## Equation Of Motion For Vortex Flow

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# Equation of Motion For Vortex Flow

Consider a fluid element ABCD rotating at a uniform velocity in a horizontal plane about an axis O.

Let r = radius of element from O

Δ r = radial thickness of the element

ΔA = Area of cross-section f element

Δθ = Angle subtended by the element at O.

The force acting on the element are

(i) Centrifugal force. mv

(ii) Pressure force P. ΔA on the face AB

(iii) Pressure force ( P + ∂P / ∂r Δr ) ΔA on the face CD.

Equating the forces in radial direction,

Net force = Time rate change of momentum

( P + ∂P / ∂r Δr ) ΔA-P. ΔA = mv

but mass = mass density x volume

m = ρ. ΔA. Δr

∂P / ∂r Δr. ΔA = ρ. ΔA. Δr v

∂P / ∂r = ρ. v

The expression ∂P / ∂r is called pressure gradient in the radial direction

As ∂P / ∂r is positive, hence pressure increases with the increase of radius r.

The pressure variation in the vertical plane is given by hydrostatic law,

∂P / ∂z = -Pg

As the pressure is the function of r and z, therefore total derivative of P.

dp = ∂P / ∂r dr +∂P / ∂z dz

Substituting the values of ∂P / ∂r and ∂P / ∂z from equation

dp = ρv

Equation gives the variation of pressure of a rotating fluid in any plane.

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Let r = radius of element from O

Δ r = radial thickness of the element

ΔA = Area of cross-section f element

Δθ = Angle subtended by the element at O.

The force acting on the element are

(i) Centrifugal force. mv

^{2}/ r acting away from the centre at O,(ii) Pressure force P. ΔA on the face AB

(iii) Pressure force ( P + ∂P / ∂r Δr ) ΔA on the face CD.

Equating the forces in radial direction,

Net force = Time rate change of momentum

( P + ∂P / ∂r Δr ) ΔA-P. ΔA = mv

^{2}/ rbut mass = mass density x volume

m = ρ. ΔA. Δr

∂P / ∂r Δr. ΔA = ρ. ΔA. Δr v

^{2}/ r∂P / ∂r = ρ. v

^{2}/ rThe expression ∂P / ∂r is called pressure gradient in the radial direction

As ∂P / ∂r is positive, hence pressure increases with the increase of radius r.

The pressure variation in the vertical plane is given by hydrostatic law,

∂P / ∂z = -Pg

As the pressure is the function of r and z, therefore total derivative of P.

dp = ∂P / ∂r dr +∂P / ∂z dz

Substituting the values of ∂P / ∂r and ∂P / ∂z from equation

dp = ρv

^{2}/ r dr- ρ g dzEquation gives the variation of pressure of a rotating fluid in any plane.

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