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04. a). A control volume is an arbitrary volume of space throughwhich fluid can flow. Let the control volume of a spherical shapedblob in a flowing fluid is 𝑉. Suppose the fluid has density 𝜌(𝑟,𝑡)and velocity field 𝑢. Then applying mass conservation to theblob,
Rate of change of mass in = − Mass out flow.
The total mass of fluid entering to the blob in a control volume𝑉 is
𝑚 = ∭𝜌 𝑉 𝑑𝑉.
The mass out flow is given by,
∬(𝜌𝑢) 𝑆 𝑑𝑆
where, 𝑆 is the surface bounded by the control volume 𝑉. Applythe Gauss Divergence Theorem to show that,
𝜕𝜌 𝜕𝑡 + ∇ ⋅ (𝜌𝑢) = 0
b). Let the rate of change of 𝑁 is given by,
𝑑𝑁 𝑑𝑡 = 𝜕 𝜕𝑡 ∫ 𝜇𝜌 𝑉 𝑑𝑉 + ∫ 𝜇𝜌𝑢 𝑆 𝑑𝑆.
Newton’s Second Law for a system in an inertial frame is, 𝑑𝑃 𝑑𝑡 = 𝐹.
where 𝑃 is the linear momentum and 𝐹 be the total force of thesystem.
Suppose a spherical shaped arbitrary fluid drop with radius 𝑎 isfalling from a fluid container which is above the ground level.Let,
𝜇 = 𝑢 = 𝑟 2𝑒𝑟 and 𝜌(𝑡, 𝑟) = 𝑡 3 .
Hence find the total force on the fluid drop at 𝑡 = 2.
Here, 𝑟 is the radius of the fluid drop and 𝑒𝑟 be theperpendicular unit vector to the surface of a sphere. State everyassumption that you made to provide the answer.