Table of contents: PDF. Back matter: PDF. Share on social. Fluid Mechanics and Its Applications ; But the torque is the tangential force times the moment arm Ro. Discussion The above is only an approximation because we assumed a linear velocity profile. It is possible to solve for the exact velocity profile for this problem, and therefore the torque can be found analytically, but this has to wait until the differential analysis chapter.
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Chapter 2 Properties of Fluids Solution A large plate is pulled at a constant speed over a fixed plate. The space between the plates is filled with engine oil. The shear stress developed on the upper plate and its direction are to be determined for parabolic and linear velocity profile cases. Therefore we conclude that the linear assumption is not realistic since it gives over prediction. Chapter 2 Properties of Fluids Solution A cylinder slides down from rest in a vertical tube whose inner surface is covered by oil.
An expression for the velocity of the cylinder as a function of time is to be derived.
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Assuming a linear velocity profile in the oil film the drag force due to wall shear stress can be expressed as. This is a first-order linear equation and can be expressed in standard form as follows:. Therefore this equation enables us to estimate dynamic viscosity of oil provided that the limit velocity of the cylinder is precisely measured. Chapter 2 Properties of Fluids Solution A thin flat plate is pulled horizontally through the mid plane of an oil layer sandwiched between two stationary plates.
The force that needs to be applied on the plate to maintain this motion is to be determined for this case and for the case when the plate. The velocity profile in each oil layer relative to the fixed wall is as shown in the figure. Stationary surface. Analysis We measure vertical distance y from the lower plate.
Then the distance of the moving plate is y from the lower plate and h — y from the upper plate, where y is variable. It is caused by the attractive forces between the molecules. The surface tension is also surface energy per unit area since it represents the stretching work that needs to be done to increase the surface area of the liquid by a unit amount.
Discussion Surface tension is the cause of some very interesting phenomena such as capillary rise and insects that can walk on water. We are to determine whether the level of liquid in a tube will rise or fall due to the capillary effect. Discussion This liquid must be a non-wetting liquid when in contact with the tube material. Mercury is an example of a non-wetting liquid with a contact angle with glass that is greater than 90o. Analysis The capillary effect is the rise or fall of a liquid in a small-diameter tube inserted into the liquid. It is caused by the net effect of the cohesive forces the forces between like molecules, like water and adhesive forces the forces between unlike molecules, like water and glass.
The capillary effect is proportional to the cosine of the contact angle, which is the angle that the tangent to the liquid surface makes with the solid surface at the point of contact. Analysis The pressure inside a soap bubble is greater than the pressure outside, as evidenced by the stretch of the soap film. Analysis The capillary rise is inversely proportional to the diameter of the tube, and thus capillary rise is greater in the smaller-diameter tube.
Discussion Note however, that if the tube diameter is large enough, there is no capillary rise or fall at all. Rather, the upward or downward rise of the liquid occurs only near the tube walls; the elevation of the middle portion of the liquid in the tube does not change for large diameter tubes. Chapter 2 Properties of Fluids Solution An air bubble in a liquid is considered. The pressure difference between the inside and outside the bubble is to be determined. Substituting, the pressure difference is determined to be: 2 0. The smaller the bubble diameter, the larger the pressure inside the bubble.
The factor 2 is due to having two surfaces in contact with air. The surface tension of the liquid is to be determined. Assumptions 1 There are no impurities in the liquid, and no contamination on the surfaces of the glass tube. Discussion Note that the gage pressure in a soap bubble is inversely proportional to the radius or diameter.
Therefore, the excess pressure is larger in smaller bubbles. A slender glass tube is inserted into kerosene. The capillary rise of kerosene in the tube is to be determined. Assumptions 1 There are no impurities in the kerosene, and no contamination on the surfaces of the glass tube.
Chapter 2 Properties of Fluids Solution The force acting on the movable wire of a liquid film suspended on a U-shaped wire frame is measured. The surface tension of the liquid in the air is to be determined.
Fluid Mechanics: Fundamentals and Applications
Assumptions 1 There are no impurities in the liquid, and no contamination on the surfaces of the wire frame. Analysis to be. Substituting the numerical values, the surface tension is determined from the surface tension force relation.
Discussion The surface tension depends on temperature. Therefore, the value determined is valid at the temperature of the liquid. A capillary tube is immersed vertically in water. The height of water rise in the tube is to be determined. Assumptions 1 There are no impurities in water, and no contamination on the surfaces of the tube.. The maximum capillary rise and tube diameter for the maximum rise case are to be determined.
Assumptions 1 There are no impurities in water, and no contamination on the surfaces of the tube. Chapter 2 Properties of Fluids Solution A steel ball floats on water due to the surface tension effect. The maximum diameter of the ball is to be determined, and the calculations are to be repeated for aluminum. Assumptions 1 The water is pure, and its temperature is constant.
When the ball floats, the net force acting on the ball in the vertical direction is zero. The height to which the water solution rises in a tree as a result of the capillary effect is to be determined. Discussion Other effects such as the chemical potential difference also cause the fluid to rise in trees.
Analysis The magnitude of the capillary rise between two large parallel plates can be determined from a force balance on the rectangular liquid column of height h and width w between the plates. The bottom of the liquid column is at the same level as the free surface of the liquid reservoir, and thus the pressure there must be atmospheric pressure. This will balance the atmospheric pressure acting from the top surface, and thus these two effects will cancel each other. The weight of the liquid column is t. Discussion The relation above is also valid for non-wetting liquids such as mercury in glass , and gives a capillary drop instead of a capillary rise.
Chapter 2 Properties of Fluids Solution A journal bearing is lubricated with oil whose viscosity is known. The torques needed to overcome the bearing friction during start-up and steady operation are to be determined. Assumptions 1 The gap is uniform, and is completely filled with oil. Substituting the given values, the torque is determined to be.
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The difference between the water levels of the two arms is to be determined. Assumptions 1 Both arms of the U-tube are open to the atmosphere. Analysis Any difference in water levels between the two arms is due to surface tension effects and thus capillary rise. Noting that capillary rise in a tube is inversely proportional to tube diameter there will be no capillary rise in the arm with a large diameter.
Then the water level difference between the two arms is simply the capillary rise in the smaller diameter arm,. Discussion Note that this is a significant difference, and shows the importance of using a U-tube made of a uniform diameter tube. Chapter 2 Properties of Fluids Solution The cylinder conditions before the heat addition process is specified.
The pressure after the heat addition process is to be determined. Assumptions 1 The contents of cylinder are approximated by the air properties. Combustion 2 Air is an ideal gas. Discussion Note that some forms of the ideal gas equation are more convenient to use than the other forms. The final temperature when half the mass is withdrawn and final pressure when no mass is withdrawn are to be determined.
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Analysis a The first case is a constant volume process. The percent increase in the absolute temperature of the air in the tire is to be determined. Noting that air is an ideal gas and the volume is constant, the ratio of absolute temperatures after and before. Therefore, the absolute temperature of air in the tire will increase by 4. Discussion This may not seem like a large temperature increase, but if the tire is originally at 20oC Chapter 2 Properties of Fluids E Solution The minimum pressure on the suction side of a water pump is given.
The maximum water temperature to avoid the danger of cavitation is to be determined. Discussion Note that saturation temperature increases with pressure, and thus cavitation may occur at higher pressure at locations with higher fluid temperatures.
A relation is to be developed for the specific gravity of the suspension in terms of the mass fraction Cs , mass and volume fraction Cs , vol of the particles. Assumptions 1 The solid particles are distributed uniformly in water so that the solution is homogeneous.