Continuity Equation

Question 1

What is the philosophy of the Continuity Equation

Answer 1

1. Apply the physical principal which is mass is conserved
2. Apply a suitable model of the flow, which is a fixed differential CV
3. Derive the equation representing the physical principal - this gives the continuity equation

Question 2

Express in physical principal in equation form

Answer 2

dm/dt = (m(dot)x - m(dot)x+dx) + (m(dot)y - m(dot)y+dy)

This then turns into:

d/dt(rho.dx.dy.1) = ((rho.u.dy.1)x-(rho.u.dy.1)x+dx) + ((rho.v.dx.1)y–(rho.v.dx.1)y+dy))

Question 3

Express continuity as 2D and 3D equations

Answer 3

2D: d rho/dt + d/dx(rho.u) + d/dy(rho.v) = 0

3D: d rho/dt + d/dx(rho.u) + d/dy(rho.v) + d/dz(rho.w) = 0

Question 4

Why is Taylor series used in the derivation of the continuity equation?

Answer 4

To relate the mass flow at the outlet to mass flow at the inlet

Question 5

Explain rate of change of Phi in terms of continuity equation

Answer 5

Rate of change of phi = net flow transport by convection + net molecular transport by diffusion + sources/sink + effects on CV surface

Question 6

Define velocity vector U

Answer 6

u = ui + vj + wk and must exist in Cartesian space

Question 7

What is continuity equation expressed as with del operator present?

Answer 7

d rho/dt + del. rho. u = 0

Question 8

Define del operator

Answer 8

del - id/dx +jd/dy + kd/dz

Question 9

State the steady flow continuity equation

Answer 9

Steady flow means nothing varies with time, ergo the time dependant variable can be ignored.
so d/dx(rho.u) + d/dy(rho.v) + d.dz(rho.w) = 0

Question 10

State incompressible flow continuity equation

Answer 10

Incompressible means that rho is constant and therefore can be taken outside the differential leaving -
du/dx + dv/dy + dw/dz = 0