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GO WITH THE FLOW Introduction to Fluid Dynamics Summer 1999 Demonstration 1: Fluid Properties, Viscosity Brian M. Argrow Aerospace Engineering Sciences OBJECTIVES Definition a fluid Fluid Properties Fluid viscosity Measurement of fluid viscosity BACKGROUND States and Properties of Substances A fluid will continuously deform under any applied stress, i.e., a fluid cannot hold a shape unless supported by some external container. This is in contrast to a solid, which offers resistance to deformation when a stress is applied. The two familiar types of fluids are liquids and gases. The primary distinction between a liquid and a gas is that a liquid can support a free surface, i.e., a boundary at which the liquid is on one side and some other fluid is on the other. A gas will not support a free surface, it will continue to expand until it is confined by some external boundary such as the walls of a container. Properties are the characteristics used to describe the state of a substance. While there are a number of defined properties, generally there are a select few used in thermodynamics (the science of energy) and fluid dynamics (the study of fluid motion) that are of particular importance. One of these properties is the mass density (usually just referred to as the density) of a substance is defined as density amount of substance mass kg , unit volum e of substance unit volum e m 3 where the SI units associated with the property are included in brackets. The definition of a fluid above included a reference to stress applied on the fluid. The stress at a point is the force per unit area applied at that point. Depending on the orientation of the surface on which the force is applied, the force may be decomposed into a part that is parallel (tangential) to the surface, and a part that is perpendicular (normal) to the surface. A fluid at rest can only support a perpendicular force since any tangential force will cause it to move. force normal force component tangential force component GO WITH THE FLOW Introduction to Fluid Dynamics Summer 1999 The fluid pressure p is defined by, pressure p normal force [Pa] or [N/m 2 ] unit area Atoms that comprise a solid always possess some degree of random motion. The energy associated with this random motion is often referred to as "heat" or "thermal energy." In a fluid, groups of atoms, i.e. molecules, are not confined to specific positions with respect to one another. For a liquid the molecules are generally in contact with each for the most part, but for a gas the molecules are generally widely spaced and spend most of the time in motion and only come in contact via intermolecular collisions. We often use imprecise adjectives such as hot, cold, warm, etc. in reference to the property called temperature. Of course these descriptors are quite useless as quantitative measures. A precise definition of temperature is temperatur e T measure of thermal energy [K] . Keeping all properties other than temperature fixed, fluid states have more collective random motion than the solid state, and thus the fluid states possess more thermal energy and are observed at correspondingly higher temperatures. As an example, water at 1-atmosphere (101,325 Pa) pressure is a solid at the freezing point T = 273 K. As the temperature increases above this value the water melts into a fluid mixture of liquid and gaseous water, finally becoming totally gaseous as it is heated beyond the boiling point. An important result from thermodynamics is that all the basic thermodynamic properties of a substance can be determined if any pair of special state variables (this includes , p, and T) are specified. So, for instance if p and T are specified, there is a unique corresponding value of . Fluid Viscosity Even though we described solids and fluids in terms of atoms and molecules, except for the special fields of rarefied gas dynamics and molecular dynamics, fluid dynamics is usually developed in terms of a continuum. A continuum description assumes that even the smallest piece of the fluid, a fluid element, is large enough that the behavior of individual molecules is unimportant and the bulk fluid properties are representative of an average of the molecular properties. In the definition of a fluid from the previous section, it was stated that a fluid continuously deforms under an applied stress. The normal component of this stress was defined to be the pressure p. Since the pressure is always in the normal direction, it does not act to deform the fluid element, it only acts to change the volume. The tangential stress component called the shear stress, however acts to deform the fluid element as shown in the figure below. The figure shows these effects on a two-dimensional fluid element pressure shear GO WITH THE FLOW Introduction to Fluid Dynamics Summer 1999 Rub your hands together and you feel the familiar sensation of friction. Because your skin is a solid, it resists the deforming shearing motion between your hands. You will also notice that if your rub your hands together quickly, you will begin to feel warmth. The friction converts some of the energy of motion between your hands into thermal energy (heat). Friction is also present in all real fluids. For most fluids, the friction, or resistance to deformation, is proportional to the rate at which the fluid is deformed (from the shear stress). This resistance is generally dependent upon the temperature of the fluid and is characterized by a property called the dynamic viscosity [Ns/m2]. This property provides a measure of how much the fluid resists the shearing motion. A large value of is characteristic of a fluid with large resistance to shearing motions relative to the low resistance of a fluid with a small value The simplest mathematical description of fluid motion around immersed bodies, for which some of the forces between the fluid and the immersed body can be computed, is called hydrodynamic theory. This theory describes a “perfect fluid,” i.e. one without viscosity (inviscid). When hydrodynamic theory was first developed it provided a means for computing the “lift force” on an immersed body. By definition, the lift is the force on a body that is perpendicular to the flow direction, so it is associated with the fluid pressure. Hydrodynamic theory fails completely however, for computing the “drag force,” that is the forced defined to be parallel and in the direction of the flow. A complete description of fluid flow and the forces on an immersed body must include the effects of viscosity. Measuring Fluid Viscosity from Drag on an Immersed Body As described, the drag force on an immersed body is in the direction of the flow; thus it works to retard the motion of a body through a fluid. The diagram below is a schematic of a sphere of radius a falling freely in a fluid. The weight of the sphere is W b gV , the buoyancy force is FB gV , and D represents the drag force acting on the sphere. Here is again the density of the fluid, b is the density of the sphere, and V is the volume of the sphere. In the schematic, the sphere is assumed to have reached its terminal velocity Ut. When it is released into the fluid, it accelerates to the terminal velocity. Once this velocity is reached, it no longer accelerates and all the forces on the sphere are in equilibrium. D FB a Ut W The drag force on immersed bodies with simple shapes can be correlated to the speed with which the body moves through the fluid. This is achieved by specifying the drag coefficient CD defined by CD drag D 1 , inertial force 2 U 2 S GO WITH THE FLOW Introduction to Fluid Dynamics Summer 1999 where D is the drag, is the density of the fluid, U is the speed of the fluid approaching the body, and S is the projected frontal area, i.e., the maximum area perpendicular to the flow direction. The subscript indicates “freestream” quantities, i.e. quantities that are measured in the undisturbed fluid far upstream of the body. In general, the overall drag force is composed of a component purely from friction and another component, called profile drag that results from the finite size and shape of the body. A number of experiments have been performed to determine CD for several geometries. These experiments show that the variation of CD depends primarily on a parameter called the Reynolds number Re, defined by Re inertial force U L , viscous force where L is some characteristic length (diameter in the case of the sphere) and the other quantities are as defined earlier. A flow with a relatively large value for Re is dominated by inertial forces, thus appears nearly inviscid. In the case of a very low-Re flow, called creeping flow or Stokes’ flow, the inertial forces can be neglected and Newton’s second law of motion reduces to Stokes’ equation for a sphere, valid for Re < 1, D 6 Ua . If the velocity (speed) V in this equation is the terminal velocity Ut of the sphere of radius a, it provides a means for computing the viscosity by writing the equation for the balance of forces on the sphere, D FB W . Or substituting with Stokes’ equation, we have finally, W FB W FB , 6 U t a 3 U t d where d is the sphere diameter. In the following experiment, use this relation to compute and compare the viscosities of a few common liquids. Experimental Procedure 1. Measure the diameter and weight of the test sphere, and then compute the volume and density. 2. Measure the density of the test fluid. a) Measure the weight of an empty beaker. b) Fill the beaker to some prescribed volume with the test fluid and record the weight of the beaker/fluid combination. c) Compute the fluid density by converting the weight of the liquid to mass and dividing by the volume of liquid. 3. Place timing marks on the cylinders containing the fluids. The speed of the falling sphere is determined by dividing the distance between the marks by the time it takes the sphere to fall that distance. 4. Compute the coefficient of viscosity