7. Fluid Mechanics

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Properties of Fluids

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All matter can be divided into two major classes 1) Solid 2) Fluids Properties of Fluids Solid requires external forces to cause it to deform. when external forces stress and deform solids, the solid will regain their original shape when these external forces are removed (Elasticity) Fluid will deform without the application of external forces. will take on the shape of the container in which they are held. -Ex. water, oil, gas etc. Fluid will continue to change shape in time even after the removal of the external forces causing the deformation.

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Fluids can be classified into two forms of matter 1) Liquids 2) Gases The difference between a liquid and gas can be defined by 1) The compressibility 2) The continuity Properties of Fluids Liquids Liquids are considered to be incompressible. Liquids only change in volume when subjected to very high pressure. Gases Gases are very compressible. Their volume can increase/decrease when subjected to slight variation in pressure.

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Liquids When a liquid is held in a container, its entire mass will arrange itself so as to be in contact with the bottom and the sides of that container, and a well-defined surface of the liquid will form. Gases A gas held in a closed container will not form a well-defined surface and will tend to fill the entire container. Properties of Fluids

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Density (  ) : Density is the mass of the fluid per unit volume m = Mass of fluid, kg V = Volume, m 3  = Density of fluid, kg/m 3 The density of water at 4 0 C = 1,000 kg/m 3 Properties of Fluids

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 = Specific weight, N/m 3 W = Weight of fluid, N (W=mg) V = volume, m 3 The specific weight of water at 4 o C = 9.81 kN/m 3 The specific weight is the weight of the fluid per unit volume Properties of Fluids Specific Weight or Unit Weight (  ) :

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The specific gravity is the ratio of the density or specific weight of the fluid to the density or specific weight of water, at a temperature of 4 o C Dimensionless Properties of Fluids Specific Gravity/Relative Gravity (S):

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Dynamic Viscosity (  ): Viscosity is the measure of a fluid’s resistance to internal shear stresses. Properties of Fluids Moving in oil would be even more difficult, as can be observed by the slower downward motion of a glass ball dropped in a tube filled with oil. It appears that there is a property that represents the internal resistance of a fluid to motion or the “fluidity,” and that property is the viscosity. Viscosity

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It can be verified experimentally that for most fluids the rate of deformation (and thus the velocity gradient) is directly proportional to the shear stress  , Fluids for which the rate of deformation is proportional to the shear stress are called Newtonian fluids. Most common fluids such as water, air, gasoline, and oils are Newtonian fluids. Blood and liquid plastics are examples of non-Newtonian fluids. Properties of Fluids

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Where,  = D ynamic (or absolute) viscosity of the fluid Unit of  = kg/m.s, or N.s/m 2 (or Pa.s) Shear Stress, Properties of Fluids Kinematic Viscosity (  ): The kinematic viscosity of a fluid is the ratio of its dynamic viscosity to its density. = kinematic viscosity, Unit = m 2 /s Newton’s law of viscosity

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where T is absolute temperature, and a, b, and c are experimentally determined constants. Properties of Fluids

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Properties of Fluids

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Properties of Fluids Surface Tension The capacity of liquids to resist tensile stresses at their surface is called surface tension. Surface of the liquid acts like a stretched elastic membrane under tension. The pulling force that causes this tension acts parallel to the surface and is due to the attractive forces between the molecules of the liquid. The magnitude of this force per unit length is called surface tension  s and is usually expressed in the unit N/m.

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Surface Tension … Properties of Fluids surface tension also can be defined as the work done per unit increase in the surface area of the liquid The excess pressure  P inside a droplet or bubble above the atmospheric pressure

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Properties of Fluids

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Capillary action Properties of Fluids This relation is also valid for non-wetting liquids (such as mercury in glass) and gives the capillary drop. In this case  > 90 o and thus cos  < 0, which makes h negative. Therefore, a negative value of capillary rise corresponds to a capillary drop

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Properties of Fluids Capillary Action ……. Example

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Capillary Action …… Solution Given, Density,  = 960 kg/m 3 Contact angle,  = 15  Note: The surface tension depends on temperature. Therefore, the value determined is valid at the temperature of the liquid. A 1.9-mm-diameter tube is inserted into an unknown liquid whose density is 960 kg/m 3 , and it is observed that the liquid rises 5 mm in the tube, making a contact angle of 15  . Determine the surface tension of the liquid. Properties of Fluids Example

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Capillary Action …… Solution Substituting the numerical values, the surface tension is determined from the surface tension force relation to be Note: The surface tension depends on temperature. Therefore, the value determined is valid at the temperature of the liquid. The surface tension of a liquid is to be measured using a liquid film suspended on a U-shaped wire frame with an 8-cm-long movable side. If the force needed to move the wire is 0.012 N, determine the surface tension of this liquid in air. Properties of Fluids Example

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A change from the liquid State to the gaseous State is known as vaporization. Properties of Fluids Vapor Pressure and Cavitation Vaporization of liquid at certain temperature takes place when pressure above the liquid surface is equal to or less than the vapor pressure ( or the saturation pressure ) of the liquid at that temperature.

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If the pressure at any point in the flowing liquid becomes equal to or less than the vapor pressure, the vaporization of the liquid starts, producing bubbles of these vapors. The bubbles of these vapors are carried by the flowing liquid into the region of high pressure where they collapse, giving rise to high impact pressure. The pressure developed by the collapsing bubbles (pitting action) is so high that the material from the adjoining boundaries gets eroded and cavities are formed on them. This phenomenon is known as cavitation . Properties of Fluids Vapor Pressure and Cavitation

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Fluids Statics

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Buoyancy Fluids Statics It is a common experience that an object feels lighter and weighs less in a liquid than it does in air. This suggests that a fluid exerts an upward force on a body immersed in it. This upward force is called the buoyant force and is denoted by F B . The buoyant force is caused by the increase of pressure in a fluid with depth . Thus, we conclude that “ the buoyant force acting on a body is equal to the weight of the liquid displaced by the body ” Note that the buoyant force is independent of the distance of the body from the free surface. It is also independent of the density of the solid body.

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Archimedes’ principle “The buoyant force acting on a body immersed in a fluid is equal to the weight of the fluid displaced by the body, and it acts upward through the centroid of the displaced volume” For floating bodies , the weight of the entire body must be equal to the buoyant force, which is the weight of the fluid whose volume is equal to the volume of the submerged portion of the floating body. That is, Therefore, the submerged volume fraction of a floating body is equal to the ratio of the average density of the body to the density of the fluid. Note: When the density ratio is equal to or greater than one, the floating body becomes completely submerged. Fluids Statics

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What happens when a body is immersed in a fluid : (2) remains at rest at any point in the fluid when its density is equal to the density of the fluid (3) sinks to the bottom when its density is greater than the density of the fluid (1) rises to the surface of the fluid and floats when the density of the body is less than the density of the fluid Fluids Statics

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Some practical examples Rise of warm air in a cooler environment and thus the onset of natural convection currents. Rise of hot air balloons or helium balloons. Rise of water vapor to high elevations. Movement of air in the atmosphere. Floating of the continents on a sea of magma. Fluids Statics

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Example Fluids Statics

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Example …… Fluids Statics

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Example …… Fluids Statics

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Stability of Immersed and Floating Bodies Fluids Statics For floating bodies such as ships, stability is an important consideration for safety

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Stability of Immersed and Floating Bodies Fluids Statics Case (a): Stable Since any small disturbance (someone moves the ball to the right or left) generates a restoring force (due to gravity) which returns it to its initial position. Case (b) : Neutrally stable Because if someone moves the ball to the right or left, it would stay put at its new location. It has no tendency to move back to its original location, nor does it continue to move away. Case (c): Unstable This is a situation in which the ball may be at rest at the moment, but any disturbance, even an infinitesimal one, causes the ball to roll off the hill — it does not return to its original position; rather it continue to move away. This situation is unstable .

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For an immersed or floating body in static equilibrium, the weight and the buoyant force acting on the body balance each other , and such bodies are inherently stable in the vertical direction . The rotational stability of an immersed body depends on the relative locations of the center of gravity G of the body and the center of buoyancy B , which is the centroid of the displaced volume. An immersed body is stable if the body is bottom-heavy and thus point G is directly below point B. A rotational disturbance of the body in such cases produces a restoring moment to return the body to its original stable position. Stability of Immersed Bodies Fluids Statics

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Thus, a stable design for a submarine calls for the engines and the cabins for the crew to be located at the lower half in order to shift the weight to the bottom as much as possible. An immersed body whose center of gravity G is directly above point B is unstable , and any disturbance will cause this body to turn upside down. A body for which G and B coincide is neutrally stable . Stability of Immersed Bodies Fluids Statics

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If the center of gravity G is directly below the center of buoyancy B , the body is always stable. But unlike immersed bodies, a floating body may still be stable when G is directly above B. This is because the centroid of the displaced volume ( B ) shifts to the side to a point B’ during a rotational disturbance while the center of gravity G of the body remains unchanged. If point B is sufficiently far, these two forces create a restoring moment and return the body to the original position. A measure of stability for floating bodies is the metacentric height GM , which is the distance between the center of gravity G and the metacenter M —the intersection point of the lines of action of the buoyant force before and after rotation. Stability of Floating Bodies Fluids Statics

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A floating body is stable if point M is above point G , and thus GM is positive Stability of Floating Bodies Fluids Statics The floating body is unstable if point M is below point G , and thus GM is negative. In this case, the weight and the buoyant force acting on the tilted body generate an overturning moment instead of a restoring moment, causing the body to capsize. The length of the metacentric height GM above G is a measure of the stability: the larger the GM , the more stable is the floating body.

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