Mechanical Properties of Fluids

Question

Derive the relation between a and γ. Assume that coefficient of linear expansions are same in all directions.

Solution

Let a rectangular cuboid be of sides L₀, B₀ and H₀ be at 0°C.
Therefore the volume of cuboid at 0°C is V_Q = L₀B₀H₀
When cuboid is heated, all of its sides will expand and hence volume.
If

straight alpha

is the coefficient of linear expansion and γ is coefficient of cubical expansion, then at temperature θ°C,

straight L space equals space straight L subscript 0 left parenthesis 1 plus αθ right parenthesis comma straight B equals straight B subscript 0 left parenthesis 1 plus αθ right parenthesis comma straight H space equals space straight H subscript 0 left parenthesis 1 plus αθ right parenthesis comma

and

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But,

straight V equals LBH

Therefore,

straight V subscript 0 left parenthesis 1 plus γθ right parenthesis space equals space straight L subscript 0 left parenthesis 1 plus αθ right parenthesis straight B subscript 0 left parenthesis 1 plus αθ right parenthesis space straight H subscript 0 left parenthesis 1 plus αθ right parenthesis

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rightwards double arrow

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Since a<<1, therefore using binomial expansion,

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rightwards double arrow

straight gamma equals 3 straight alpha

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Mechanical Properties Of Fluids

Derive the relation between a and γ. Assume that coefficient of linear expansions are same in all directions.

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