficient of the
term in the original PDE, substituting in given values for using ICs or solution to ODE
Space-time diagram is a plot of t versus x, used to show characteristics