HOMEWORx - 1 1) What is the primary reason for nondimensionaliaing an equation? 2) List the three primary purposes of dimensional analysis? 3) The pressure difference P, across apartial blockage in an artery (called a stenosis) is approximated by the equation p=KvDV+Ks(AiAe1)2V2 where, V is the blood velocity, is the blood viscosity, D the artery diameter, A0 the area of the unobstructed artery, and A1 is the area of the stenosis. Determine the dimensions of Ky and KU. Would this equation be valid in any system of units? 4) The volume rate of flow, Q, through a plpe containing a slowly moving liquid is given by the equation Q=8IR4P Where R is the pipe radius, P the pressure drop alone the pipe, is viscosity and I the length of the pipe. What are the dimensions of constant a ? Would you classify thia equation as a general homogenous equation? Explain. 5) It is known that the variation of pressure, P, within a static fluid is dependent upen the specific weight of the fluid and the elevation difference, z. Using dimensional analysis, find the form of the hydrostatic equation for pressure variation. 6) Water sloshes back and forth in a tank as shown in the figure. The frequency of the sloshing. w, is assumed to be a function of acceleration of gravity, g, the average depth of water, h, and the length of the tanik, L. Develope a suitable set of dimensionless parameters for this problem using, g and I as repeating variables. 7) At a sudden contraction in a pipe the diameter changes from D1 to D2. The pressure drop, P, which developes accross the contraction is a function of D2 and D2 as well as the velocity, V, in the larger pipe, and the fluid density, p,and viscosity, . Use D5,V and as repeating variables to determine a suitable set of dimensionless variables. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable? 8) As shown in the figure, a jet of liquid directed against a block can tip over the block. Assume that the velocitiy, V, needed to tip over the block is a function of the fluid density, p, the diameter of the jet, D, the weight of the block, W, the width of the block, b, and the distance, d, between the jet and the bottom of the block. Determine a set of dimensionless parameters for this system. Form the dimensionless parameters by inspection. 9) The water velocity at acertain point along a 1:10 scale model of a dam spillway is 3m/s. What is the corresponding prototype velocity if the model and prototype operate in accordance with froude number similarity? 10) An East Coast estuary has a tidal period of 12.42h (the semidiurnal lunat tide) and tidal currents of approximately 80cm/s. If a one-five-hundredth-scale model is constructed with tides driven by a pump and storage apparatus, what should the period of the model tides be and what model current speeds are expected? 11) A prototype ocean-platform piling is expected to encounter currents of 150cm/s and waves of 12-s period and 3-m height. If a one-fifteenth-scale model is tested in a wave channel, what current speed, wave period, and wave height should be encountered by the model? HOMEWORx - 1 1) What is the primary reason for nondimensionaliaing an equation? 2) List the three primary purposes of dimensional analysis? 3) The pressure difference P, across apartial blockage in an artery (called a stenosis) is approximated by the equation p=KvDV+Ks(AiAe1)2V2 where, V is the blood velocity, is the blood viscosity, D the artery diameter, A0 the area of the unobstructed artery, and A1 is the area of the stenosis. Determine the dimensions of Ky and KU. Would this equation be valid in any system of units? 4) The volume rate of flow, Q, through a plpe containing a slowly moving liquid is given by the equation Q=8IR4P Where R is the pipe radius, P the pressure drop alone the pipe, is viscosity and I the length of the pipe. What are the dimensions of constant a ? Would you classify thia equation as a general homogenous equation? Explain. 5) It is known that the variation of pressure, P, within a static fluid is dependent upen the specific weight of the fluid and the elevation difference, z. Using dimensional analysis, find the form of the hydrostatic equation for pressure variation. 6) Water sloshes back and forth in a tank as shown in the figure. The frequency of the sloshing. w, is assumed to be a function of acceleration of gravity, g, the average depth of water, h, and the length of the tanik, L. Develope a suitable set of dimensionless parameters for this problem using, g and I as repeating variables. 7) At a sudden contraction in a pipe the diameter changes from D1 to D2. The pressure drop, P, which developes accross the contraction is a function of D2 and D2 as well as the velocity, V, in the larger pipe, and the fluid density, p,and viscosity, . Use D5,V and as repeating variables to determine a suitable set of dimensionless variables. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable? 8) As shown in the figure, a jet of liquid directed against a block can tip over the block. Assume that the velocitiy, V, needed to tip over the block is a function of the fluid density, p, the diameter of the jet, D, the weight of the block, W, the width of the block, b, and the distance, d, between the jet and the bottom of the block. Determine a set of dimensionless parameters for this system. Form the dimensionless parameters by inspection. 9) The water velocity at acertain point along a 1:10 scale model of a dam spillway is 3m/s. What is the corresponding prototype velocity if the model and prototype operate in accordance with froude number similarity? 10) An East Coast estuary has a tidal period of 12.42h (the semidiurnal lunat tide) and tidal currents of approximately 80cm/s. If a one-five-hundredth-scale model is constructed with tides driven by a pump and storage apparatus, what should the period of the model tides be and what model current speeds are expected? 11) A prototype ocean-platform piling is expected to encounter currents of 150cm/s and waves of 12-s period and 3-m height. If a one-fifteenth-scale model is tested in a wave channel, what current speed, wave period, and wave height should be encountered by the model