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Buoyancy and Drag on Immersed Bodies

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Buoyancy and drag are two forces that commonly arise when considering the motion of an object through a fluid. The prediction and characterization of these forces is critical to solving many mechanical problems, such as engineering vehicles, or understanding the motion of swimming and flying organisms. As your intuition might suggest, the buoyant force acts vertically upward on the object in direct opposition to gravity. Likewise, the drag force tends to slow an object down relative to the surrounding fluid, acting in opposition to the relative motion of the object. In this video, these two forces will be examined in greater detail to show how they arise and how to determine their magnitude. Their effect on small bubbles and droplets rising in a fluid will then be illustrated by an experiment before finishing with a discussion of other applications.

To begin, let's have a closer look at buoyancy. When an object is fully immersed in a fluid, the magnitude of the buoyant force is simply the product of the surrounding fluid density, the volume of the object, and the acceleration due to gravity. This is equivalent to the weight of the fluid displaced by the object, as stated by Archimedes' Principle. Of course, the gravitational force, which is the average density of the object times it's volume and acceleration due to gravity, is still pulling downward in opposition to the buoyant force. So, if the average density of the object equals the density of the fluid, the sum of the buoyant and gravitational forces will equal zero, and the object will be neutrally buoyant. Likewise, if the object is more dense, it will sink, and if it is less dense, it will float. Once the object begins moving however, it will encounter another force, drag. Drag is due to frictional resistance caused by the motion of the object through the fluid, and acts against the direction of motion as indicated by the velocity vector "U". Calculating the magnitude of the drag force is more complicated, but in general, it can be modeled as 1/2 the product of the fluid density, the projected area of the body and the direction of motion, the drag coefficient, and the relative velocity squared. The drag coefficient captures the effect of the shape of the object and since it depends on the Reynolds Number, also takes into account the relative magnitude of inertial and viscous fluid forces on the body. The Reynolds Number is determined by multiplying the relative velocity and characteristic length scale of the object, by the ratio of the fluids density and viscosity, but in general, there is no simple equation for the drag coefficient, and it must be determined empirically or numerically. Now, consider all three of these forces acting on a spherical object in a dense fluid. The buoyant force will counter the force of gravity, and accelerate the object upwards. But as velocity increases, so will the drag. Eventually, the object will reach a constant velocity, called the Terminal Velocity, where all three forces are in balance. If the density of the fluid and the mass diameter and terminal velocity of this sphere are known, then the drag coefficient can be calculated. Now, let's test these principles by measuring the drag coefficient of small air bubbles in oil droplets rising in glycerin, and comparing the results to theory. For low Reynolds Number bubbles and droplets, the drag coefficient should be 16 divided by the Reynolds Number.

To perform these tests, you'll need a clear liquid tank with an injection port. Follow the instructions in the text to assemble the tank. When construction of the tank is complete, set it up so that the injection port is easily accessible, and the fill it with glycerin to a depth of approximately 25 cm by slowly pouring a film against the inside wall. This technique will help to reduce bubble entrainment in the container. Some gas will inevitably get entrained and will need time to rise out of the glycerin, so use this time to set up the camera and backlight. Affix the camera to a tripod, facing the container squarely and high enough that the upper portion of the liquid is in view. Opposite the camera, mount a bright light source, and if necessary, insert a diffuser sheet between the light and the container to achieve more even illumination. Now, carefully insert a ruler vertically into the glycerin above the injection port, with the markings facing the camera. Adjust the field of view to span a vertical height of approximately 150mm, and the focus the camera on the markings. Record a brief video of the ruler for calibration, and then carefully extract it from the tank. Do not adjust the position or field of view of the camera for the remainder of the experiment or the calibration will be invalid. Finally, prepare two syringes with thin needles. The first syringe will just contain air, but fill the second with a mixture of a low viscosity vegetable oil and an oil based food coloring. You are now ready to perform the experiment. Use the first syringe to inject an air bubble, and record it with the camera as it rises. Repeat this process 10 to 15 times, and with a variety of bubble sizes. Now, repeat the procedure with the colored oil and record 10 to 15 droplets of varying size.

Transfer all of the video files from the camera to a computer with software capable of exporting individual frames from the videos as images. Open the calibration video of the ruler first and export one frame. Use this image to determine the scaling factor in terms of meters per pixel. After you have the scaling factor, you can process the rest of the videos. Export one frame with the bubble or droplet near the bottom of the view and measure the horizontal diameter in pixels. Next, measure the vertical distance in pixels from the top of the image to the top edge of the bubble or droplet. Finally, record the timestamp for this frame. Now, export a second frame with the bubble or droplet near the top of the view, but still completely within the glycerin. Once again, measure the horizontal diameter, the vertical distance, and the timestamp. You now have two horizontal diameters and vertical positions corresponding to the two measurement times. Take the average of the diameter measurements, and then use the scaling factor to convert this value from pixels to meters. Now, take the difference in vertical height between the two frames. Use the scaling factor once again to convert this distance from pixels to meters. The time taken to rise this distance is found by taking the difference between timestamps for the two frames. Now that the changes in position and time are known, the terminal velocity is easily determined by taking the ratio of the two. Use these results to calculate the drag coefficient with the equation that was derived earlier. Look up published values for the fluid densities and the acceleration due to gravity. Recall that the theoretical treatment predicts a relationship between the drag coefficient and the Reynolds number. Calculate the Reynolds number using your measurements and the published values for the density and viscosity of glycerin. We will use this result soon to compare the measurements with theory, but for a meaningful comparison, the measurement uncertainty must also be known. Propagate your uncertainties as described in the text to determine the final uncertainty in the drag coefficient and the Reynolds number. Once you have finished analyzing all of the videos, take a look at the results.

First, compare the videos from air bubbles of different sizes. At these low velocity and length scales, strong surface tension forces result in nearly spherical bubbles, but the smaller bubbles rise at lower velocities due to relatively stronger drag forces. The largest bubbles approach a Reynolds number of two resulting in somewhat flattened tails in the wake region. Now, compare the videos of different sizes of oil droplets. As with the bubbles, the droplets remain nearly spherical, and the smaller droplets rise at lower velocities due to stronger drag forces. The largest oil drops only approach a Reynolds number of 0.2 however due to their greater weight, and they form slightly teardrop shapes, likely due to the high inertia of the oil circulating inside the droplets. Finally, plop the measured drag coefficient as a function of Reynolds number for the bubbles and droplets, and compare this to the theoretical prediction. Overall, qualitatively close agreement is observed with the theory with most measured drag coefficient values matching within experimental uncertainty.

Buoyancy and drag are forces that impact an enormous variety of industrial processes and mechanical systems. Boiling water reactors, BWRs, are a type of steam generator in nuclear power plants. In these reactors, vertical bundles of radioactive fuel rods heat upward flowing high-pressure water to produce steam. This video shows a scaled down experiment of liquid gas flow along transparent cylinders representing the fuel rods. Concepts such as buoyancy and drag must be considered to predict the behavior of two phase flow in these fuel assemblies and ensure safe operation. If gas bubbles are not removed quick enough by buoyancy and fluid flow, the fuel rods surfaces can dry out, leading to over-heating and failure. Vehicles such ass cars, planes, and boats experience significant drag forces. For example, at highway speeds a typical sedan may require horsepower or 30 kW, just to overcome aerodynamic resistance. Careful design on vehicle shape and intake exhaust pathways can control airflow around a vehicle and reduce drag. Thereby, increasing efficiency.

You've just watched Jove's Introduction to Buoyancy and Drag. You should now understand how and when these forces arise and how they can effect the motion of objects in a fluid. You have seen how to calculate these forces based on physical properties and a method for determining the drag coefficient of an object by measuring its terminal velocity. Thanks for watching.

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