Bill is working on an electrical circuit problem. He remembers from his electrical engineering class that voltage drop
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Fluid Mechanics: Fundamentals and Applications
- 5.13 The torque due to the frictional resistance of the oil film between a rotating shaft and its bearing is found to be dependent on the force F normal to the shaft, the speed of rotation N of the shaft, the dynamic viscosity of the oil, and the shaft diameter D. Establish a correlation among these variables by using dimensional analysis.arrow_forward(a) Discuses three necessary conditions for complete similarity between a model and a prototype in dimensional analysis. (b) In the design and development competition in a University, a team of students has planned to develop a prototype of submarine that would travel fully submerged at a certain speed in a fresh lake water. The team has designed a model which has one-sixth of a prototype size to test in the university wind tunnel to study the effect of various parameters.What will happen to the length of the model if the temperature of the blowing wind increases as a result of the season change in summer compared to winter?Considering the necessary assumptions discuss the reason.arrow_forwardCold water enters a pipe, where it is heated by an external heat source . The inlet and outlet water temperatures are Tin and Tout, respectively. The total rate of heat transfer Q. from the surroundings into the water in the pipe is Q. = m·cp(Tout − Tin) where m . is the mass flow rate of water through the pipe, and cp is the specific heat of the water. Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.arrow_forward
- Some children are playing with soap bubbles, and you become curious as to the relationship between soap bubble radius and the pressure inside the soap bubble . You reason that the pressure inside the soap bubble must be greater than atmospheric pressure, and that the shell of the soap bubble is under tension, much like the skin of a balloon. You also know that the property surface tension must be important in this problem. Not knowing any other physics, you decide to approach the problem using dimensional analysis. Establish a relationship between pressure difference ΔP = Pinside − Poutside, soap bubble radius R, and the surface tension ?s of the soap film.arrow_forwardWrite the primary dimensions of each of the following variables from the field of solid mechanics, showing all your work: (a) moment of inertia I; (b) modulus of elasticity E, also called Young’s modulus; (c) strain ? ; (d) stress ?. (e) Finally, show that the relationship between stress and strain (Hooke’s law) is a dimensionally homogeneous equation.arrow_forwardDimensional analysis can be used in problems other than áuid mechanics ones. The important variablesaffecting the period of a vibrating beam (usually designated as T and with dimensions of time) are the beamlength `, area moment of inertia I, modulus of elasticity E, material density , and Poissonís ratio , so thatT = f cn(`; I; E; ; )Recall that the modulus of elasticity has typical units of N/m2 and Poissonís ratio is dimensionless.(a) Find dimensionless version of the functional relationship.(b) If E and I must always appear together (meaning that EI is e§ectively a single variable), Önd a dimensionless version of the functional relationship.arrow_forward
- The Stokes-Oseen formula for drag force on a sphere at low speed is given asD = 3dV +916V 2d2, where D is drag, V is velocity, is density, d is the sphere diameter, and is the viscosity coe¢ cient.(a) Using the formula given, Önd the dimensions of the viscosity coe¢ cient. (Donít simply look upthe dimensions; use the formula to show them.) Be sure to show your work. Find the primaryunits of viscosity in SI and British units.(b) Verify that the Stokes-Oseen formula is dimensionally homogeneous.arrow_forwardThe Stokes number, St, used in particle dynamics studies,is a dimensionless combination of five variables: accelerationof gravity g , viscosity μ , density ρ , particle velocity U ,and particle diameter D . ( a ) If St is proportional to μand inversely proportional to g , find its form . ( b ) Showthat St is actually the quotient of two more traditionaldimensionless groups.arrow_forwardIn the study of turbulent flow, turbulent viscous dissipation rate ? (rate of energy loss per unit mass) is known to be a function of length scale l and velocity scale u′ of the large-scale turbulent eddies. Using dimensional analysis (Buckingham pi and the method of repeating variables) and showing all of your work, generate an expression for ? as a function of l and u′.arrow_forward
- Consider fully developed flow between two infinite parallel plates separated by distance h, with the top plate moving and the bottom plate stationary. The flow is steady, incompressible, and two-dimensional in the xy-plane. a) Use the first principle (dimensional analysis) to generate a dimensionless relationship for the x-component of fluid velocity u as a function of fluid viscosity μ, top plate speed v, distance h, fluid density ρ, and distance y. b) Name the common dimensionless number formed in (a). Hint: modifying the dimensionless number if necessary.arrow_forwardWhen small aerosol particles or microorganisms move through air or water, the Reynolds number is very small (Re << 1). Such flows are called creeping flows. The drag on an object in creeping flow is a function only of its speed V, some characteristic length scale L of the object, and fluid viscosity µ. Use dimensional analysis to generate a relationship for the drag force FD as a function of the independent variables.arrow_forwardWrite the primary dimensions of each of the following variables, showing all your work: (a) specific heat at constant pressure cp; (b) specific weight; (c) specific enthalpy h. and (d) specific gas constantarrow_forward
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning