User:Pstasiak

About me

An example of interference patterns for classical oscillators. The image shows three phase shifted waves and their resulting interference patterns. Image produced using the matlab software.

I am a third year mathematics student at Newcastle University on an integrated masters course, with a keen interest in studying applied mathematics. Some of these areas include but are not limited to:

• Quantum mechanics
• Fluid dynamics
• Variational methods & Lagrangian dynamics
• Special and general relatvity
• Methods for ordinary and partial differential equations

In my spare time I like to play chess both competitively and leisurely and I enjoy playing and producing various genres of electronic music.

Favourite equations

Fluid dynamics

Listed here are some of my favourite equations that I have come across during my time at university. The first equation is the Navier-Stokes equation for an incompressible viscous fluid, which predicts the flow of a viscous fluid under an applied pressure gradient

${\frac {\partial {\vec {u}}}{\partial t}}+({\vec {u}}\cdot \nabla ){\vec {u}}=-{\frac {1}{\rho }}\nabla p+\nu \nabla ^{2}{\vec {u}}$

where we wish to solve for the fluid velocity ${\vec {u}}$, where $\rho$ is the fluid denisty, $p$ is the pressure and $\nu$ is the kinematic viscosity. The Navier-Stokes equation can be expressed in dimensionless form by considering an object of size $D$ moving at a speed of $U$.

• Characteristic length $D$
• Characteristic speed $U$
• Characterstic time $T=D/U$

An example of an interference pattern for two counter-propagating classical waves, produced using the matlab software.

The dimensionless variables in this case becomes

• $x'=x/D$
• $t'=t/T$
• ${\vec {u}}'={\vec {u}}/U$
• $p'=p/(\rho U^{2})$

This gives the dimensionless Navier-Stokes equation for an incompressible viscous fluid

${\frac {\partial {\vec {u}}'}{\partial t'}}+({\vec {u}}'\cdot \nabla '){\vec {u}}'=-\nabla 'p'+{\frac {1}{Re}}\nabla '^{2}{\vec {u}}'$

where here $Re={\frac {UD}{\nu }}$ is the Reynolds number.

Quantum mechanics

Another one of my favourite equations is the time-dependent Schrödinger equation, given here in 1-dimension

$i\hbar {\frac {\partial }{\partial t}}\Psi (x,t)=-{\frac {\hbar }{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi (x,t)+V(x,t)\Psi (x,t)$

where we wish to solve for the wavefunction $\Psi (x,t)$, where $\hbar =h/2\pi$ is Planck's constant and $V(x,t)$ is the potential.