Learning Management System - Virtual University of Pakistan



Leibniz Theorem:The Leibniz integral rule gives a formula for differentiation of a?definite integral?whose limits are functions of the differential variable.In one dimension form i.e. when G=G(x,t):ddta(t)b(t)G(x,t)dx=a(t)b(t)?G?t dx+Gbt,tdbdt-Gat,tdadt In three dimension form i.e. when G=G(x,y,z,t):ddtCVG(x,y,z,t)dV=CV?G?t dV+CSGVCS.n dA Here A is the area on a control surface (CS) with unit outer normal n and CV is the control volume.Reynolds theorem:By Reynolds Transport Theorem (RTT) the time rate of change of some property of a system is equal to the time rate of change of that ?control volume plus the net flux of that property out of the control volume by mass crossing that surface.Derivation of Reynold’s Transport Theorem in terms of relative velocity:By Leibniz theorem we know that ddtCVG(x,y,z,t)dV=CV?G?t dV+CSGVCS.n dA (1)For RTT, put G=ρb in eq. (2) we getddtCVρbdV=CV?(ρb)?t dV+CSρbVCS.n dA (2)Applying the Leibniz Theorem on some material volume (a system of fix identity moving with the fluid flow) say Bsys we have,dBsysdt=CV?ρb?t dV+CSρbV.n dAAs material surface moves with the fluid so absolute velocity of moving fluid will be equal to the velocity of the fluid VA = V here. So CV?ρb?t dV=dBsysdt-CSρbV.n dA (3)Putting this value from eq. (3) to eq. (2) we get,ddtCVρbdV=dBsysdt-CSρbV.n dA+CSρbVCS.n dA dBsysdt=ddtCVρbdV+CSρb(V-VCS ).ndA (4)Since V-VCS=Vr where Vr is fluid velocity relative to the coordinate system moving with the control volume, so eq. 4 becomes, dBsysdt=ddtCVρbdV+CSρbVr .ndA (5)which is the general form of Reynold’s Transport Theorem for not fixed control volume. If the control volume is fixed then we have,dBsysdt=ddtCVρbdV+CSρbV .ndA (6)which is the general form of Reynold’s Transport theorem for fixed control volume. Eq. 6 can be interpreted as“Time rate of change of some property of a system is equal to the time rate of change of that control volume plus the net flux of that property out of the control volume by mass crossing that surface.” ................
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