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We have already derived the continuity equation in three dimensions in differential form. It is also possible to get continuity equation using the same system for all flows in a cylindrical polar coordinate system as shown in Fig. 4.7

Continuity Equation in a Cylindrical Polar Coordinate System” = C

Suppose the coordinate system of an elementary control volume be (r, θ, z)

In the direction of r, the Rate of mass enters the system can be given by ρvr rdθdz

Rate of mass leaves the system be (ρ + ∂ρ/∂r  dr)(v_r  + (∂v_r)/∂r  dr) rdθdz

Since,

For control volume:

Rate of mass entering control volume = Rate of mass leaves + Rate of accumulation of mass

Substituting the values, we get-

ρvrrdθdz    =    (ρ + ∂ρ/∂r  dr )(v_r  + (∂v_r)/∂r  dr )rdθdz    +    Rate of accumulation of mass

Or,

Rate of accumulation

of mass    =    ρ [vrrdθdz] – (ρ + ∂ρ/∂r  dr )(v_r  + (∂v_r)/∂r  dr )rdθdz

=    ρ [vrrdθdz] – [ρvr + ρ (∂v_r)/∂r  dr + vr∂ρ/∂r  dr + ∂ρ/∂r  .  (∂v_r)/∂r(dr)2] [rdθdz + drdθdz]

Neglecting higher order terms, so ∂ρ/∂r  .  (∂v_r)/∂r(dr)2 = 0

Or,
=    ρ [vr rdθdz] – [ρvr + ρ (∂v_r)/∂r  dr + vr∂ρ/∂r  dr] [rdθdz + dr dθdz]

=    ρ [vr rdθdz] –[ρvrrdθdz+ ρ (∂v_r)/∂r rdrdθ dz + vr∂ρ/∂r rdrdθ dz

+ ρvrdrdθdz]

(Other higher order terms are neglected)

=    ρvrdr dθdz – (ρ (∂v_r)/∂r  + v_r  ∂ρ/∂r)rdrdθdz

Or,

Rate of accumulation of mass    =    – ρvrdr dθdz– ρ (∂v_r)/∂rrdr dθdz

Now,

Rate of accumulation of mass in θ direction can be given by,

Rate of mass enters beρv_θdrdz

Rate of mass leaves be (ρ + ∂ρ/r∂θ  dθ)(v_θ  + (∂v_θ)/r∂θ  dθ)drdz

So,

Rate of accumulation of mass = ρv_θdrdz– (ρ + ∂ρ/r∂θ  dθ)(v_θ  + (∂v_θ)/r∂θ  dθ)drdz

= ρv_θdrdz–[ρv_θ+ρ (∂v_θ)/r∂θ+v_θ  ∂ρ/r∂θ  dθ+0] dr dz

(Neglecting higher order terms, so we can write zero ‘0’)

= ρv_θdrdz–ρv_θdrdz + (ρ (∂v_θ)/r∂θ+v_θ  ∂ρ/r∂θ  dθ)drdz

Or,

Rate of accumulation of mass =– (∂(ρv_θ))/r∂θrdr dθdz

Similarly,

Rate of accumulation of mass in z direction can be given by,

Rate of mass enters be ρv_zrdθdr

Rate of mass leaves be (ρ + ∂ρ/r∂z  dz)(v_z  + (∂v_z)/∂z  dz)rdθdr

So,

Rate of accumulation of mass = ρv_zrdθdr– (ρ + ∂ρ/r∂z  dz )(v_z  + (∂v_z)/∂z  dz)r dθdr

= ρv_zrdθdr–[ρv_z+ v_z  ∂ρ/r∂z  dz+ρ (∂v_z)/∂z  dz+0]r dθdr

(Neglecting higher order terms, so we can write zero ‘0’)

= ρv_zrdθdr– ρv_zrdθdr –(ρ (∂v_z)/∂z+v_z  ∂ρ/∂z)r dθdrdz

Or,

Rate of accumulation of mass = –(∂(ρv_z))/∂zrdr dθdz

Adding all above equations of rate of accumulation of mass, we get the rate of accumulation of an elementary control volume –

= –ρvrdr dθdz– ρ (∂v_r)/∂rrdr dθdz– (∂(ρv_θ))/r∂θrdr dθdz– (∂(ρv_z))/∂zrdr dθdz

Or,

∂ρ/∂trdr dθdz = – [(ρv_r)/r  – ρ (∂v_r)/∂r– (∂(ρv_θ))/r∂θ– (∂(ρv_z))/∂z]rdr dθdz

Or,

∂ρ/∂t = – [(ρv_r)/r  – ρ (∂v_r)/∂r– (∂(ρv_θ))/r∂θ– (∂(ρv_z))/∂z]

= –[(∂(〖ρv〗_r))/∂r  – 1/r.(∂(ρv_θ))/∂θ–(∂(ρv_z))/∂z  – (ρv_r)/r]
Or,

∂ρ/∂t + [(∂(〖ρv〗_r))/∂r+1/r.(∂(ρv_θ))/∂θ+(∂(ρv_z))/∂z+(ρv_r)/r] = 0

This is the three-dimensional Continuity Equation for unsteady compressible flow.

Note:
The above Continuity Equation can be formed for two-dimensionalcoordinate system (r, θ) for different fluid flow-

    For Unsteady Compressible Flow:

The equation can be written as-

∂ρ/∂t + [(∂(〖ρv〗_r))/∂r+ 1/r.(∂(ρv_θ))/∂θ+ (ρv_r)/r] = 0

    For Steady Compressible Flow:

The equation can be written as,

(∂(〖ρv〗_r))/∂r+ 1/r.(∂(ρv_θ))/∂θ+ (ρv_r)/r=  0

    For Unsteady Incompressible Flow:

The equation can be given by,

(∂(v_r))/∂r+ 1/r.(∂(v_θ))/∂θ+ v_r/r=  0

    For Steady Incompressible Flow:

Equation should be-

(∂(v_r))/∂r+ 1/r.(∂(v_θ))/∂θ+ v_r/r=  0