ds △t S
The total change in volume of the whole control volume over the time increment At is obviously a g/ven as bellow △ S Step 5. If the integral above is divided by At. the result is physically the time rate change of the control volume DV 1 = △t)·dS= Dt S
The total change in volume of the whole control volume over the time increment is obviously given as bellow t ( ) S V t dS Step 5. If the integral above is divided by .the result is physically the time rate change of the control volume t ( ) = = S S V t dS V dS Dt t DV 1
Step 6. Applying Gauss theorem, we have V·V Dt Step 7 As the moving control volume approaches to a infinitesimal volume sy, Then the above equation can be rewritten as D(SV Dt
Step 6. Applying Gauss theorem, we have = V VdV Dt DV Step 7. As the moving control volume approaches to a infinitesimal volume, . Then the above equation can be rewritten as V ( ) = V VdV Dt D V
Assume that sy is small enough such that vv is the same through out Sy. Then, the integral can be approximated as (v v sy, we have D() V·VδV 1D(7) O厂 Dt Sy Dt ◆ Definition of v.V: Vv is physically the time rate of change of the volume of a moving fluid element of fixed mass per unit volume of that element
Assume that is small enough such that is the same through out . Then, the integral can be approximated as , we have V V V ( V )V ( ) V V Dt D V = ( ) Dt D V V V 1 = or Definition of : is physically the time rate of change of the volume of a moving fluid element of fixed mass per unit volume of that element. V V
o Another description of fv ds and vv Assume S is a control surface corresponding to control Wolume v, which is selected in the space at time t At time t, the fluid particles enclosed by S at time t will have moved to the region enclosed by the surface S The volume of the group ofparticles with fixed identity enclosed by s at time t is the sum of the volume in region A and B. And at time t, this volume will be the sum of the volume in region b and C As time interval approaches to zero, S, coincides with S If s is considered as a fixed control volume, then, the region in A can be imagined as the volume enter into the control surface, C leave out
Another description of and : S V dS V Assume is a control surface corresponding to control volume , which is selected in the space at time . At time the fluid particles enclosed by at time will have moved to the region enclosed by the surface . The volume of the group of particles with fixed identity enclosed by at time is the sum of the volume in region A and B. And at time , this volume will be the sum of the volume in region B and C. As time interval approaches to zero, coincides with . If is considered as a fixed control volume, then, the region in A can be imagined as the volume enter into the control surface, C leave out. V S t 1 t S t S1 S t 1 t S1 S S