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а. Using the definition of bulk modulus, derive an expression for the density field, p, in terms of pressure, p , and the given parameters. Leave all parameters as algebraic. (At this point, your expression should not include z .)

а. Using the definition of bulk modulus, derive an expression for the density field, p, in terms of pressure, p , and the given parameters. Leave all parameters as algebraic. (At this point, your expression should not include z .)

Structural Analysis
Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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For many applications, it is accurate enough to treat water as incompressible. But for a more detailed understanding of fluid flows, compressibility must be taken into account. Fortunately, the bulk modulus of water is close to constant, so let's model the pressure in a motionless water column assuming that the bulk modulus is a constant, K, = 2.1 GPa. The pressure and density at the top surface of the water column are 6. Po =1000 hPa and P, =1000 kg/m², respectively, and the downward acceleration due to gravity is g =9.8 m/s. Take depth, z, to be positive downward with z = 0 at the top %3D %3D %3D surface. Using the definition of bulk modulus, derive an expression for the density field, p, in terms of pressure, p, and the given parameters. Leave all parameters а. as algebraic. (At this point, your expression should not include z .) b. Starting from hydrostatic balance and the expression in part (a), derive an expression for the pressure field, p(z), in terms of z and the given parameters. Leave all parameters as algebraic. с. Using your answers to parts (a) and (b), obtain an expression for the density field in terms of z and the given parameters. Leave all parameters as algebraic. d. Using software like MATLAB, plot the density versus depth over the range 0<z<10* m for the parameter values given above. Place depth on the vertical axis with values increasing downward. Plot the percent error of the compressible solution of the density field compared to the value typically used when assuming incompressibility (po). е.
Transcribed Image Text:For many applications, it is accurate enough to treat water as incompressible. But for a more detailed understanding of fluid flows, compressibility must be taken into account. Fortunately, the bulk modulus of water is close to constant, so let's model the pressure in a motionless water column assuming that the bulk modulus is a constant, K, = 2.1 GPa. The pressure and density at the top surface of the water column are 6. Po =1000 hPa and P, =1000 kg/m², respectively, and the downward acceleration due to gravity is g =9.8 m/s. Take depth, z, to be positive downward with z = 0 at the top %3D %3D %3D surface. Using the definition of bulk modulus, derive an expression for the density field, p, in terms of pressure, p, and the given parameters. Leave all parameters а. as algebraic. (At this point, your expression should not include z .) b. Starting from hydrostatic balance and the expression in part (a), derive an expression for the pressure field, p(z), in terms of z and the given parameters. Leave all parameters as algebraic. с. Using your answers to parts (a) and (b), obtain an expression for the density field in terms of z and the given parameters. Leave all parameters as algebraic. d. Using software like MATLAB, plot the density versus depth over the range 0<z<10* m for the parameter values given above. Place depth on the vertical axis with values increasing downward. Plot the percent error of the compressible solution of the density field compared to the value typically used when assuming incompressibility (po). е.
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