## Continuity Equation in a Polar Form

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# Continuity Equation in a Polar Form

Consider a two-dimensional incompressible flow field. The two dimensional polar coordinates are ‘r’ and θ angle subtended by elements is 'dθ' . Consider a fluid elements ABCD between radius ‘r’ and (r + dr) dθ .

The components of velocity in radial direction Vr ad tangential direction V

Assume thickness of element perpendicular to the plane of paper is unity.

Consider flow in radial direction:

Mass if fluid entering the face AB during the time dt,

Fluid influx = Density x Velocity in r-direction x Area x Time

ρ . Vr (rdθ x 1) dt Thickness = 1

Mass fo fluid leaving the face CD during the same time dt is,

Fluid efflux = ρ ( Vr + ∂ vr / ∂r dr ) (r + dr) dθ . dt

Mass accumulated in element = Fluid influs-Fluid efflux

ρVr . rdθ . dt - [ ∂vr rdθdt + ρ ∂vr / ∂r . r. drdθ . dt + ρVr . dr . d . dt + ρ dvr / dr . d2 rdθdt ]

Neglect d2r is too small,

= - [ ρV

= - ρ [ V

= - ρ [ V

This is written in this form because is the volume f elemtn.

Consider flow in tangential direction,

= [Mass of fluid entering BC- Mass of fluid leaving CD] dt

= [ ρ V

= [ ρ V

= - ∂V

= - ∂ / ∂θ V

= - 1/r V

According law of conservation of mass, “The total gain mass be equal to the rate of change of luid mass in element ABCD”.

Total gain of mass = Mass accumulate due to radial + Mass accumulate due to tangential

= - [ ρVr + ρ dV / dr ] drdθ . dt + [ ρ ∂V

But, mass of fluid element = ρ x volume of fluid element.

= ρ [ rdθ . drdt ]

Rate of increases of fluid mass in element with time,

= ∂ / ∂t [ ρ rdθ + (r+dr) d / 2 dr dt

= ∂ / ∂t ρ ( rdθ / 2 + rdθ / 2 + drdθ / 2) drdt

dr x dθ = 0

By law of conservation,

∂ / ∂t ( ρ. r. dθ. dr) dt = - [ ρVr + ρr dVr / dr. r + ρ ∂Vθ / ∂θ ] dr. dθ .dt

Consider dt = 1,

∂ / ∂t ( ρ. r. dθ. dr) + [ ρVr + r dVr / dr. r + ρ ∂Vθ / ∂θ ] dr. dθ .dt = 0

For steady flows ∂ / ∂t = 0

[ ρV

For incompressible flow, ρ = constant.

V

Or

∂ / ∂r (r. V

Which is the equation of continuity in polar co-ordinates for two dimensional, steady incompressible flow.

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θ ρ ∂

The components of velocity in radial direction Vr ad tangential direction V

_{θ}.Assume thickness of element perpendicular to the plane of paper is unity.

Consider flow in radial direction:

Mass if fluid entering the face AB during the time dt,

Fluid influx = Density x Velocity in r-direction x Area x Time

ρ . Vr (rdθ x 1) dt Thickness = 1

Mass fo fluid leaving the face CD during the same time dt is,

Fluid efflux = ρ ( Vr + ∂ vr / ∂r dr ) (r + dr) dθ . dt

Mass accumulated in element = Fluid influs-Fluid efflux

ρVr . rdθ . dt - [ ∂vr rdθdt + ρ ∂vr / ∂r . r. drdθ . dt + ρVr . dr . d . dt + ρ dvr / dr . d2 rdθdt ]

Neglect d2r is too small,

= - [ ρV

_{r}. d_{r}. dθ . dt + ρ dV / d_{r}. r. d_{r}. dθ . dt= - ρ [ V

_{r}+ dV_{r}/ d_{r}] d_{r}. dθ .dt= - ρ [ V

_{r}/ r + dV / d_{r}] rd_{r}. dθ .dtThis is written in this form because is the volume f elemtn.

Consider flow in tangential direction,

= [Mass of fluid entering BC- Mass of fluid leaving CD] dt

= [ ρ V

_{θ}. dr x - ρ (V_{θ}+ ∂V_{θ}/ ∂θ . dθ) dr ] dt= [ ρ V

_{θ}drdt - ρ V_{θ}drdt - ∂V_{θ}/ ∂θ . dθ. drdt ]= - ∂V

_{θ}/ ∂θ ρdθ . drdt= - ∂ / ∂θ V

_{θ}rdθ . drdy / r ( Multiplying and divide by r)= - 1/r V

_{θ}/ ∂θ (rdr/ dθ / dt)According law of conservation of mass, “The total gain mass be equal to the rate of change of luid mass in element ABCD”.

Total gain of mass = Mass accumulate due to radial + Mass accumulate due to tangential

= - [ ρVr + ρ dV / dr ] drdθ . dt + [ ρ ∂V

_{θ}/ ∂θ . dθ. dr .dt ]But, mass of fluid element = ρ x volume of fluid element.

= ρ [ rdθ . drdt ]

Rate of increases of fluid mass in element with time,

= ∂ / ∂t [ ρ rdθ + (r+dr) d / 2 dr dt

= ∂ / ∂t ρ ( rdθ / 2 + rdθ / 2 + drdθ / 2) drdt

dr x dθ = 0

By law of conservation,

∂ / ∂t ( ρ. r. dθ. dr) dt = - [ ρVr + ρr dVr / dr. r + ρ ∂Vθ / ∂θ ] dr. dθ .dt

Consider dt = 1,

∂ / ∂t ( ρ. r. dθ. dr) + [ ρVr + r dVr / dr. r + ρ ∂Vθ / ∂θ ] dr. dθ .dt = 0

For steady flows ∂ / ∂t = 0

[ ρV

_{r }+ ρ ∂ / ∂r (V_{r}) r + ∂ / ∂θ ] dr. dθ = 0For incompressible flow, ρ = constant.

V

_{r}+ ∂ / ∂r (Vr) + ∂ / ∂θ . (V_{θ}) = 0Or

∂ / ∂r (r. V

_{r}) + ∂ / ∂ . V_{θ}= 0Which is the equation of continuity in polar co-ordinates for two dimensional, steady incompressible flow.

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