Consider a domain where density varies based on the given equation The velocity

Question

Consider a domain where density varies based on the given equation The velocity field in the domain is defined by the following equation In the above equations, t denotes time and (x,y,z) denotes position inside the domain. 1. Consider the questions below at a given position Xo = (5,30,11) in the domain at a time to = 30 s a. Find the time rate of change of density of a material particle lowing through the domain at point xo at time to. [5] Is this a steady flow? Why, or why not? [1] Based only on your theoretical understanding of fluid mechanics, is this a compressible flow? Why, or why not? [1] b. c. Mach number of this flow. Based on Mach number data, would you predict the flow to be compressible? 13+1] Find the vorticity vector at point xo at time to. Is this flow rotational at that point and time? If yes, what is the angular velocity of that rotation? [3+1/2+1/21 e.

Explanation / Answer

1. given density variation

rho = (2x - 3y)e^(-t^2)

and velocity field

v = 2tx i + 3x^2 j + (2z - y) k

where i j , k are unit vectors along x , y and z directions respectively

also, xo = 5

yo = 30

zo = 11

to = 30 s

a. d(rho)/dt = (2x - 3y)*e^(-t^2)*(-2t) = 2t(3y - 2x)e^(-t^2)

so at xo, yo, zo, to

d(rho)/dt = 3*30(3*30 - 2*5)e^(-30^2) = 9.824*10^-388

b. for a steady flow, d(rho)/dt = 0

i.e. there is no time dependance of density

but as we can see

d(rho)/dt = 2t(3y - 2x)e^(-t^2) != 0

hence

this is not a steady flow

c. as density is a fuinction of time, this means that density changes with time, hence for a flow with time dependent density changes, it is a compressible flow

d. if the material is assumed to be air, ideal gas at 30 C

then , v = 2tx i + 3x^2 j + (2z - y) k

at xo, yo, zo, to

vo = 2*30*5i + 3*25 j + (2*11 - 30)k

vo = 300i + 75j - 8k

so |vo| = 309.336 m/s

now speed of sound in air at 30 C = 334 m/s

hence

mach number = m = |vo|/334 = 0.926

based on mach number the flow is compressible as at mach number close to 1, air at normal standard temperature becomes a compressible flow problem

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