# math

## From Newton's Second Law to the Equations of Change for a fluid

Most often when you see the derivation for the general form of the equations of change for fluid dynamics it is done by considering some differential element. What follows is a way of arriving at the same result by considering Newton's second law for an arbitrary volume of fluid moving …

tags: fluid dynamics,

## Using the Bernoulli equation to derive Toricelli's equation

Bernoulli's equation for fluid flow along a streamline is a really useful go-to equation for explaining various fluid mechanics phenomena, and if you extend it by taking into account friction you end up with the engineering Bernoulli equation or a mechanical energy balance, which is practically the default starting point …

tags: fluid dynamics,

## Flow constrained by concentric spheres

The last few flow problems I toyed with used a simple momentum balance as the starting point, time to move on to other ways to solve flow problems such as the continuity equation and the equations of motion for the fluid (e.g. the Navier-Stokes equations ).

Today I'm going to …

tags: fluid dynamics,

## Not no-slip -- low density gas flow

Continuing on with tube flow, what happens when the fluid moving through the tube is low density and thus the no-slip boundary condition breaks down? (from 2B.9 Transport Phenomena)

I want the mass flowrate for a low density gas moving through a tube in slip flow.

First off we …

## Liquid in and on pipes

One of the nice things about setting up the math for simple fluid flow problems is that you can recycle the initial bits for various other uses. If you set up a balance based on a particular geometry of a differential volume then a wide variety of possible flow cases …

tags: fluid dynamics, pipe flow,

## The ideal gas law from Maxwell's velocity distribution

Recently I've been playing around with finding the relations between the microscopic particulars of an ideal gas and the macroscopic observables we all know and love -- in particular the relation between temperature and the average kinetic energy (velocity) of the particles in the gas. We can take the idea further …

tags: kinetic theory, gas,

## Mathematical interlude

Last time I poked around with calculating the mean velocity of the particles in a gas from the Maxwell distribution of velocities. While doing so I bashed my way through some integrals that, on reflection, are much easier when I remembered that they are in the back of some of …

## Average velocity of the molecules in a gas

In this problem I aim to show that the expression for the average velocity of particles in a gas can be derived from the Maxwell velocity distribution (problem 1C.1(a) Transport Phenomena).

In the kinetic molecular theory of gases, velocities are randomly distributed and have an average magnitude given …

tags: kinetic theory, gases,

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