Derivation Of Continuity Equation 72k1t

7. Derivation of the Continuity Equation in Cartesian Coordinates [Previous Article: How Euler Derived the Continuity Equation] The continuity equation is an expression of a fundamental conservation principle, namely, that of mass conservation. It is a statement that fluid mass is conserved: all fluid particles that flow into any fluid region must flow out. To obtain this equation, we consider a cubical control volume inside a fluid. Mass conservation requires that the the net flow through the control volume is zero. In other words, all fluid that is accumulated inside the control volume (due to compressibility for example) + all fluid that is flowing into the control volume must be equal to the amount of fluid flowing out of the control volume. Accumulation + Flow In = Flow Out

The mass of the control volume at some time t is The time rate of change of mass in the control volume is

Now we can compute the net flow through the control volume faces. Starting with the x direction, the net flow is

Similarly, the net flow through the y faces is

while that through the z faces is

Upon adding up the resulting net flow and diving by the volume of the fluid element (i.e. dxdydz), we get the continuity equation in Cartesian coordinates

8. Derivation of the Continuity Equation in Cylindrical Coordinates This is one of my favorite derivations. Although it would sound a bit intimidating at first, as none of the standard textbooks carry out the derivation in curvilinear coordinates; it is rather easy to obtain. And guess what? the math is quite rewarding! So we first have to start by selecting a convenient control volume. The idea here is to pick a volume whose sides are parallel per say to the coordinates. For cylindrical coordinates, one may choose the following control volume

Again, as we did in the previous post, we need to for all the fluid that is accumulating, and flowing through this control volume, namely: Rate of Rate of Flow In = Accumulation + Rate of Flow Out or

Accumulation + Flow Out - Flow In = 0 First, let’s get some basics laid out. The velocity field will be described as I always prefer to use u, v, and w instead of ur, utheta, and uz to save on subscripts, although the latter nomenclature is a bit more descriptive… we’ll get used to it. Now, by construction, the volume of the differential control volume is while the mass of fluid in the control volume is The rate of change of mass or accumulation in the control volume is then

For the net flow through the control volume, we deal with it one face at a time. Starting with the r faces, the net inflow is while the outflow in the r direction is

So that the net flow in the r direction is

Being O(dr^2), the last term in this equation can be dropped so that the net flow on the r faces is

The net flow in the theta direction is slightly easier to compute since the areas of the inflow and outflow faces are the same. At the outset, the net flow in the theta direction is

We now turn our attention to the z direction. The face area is that of a sector of angle d\theta:

then, the inflow at the lower z face is while the outflow at the upper z face is

Finally, the net flow in the z direction is

Now we can put things together to obtain the continuity equation

dividing by dV and rearranging the r components of the velocity

9. Derivation of the Continuity Equation in Spherical Coordinates We start by selecting a spherical control volume dV. As shown in the figure below, this is given by where r, θ, and φ stand for the radius, polar, and azimuthal angles, respectively. The azimuthal angle is also referred to as the zenith or colatitude angle. The differential mass is

We will represent the velocity field via

In an Eulerian reference frame mass conservation is represented by accumulation, net flow, and source in a control volume.

Accumulation The accumulation term is given by the time rate of change of mass. We therefore have

The net flow through the control volume can be divided into that corresponding to each direction.

Radial Flow Starting with the radial direction, we have The inflow area Ain is a trapezoid whose area is given by The key term here is the sine term. Note that the mid segment is the average of the bases (parallel sides). Upon expansion of Ain, and in the limit of vanishing dθ, we have substitution into Ain yields where high order have been dropped. The outflow in the radial direction is

but Where and By only keeping the lowest (second & third) order in the resulting expression, we have Note, that in the expression for Aout, we kept both second order and third order . The reason for this is that this term will be multiplied by "dr" and therefore, the overall order will be three. In principle, one must carry all those until the final substitution is made, and only then one can compare and keep those with the lowest order. At the outset, the net flow in the radial direction is given by

Polar Flow (θ) The inflow in the polar direction is where The outflow in the θ direction is

Where Upon expansion, and keeping both second and third order , we get Finally, the net flow in the polar direction is

Azimuthal Flow (φ) The inflow in the azimuthal direction is given by with while the outflow is

and At the outset, the net flow in the polar direction is

Continuity Equation Now, by collecting all mass fluxes we have

which, upon dividing by dV and combining , reduces to

which is the continuity equation in spherical coordinates.