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Fluids Unit 1 – Topic 2 (materials)
Learning Objectives Specification points: (23) be able to use the equation densityρ = m (24) understand how to use the relationship upthrust = weight of fluid displaced (25) (a) be able to use the equation for viscous drag (Stokes’ Law), F = 6πηrv. (b) understand that this equation applies only to small spherical objects moving at low speeds with laminar flow (or in the absence of turbulent flow) and that viscosity is temperature dependent. To be able to: • Understand and use the density formula. • State that an upthrust is provided by the fluid displaced by a submerged or floating object. • Calculate the upthrust in terms of the weight of the displaced fluid. • Recall and apply the principle that, for an object floating in equilibrium, the upthrust is equal to the weight of the new object to new situations or to solve related problems. • Define what viscosity is • Understand and use stokes law.
Success criteria • Successfully………… • A – Apply Stokes’ law in terminal velocity conditions to calculate missing variables and/or to explain different relevant scenarios. • B/C – Using viscosity terminology , such as rate of flow, to explain certain scenarios and solve problems. • C/D – Use Stokes’ law to calculate viscous drag. • E – Use the density formula to calculate density.
Density Fluids What is a fluid? A fluid is any substance that can flow. Usually that’s liquids and gases For example; water, air , oil Density The density of a substance is the relationship between the mass of the substance and how much space it takes up (volume). Density is measure of the mass per unit volume of a substance Unit of density(ρ): kilograms per cubic meter (in SI units) (kg/m3)
Upthrust When an object is submerged in a fluid, upthrust is the upward force that a liquid or gas exerts on a body floating in it. According to Archimedes' Principle, the upthrust on an object in a fluid is equal to the weight of the fluid displaced. So the volume displaced multiplied by the density of the fluid. Upthrust = Weight of Fluid Displaced The volume of this amount of liquid is equal to the volume of the object itself. The weight of fluid displaced and therefore the upthrust will be bigger if the density of the liquid is large.
When do things float? • Any object which is less dense than water will float • When floating, an object will sink until it has displaced an equal weight of water to its own total weight. • If weight is then added to the object, it will sink until it has displaced enough water to again equal its own weight. • If weight is removed, the upthrust will now be greater than its weight, and it will accelerate upwards until the upthrust is again equal to its weight. Ensure that you use the terms weight and mass correctly! Weight is the force, mass is the amount of matter. • Therefore, the factors affecting upthrust are: • density of the fluid • volume of the object immersed Unit of Upthrust: newton (N) or kgm/s2
Worked example A ball of mass 10 kg is held under the surface of a pool. The instant its released, it has an instantaneous acceleration of 4 m/s2 towards the bottom of the pool. What is the volume of the ball? • The net force on the ball is expressed as: • Fnet= ma • Since its accelerating downward, the force of gravity is larger then the buoyant force(upthrust). Therefore; • Fnet = Weight- Upthrust • Therefore; • Fnet = mg - (Vball * ρwater * g) • (10*4 ) = (10g) – (Vball* 1000 * g) • Rearrange; • (10g- 40)/1000g = Vball Answer: 0.0059m3
Fluid movement Fluid flows can be divided into two different types: laminar flow and turbulent flow Laminar flow ( streamline flow) The term streamline flow is descriptive of the flow because, in laminar flow, there is virtually no mixing between layers, fluid particles move in definite and observable paths or streamlines. • In summary, • Laminar flow is characterized by : • the velocity of the fluid at a point is constant • No abrupt change in the direction/speed of flow • The lines of flow do not cross • layers/flowlines/streamlines are parallel Labelling and drawing fluid flow diagrams is not needed for the new specification.
Fluid movement Turbulent flow Turbulent flow, type of fluid flow in which the fluid undergoes irregular fluctuations, or mixing, in contrast to laminar flow, in which the fluid moves in smooth paths or layers. In turbulent flow the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction. Turbulence is also characterized by eddies, and apparent randomness. • In summary, • Turbulent flow is characterized by: • the velocity at a point keeps on changing • Abrupt changes in speed/direction • Eddies/whirlpools form • Layers/flowlines/streamlines mix/cross Labelling and drawing fluid diagrams is not needed for the new specification.
Viscosity A fluids resistance to flow • In a fluid, each 'layer' exerts a force of friction on another 'layer’. • This frictional force is also present when solid object moves through a liquid. This force is termed Viscous Drag. • Viscous Drag is greater in Turbulent Flow than Laminar Flow. We say a thick fluid has high viscosity or is very viscous, while a thin fluid would have low viscosity
Viscosity The size of the Viscous Drag in a fluid depends on the (coefficient of) Viscosity of that fluid. (coefficient of) Viscosity is given the letter η(Eta) and is measured in /Pa s. The greater the Viscosity, the greater the Viscous Drag, therefore the lower the rate of flow (rate of flow decreases as viscosity increases). Coefficient of viscosity; a numerical value given to a fluid to indicate how much it resists flow • In most liquids, Viscosity decreases as temperature increases, (temperature is inversely proportional to viscosity) • whereas in most gases, Viscosity increases as temperature increases.
Stokes’ law It is possible to calculate the drag force exerted on a spherical object in a fluid using Stoke's Law: F = 6πηrv • r is the radius of sphere (m) • η is the viscosity (Pa s) • v is velocity (m/s) • Conditions: • Laminar flow, • slow moving(reach its terminal velocity) • Small smooth spherical object
Terminal velocity Consider a sphere falling through a viscous fluid. As the sphere falls , its velocity increases until it reaches a velocity known as the terminal velocity. Terminal velocity is achieved, therefore, when the speed of a moving object is no longer increasing or decreasing; the object’s acceleration is zero At this velocity the frictional drag (due to viscous forces ) and upthrust is balanced by the gravitational force and the velocity is constant • This means that the forces acting on the object are balanced. An equation can be formed by equating Weight with Upthrust and Viscous Drag • weight – upthrust – viscous drag = 0 • or • upthrust + viscous drag = weight
Terminal velocity Terminal velocity graphs Velocity-time graph Acceleration-time graph
Terminal velocity weight = upthrust + viscous drag • It can be expressed as the following; • This formula can be rearranged to solve for v ( terminal velocity);
Terminal velocity When carrying out an experiment to determine the viscosity, by dropping a spherical object in a fluid for example, a graphical method can be used to determine a value • Some measurements need to be taken/ obtained first, such as; • radius of object • density of object ( mass can be rather obtained) • density of fluid • Calculate the velocity of object ( terminal velocity) • Then, • Plot a graph of v against in (y –axis) and (x- axis) respectively • measure the gradient Hence, the coefficient of viscosity can be obtained using the following formula;
Success criteria Which grade are you on? • Successfully………… • A – Apply Stokes’ law in terminal velocity conditions to calculate missing variables and/or to explain different relevant scenarios. • B/C – Using viscosity terminology , such as rate of flow, to explain certain scenarios and solve problems. • C/D – Use Stokes’ law to calculate viscous drag. • E – Use the density formula to calculate density.
