# Fluid Mechanics and order of magnitude calculation

• Niles
In summary, the conversation discusses the calculation of terms with a Mach number of M^3 being discarded, and the expression uu\partial_t \rho \approx \rho_0 uu\nabla u being thrown away due to it being \mathcal O(M^3). However, there is a question about whether the derivative of u is necessarily on the same order as Ma, and if the problem assumes M << 1 and no viscous effects. The possibility of a more complicated order-of-magnitude analysis is also mentioned.
Niles
Hi

In my lecture notes we making some calculations and all terms $\mathcal O(M^3)$ are to be thrown away. Here M is the Mach number. Now, there is the expression (u denotes the velocity):
$$uu\partial_t \rho \approx \rho_0 uu\nabla u$$
which in my notes are thrown away because they claim it is $\mathcal O(M^3)$. But is it really true, I mean the derivative of u will not necessarily be on the same order as Ma, right?

That's right. The gradient also contains a length scale in each direction. In many cases one simply asserts on physical grounds that du/dx is same order as u (so the flow is sufficiently "smooth") or that there is some characteristic length scale of order one. Does the problem assume M << 1 and also no viscous effects?

I'd also postulate that there is some more complicated order-of-magnitude analysis that can be done here a la that done in deriving Prandtl's boundary layer equations, but it would be difficult to carry that out without more information from the OP on what assumptions were made and what the physical situation is.

## What is fluid mechanics?

Fluid mechanics is a branch of physics that studies the behavior of fluids, which include liquids and gases, and their interactions with forces and other substances.

## What types of problems can be solved using fluid mechanics?

Fluid mechanics can be used to solve problems related to fluid flow, such as calculating the velocity and pressure of fluids in pipes or channels, predicting the lift and drag forces on objects moving through fluids, and understanding the behavior of fluids in various engineering applications.

## What is the purpose of order of magnitude calculation in fluid mechanics?

Order of magnitude calculation is used in fluid mechanics to estimate the approximate values of variables in a problem, without the need for precise calculations. This helps in quickly identifying the dominant factors influencing the behavior of fluids in a system.

## How is order of magnitude calculation performed in fluid mechanics?

Order of magnitude calculation involves identifying the relevant physical quantities and estimating their values using basic principles and assumptions. These values are then used to obtain a rough estimate of the final result, which is usually within one or two orders of magnitude of the actual value.

## What are some common applications of fluid mechanics and order of magnitude calculation?

Fluid mechanics and order of magnitude calculation are used in a wide range of fields, including aerospace engineering, chemical engineering, civil engineering, environmental science, and geophysics. Some specific applications include designing efficient airfoils for aircraft, predicting the flow of pollutants in rivers and oceans, and studying the behavior of fluids in the human body.

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