User:Pstasiak

About me
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.

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 $$T = D/U$$

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.