1.5. Example Problem: Pipe Flow

So, let us take an example problem. It is a very, very famous problem, which is an example of the pipe flow. And what we have to determine? We have to determine, pressure drop per unit length. So, we will explore, in principle, how these things can be done using experiments, for example. So, the pressure drop per unit length of the pipe that develops along it, is a result of friction. And this phenomenon cannot be explained analytically without the use of experimental data.

The example problem addresses the concept of pressure drop per unit length in pipe flow, which results from friction between the fluid and the walls of the pipe. Essentially, the problem illustrates the challenges in understanding fluid behavior analytically without experiments.

Imagine trying to understand how a car slows down while driving on a dirt road versus a smooth highway. The difficulty in predicting the exact slowdown due to different friction levels parallels how fluid behavior in pipes can be complex and often requires experimental data to get accurate results.

So, what do we do? First, we determine the important variables of the flow related to the pressure drop. We must know what are the flow parameters, for example? Flow parameters or important variables here, what we call. That is related to this pressure drop. So, it has, I mean, when it comes to our mind, we can write that pressure P per unit length. So, this is length, can be a function of, this is D, that is, it is a function of diameter of the pipe, ρ is the density of the fluid, viscosity of the fluid and of course, it should also depend upon the velocity of the flow.

In this step, we identify the key variables affecting the pressure drop along a pipe: the diameter of the pipe (D), the density of the fluid (ρ), the viscosity of the fluid (µ), and the velocity of the flow (V). Understanding how these variables interact is crucial for analyzing and predicting fluid behavior in pipes.

Think about drinking through a straw. If the straw is thin (small diameter), it's harder to drink compared to a wider straw because of higher friction. Similarly, the fluid's density and viscosity, as well as how fast you're trying to drink, will affect how much effort it takes to drink.

So, normally if we want to conduct experiments, it is, I mean, it is very logical that we can vary one variable at a time and hold the other constant.

In experimental settings, we typically change one variable while keeping others constant to isolate the effect of that variable on the outcome. For example, if we are studying the effect of pipe diameter on pressure drop, we would fix the fluid density, viscosity, and flow velocity while varying the pipe diameter and measuring the resultant pressure drop.

This is like testing different brands of batteries in a flashlight. To see which one lasts longer, you would keep the same flashlight and bulb but try out different batteries one at a time. This way, you can clearly see the impact of each battery type.

So, what we see, we have now done 5 experiments, for each plot and the while the other things were constant. Therefore, the total number of experiments have been done as 20.

By conducting multiple experiments varying different parameters (e.g., density, diameter, velocity), the number of experiments increases significantly. Here, 5 experiments for each variable with other constants leads to a cumulative total of 20 experiments, which helps in accurately charting the relationships among the variables.

Consider baking cookies. If you change the type of chocolate chips but keep the same dough recipe, you are doing a set of experiments to find which cookie recipe is best, accumulating results to understand which chocolate flavor you prefer.

So, total combination because there are 4 variables so, what is going to be our number of experiments? 10 into 10 into 10 into 10. So, there will be 10,000 experiments that we have to conduct and, you know, conducting 10,000 experiments to make it more widely applicable is not a very good idea.

When considering more variables, the number of experiments required increases exponentially. In this case, trying to test 10 different values for each variable results in 10,000 total experiments, which is impractical due to resource constraints.

Imagine trying to taste test 10 different flavors of ice cream at 10 different temperatures, and you want to do this for each flavor of sprinkles. The logistics can become overwhelming very quickly, similar to testing a vast array of experimental conditions in lab settings.

Fortunately, to avoid this, to have, to do, to be able to do less number of experiments, there is a µch, µch simpler approach called dimensionless groups or the dimensional analysis.

Dimensional analysis simplifies experimental investigations by allowing us to reduce multiple variables into dimensionless groups. This means we can assess the relationships between these key variables without needing to conduct every possible experiment, saving time and resources.

Think of learning to ride a bike. Instead of practicing how to balance and pedal all by trying different bikes (over 10,000 tests!), you learn that balance is the key skill, which can apply broadly across many bikes rather than needing to practice on every type.

We have not only reduce the number of variables from 5 to 2. But the dimensionless plot which we have got in the last slide is independent of the system of units used.

Using dimensional analysis, we've expedited our experiments from managing 5 variables down to 2 essential dimensionless groups. This also means our findings remain relevant across different measurement systems (like SI or CGS), making results universally applicable.

This is similar to how GPS devices can navigate anywhere in the world, irrespective of local street names or measurements. It simplifies complex mapping into actionable directions no matter where you are.

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