## Continuity Equation In Three Dimensions

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# Continuity Equation in Three-dimensions

Continuity equation is based on the principle of conservation of mass. It states, “When a fluid flowing though the pipe at any section, the quantity of fluid per second remains constant”.

Consider a fluid element of lengths dx, dy, dz in the direction of x, y, z.

Let u, v, w are the inlet velocity components in x,y,z direction respectively.

Let ρ is mass density of fluid element at particular instate.

Mass of fluid entering the face ABCD (In flow)

= Mass density x Velocity x-direction x area of ABCD

= ρ x u x (∂y x ∂z)

Then mass of fluid leaving the face EFGH (out flow) = (ρu∂y . ∂z) + ∂ / ∂x (ρu∂y∂z)

Rate of increases in mass x-direction = Outflow – Inflow

= [ (ρudydx) + ∂ / ∂

Rate of increases in mass x direction = ∂ / ∂

Similarly,

Rate of increase in mass y-direction = ∂ / ∂y ρ v ∂x ∂y ∂z

Rate of increases in mass z-direction = ∂ / ∂

Total rate of increases in mass = (7.7.3) + (7.7.4) + (7.7.5)

= ∂

By law of conservation of mass, there is no accumulation of mass, and hence the above quantity must be zero.

∂x . ∂y. ∂z [ ∂ρu / ∂

∂ (ρu) / ∂

If fluid is incompressible, then is constant

∂

This is the continuity equation for three-dimensional flow.

NOW, for tow-dimensional flow, the velocity component w = 0

Hence continuity equation is, ∂

For more help in

Consider a fluid element of lengths dx, dy, dz in the direction of x, y, z.

Let u, v, w are the inlet velocity components in x,y,z direction respectively.

Let ρ is mass density of fluid element at particular instate.

Mass of fluid entering the face ABCD (In flow)

= Mass density x Velocity x-direction x area of ABCD

= ρ x u x (∂y x ∂z)

Then mass of fluid leaving the face EFGH (out flow) = (ρu∂y . ∂z) + ∂ / ∂x (ρu∂y∂z)

Rate of increases in mass x-direction = Outflow – Inflow

= [ (ρudydx) + ∂ / ∂

_{x}(ρudydz) d_{x}] - (ρudydz)Rate of increases in mass x direction = ∂ / ∂

_{x}ρ u d_{x}d_{y}d_{z}Similarly,

Rate of increase in mass y-direction = ∂ / ∂y ρ v ∂x ∂y ∂z

Rate of increases in mass z-direction = ∂ / ∂

_{z }ρ w ∂_{x }∂y ∂_{z}Total rate of increases in mass = (7.7.3) + (7.7.4) + (7.7.5)

= ∂

_{x}∂_{y}∂_{z}[ ∂ρu / ∂_{x}+ ∂ρv / ∂_{y}+ ∂ρw / ∂_{z}]By law of conservation of mass, there is no accumulation of mass, and hence the above quantity must be zero.

∂x . ∂y. ∂z [ ∂ρu / ∂

_{x}+ ∂ρv / ∂_{y}+ ∂ρw / ∂_{z}] = 0∂ (ρu) / ∂

_{x}+∂ (ρv) / ∂_{y}+ ∂ (ρw) / ∂_{z}= 0........for compressible fluidIf fluid is incompressible, then is constant

∂

_{u}/ ∂_{x}+ ∂_{v}/ ∂_{y }+ ∂_{w}/ ∂_{z}= 0This is the continuity equation for three-dimensional flow.

NOW, for tow-dimensional flow, the velocity component w = 0

Hence continuity equation is, ∂

_{u }/ ∂_{x}+ ∂_{v}/ ∂_{y}= 0For more help in

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