Tuesday, July 5, 2011

MrsDrPoe: Simplifying Assumptions

Most "real-life" fluid mechanics problems are super complex and are governed by a multitude of non-linear differential equations, which means that it would take you, or me, or Einstein lifetimes to determine a flow "solution" to these equations by hand...and that's only if an analytical solution is actually possible. With that in mind, we often examine a problem and try to make as many simplifying assumptions as possible to pare it down to a system of equations that we can handle. This is particularly important when investigating fluid mechanics for the first time. So today we will examine a few of the standard simplifying assumptions that we'll apply as we continue our study of fluids.

While the no-slip condition isn't exactly a simplifying assumption, it is typically employed as a simple boundary condition. Because of a fluid's viscosity, or resistance to flow, fluid molecules immediately adjacent to a solid surface cling to the surface. Thus, the velocity of the fluid at the surface boundary is zero since those molecules are not moving or not "slipping" along the surface. If we make an assumption that the fluid is inviscid (having no viscosity), the no-slip condition does not hold, and the fluid at the solid surface has a non-zero velocity.

If we assume that the characteristics of a flow or the properties of a fluid are not changing with respect to time, we are making the steady or steady-state assumption. This assumption eliminates any time derivatives present in the governing equations since these mathematical terms represent physical changes in time. If this assumption CANNOT be made, the flow is termed unsteady.

We can reduce the number of spatial dimension that we consider in a problem by making the assumption that the flow is laminar. If you go to your bathroom sink and turn on the faucet slightly, you will see an example of laminar flow. This flow is characterized by order; the water from the faucet is orderly flowing downward into the sink. If you turn the faucet on the whole way, you will see an example of turbulent flow. This flow is characterized by statistical chaos; the water is predominantly flowing in the downward direction, but there are spurts, sputters, and rotations in the flow that give it velocity components in all three spatial directions. If we examine the governing equations for the laminar flow from the faucet that is only in the downward direction, we can eliminate any terms with either of the other two velocity components. Furthermore, for most laminar flow cases, the derivatives with respect to the third spatial direction can be eliminated since we can often define the flow in just two directions.


We have previously mentioned this term, but I would like to expand on the idea here. Technically there are no incompressible fluids; however, in many instances, we can make the assumption that a fluid is incompressible without incurring much error. One benefit to this assumption is that it makes the math easier, since the density is considered a constant not changing with space or time. Thus, density can be pulled out of any derivative terms in the governing equations, AND the continuity and momentum equations can be solved independently of one another (we'll get back to this soon).

Be sure to keep these in mind as we journey through the world of fluid mechanics. My students would often get nervous at the thought of eliminating terms from equations, but as long as we do so by making these assumptions (when they are valid), we are making fluid problems solvable. Simplifying assumptions are your friend!


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