Tuesday, August 2, 2011

MrsDrPoe: Conservation of Momentum, Part I

It's another Thesis Tuesday on the blog! Today we'll be looking at the second conservation equation that governs fluid mechanics: Conservation of Momentum (AKA Newton's 2nd Law or the Linear Momentum Equation).

In Lagrangian terms, this equation simply states that the sum of the forces acting on an object is equal to the mass of the object times its acceleration (or the time rate of change of its linear momentum). Recall that we wish to employ an Eulerian methodology; therefore, we must use Reynolds Transport Theorem to convert this equation.

For this case, our extensive (mass dependent) property is linear momentum, and our intensive (mass independent) property is linear momentum per unit mass, or density times velocity. Combining the result from the Reynolds Transport Theorem with the Lagrangian form of the conservation of momentum equation, we get the sum of the forces acting on the control volume is equal to:

sum(F) = d/dt(int(rho*V)dV) + int(rho*V*(V.n))dA

Here, F is a force, rho is the density, V is the velocity, and n is the unit outward normal vector.

The first thing we should notice about this equation is that it is a vector equation and can thus be written as 1, 2, or 3 equations depending on how many dimensions our problem has. For a 3D problem:

sum(Fx) = d/dt(int(rho*u)dV) + int(rho*u*(V.n))dA

sum(Fy) = d/dt(int(rho*v)dV) + int(rho*v*(V.n))dA

sum(Fz) = d/dt(int(rho*w)dV) + int(rho*w*(V.n))dA

The second thing we should notice is the similarity between these equations and the conservation of mass equation. Each one has something equal to a term tracking how our intensive property changes with time in the control volume plus a term tracking how much of the intensive property is crossing the control surface (into or out of the control volume).

We will employ the following methodology to a fixed, non-deforming control volume:

1) Choose the best CV, establishing all bounds and known conditions
2) Apply the conservation of mass equation
3) Determine all forces acting on the CV
4) Evaluate the right side of the conservation of momentum equation in terms of unknown and known variables
5) Combine both sides of the conservation of momentum equation and solve for the unknowns

Now, let that stew, and tune in next Tuesday for some fun examples.