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.

MrsDrPoe: Taking a Different View on Physics

Welcome to Thesis Tuesday on the blog! Today, as we continue our discussion of basic fluid mechanics, we will examine specifically how fluid mechanics differs from classical physics.

In a typical physics class, we examine moving objects from a Lagrangian viewpoint, meaning that we tag and track information for an object. For a solid mechanics application, like an accelerating car, this viewpoint makes sense- we tag the car and we can track its velocity, acceleration, etc. For fluid mechanics, however, this method entails the tagging and tracking of individual fluid particles, which allows us to know how properties of each particle change with time, but not how properties change with time in relation to position in the fluid. Thus, we employ the Eulerian viewpoint, which provides us with a functional relationship for a fluid property as a function of space and time (i.e. a field).

From these differences stem the terms control system and control volume. A control system is also known as a closed system, meaning that the object we are examining is completely enclosed in the system- nothing flows in or out. This pairs with a Lagrangian "particle-tracking" viewpoint. The control system can move with the particle (ex: a car), but the particle is always contained. A control volume is also known as an open system, where the object we are examining is free to exit, enter, or remain inside the system. This pairs with an Eulerian "field" or "flow" viewpoint. A control surface is the boundary of a control volume.

From basic calculus, we know that, given the functional form of a velocity, we can determine the acceleration by taking the derivative of velocity function. When dealing with a velocity field (V), however, we must take the material derivative in order to find the acceleration field. The material derivative has both a local part and a convective part. The local term (dV/dt) tells us how the velocity at a certain point in the field changes with time, while the convective term (u[dV/dx] + v[dV/dy] + w[dV/dz]) tells us how the velocity changes with respect to position in the flow. For example, if we sit at a restaurant and watch traffic directly in front of us, we can see how the velocity or the "flow" of cars changes with time at that specific location. If we drive down the road examining the speed of our car (or another) as we go from point A to point B, we can see how the velocity of the "flow" of cars changes with position.

Introduction of the material derivative allows for a brief discussion of the Reynolds Transport Theorem. This is a mathematical formulation that allows us to convert the physical laws from a Lagrangian viewpoint to an Eulerian viewpoint. The material derivative of an extensive property (B, property dependent on the amount of mass present) is representative of a Lagrangian viewpoint. This is set equal to two integral terms involving the corresponding intensive property (b, property independent on the amount of mass present; B = mb). The first integral term represents how the amount of b in the control volume changes with time; the second represents the net amount of b that crosses the control surface.

I hope you'll forgive me for looking more at the math behind the scenes today, but this will (hopefully) help those of you with a basic physics background see how fluid mechanics takes a different view of the subject.