Reynolds transport theorem

The Reynolds theorem or Reynolds transport theorem (also known as the Leibniz-Reynolds transport theorem) is the generalised form of the Leibniz integral rule in three dimensions. The name is given as such because of Osborne Reynolds.

The Reynolds transport theorem states that the sum of the rate of change of property per unit time for a control volume and rate of efflux of the property are equal to the rate of the change of the extensive property of the system with respect to time. The main function of the Reynold transport theorem is to help to drive the laws of conservation like conservation of momentum, conservation of linear momentum, conservation of mass, conservation of kinetic energy, etc.

Generalised form of the Reynolds equation

The generalised form of the Reynolds equation is given as,

dNdtsystem=c.v.t()dV+c.s.vbndA+c.s.vrndA

Where η represents the intensive property related to the extensive property N, (N per unit mass), t represents the time, cv represents control volume, cs represents control surface, ρ represents the fluid density, V represents the volume, vb represents the velocity of the boundary of the control volume (the control surface), vr represents the velocity of the fluid with respect to the control surface, n represents the outward pointing normal vector on the control surface, and A represents the area.

Kinematic transport theorem

Newton’s second law needs a balance between the rate of change of particle momentum and the applied forces in particle physics. In continuum mechanics, the rate of change in a fluid volume must be calculated. Different control options are now available, such as a geometric volume that is fixed in space or moves according to a set of rules, or a material volume made up of the same fluid body. Now, the rate can be determined by using the kinematic transport theorem. Firstly, derive the theorem for any moving volume, V(t) bounded by S(t), therefore, the derivation can be written as,

Theorem 1: Let F(x,t) be some fluid property per unit volume, then,

ddtvFdv=vFtdV+sFUndS

Where Un is the normal component of the velocity of a point on S. Further,

v F(x,t)dVt+dt=V(t+dt)F(x,t+dt)dV

        =V(t+dt)F(x+t)+Ftdt+O(dt)2dV

Now, because of the movement of S(t) the volume change for a surface element dS is Un dt dS,

V(t+dt)=V(t)+V=V(t)+S(t)UndSdt

VF(x,t)dVt+dt=V(t)F(x,t)+Ftdt+O(dt)2dV+S(t)UndSdtG(x,t)+Ftdt+O()dt2dV

=V(t)F(x,t)dV+V(t)FtdV+S(t)UndSF(x,t)dt+O(dt)2

ddtVFdV=dt01dtVF(x,t)dVt+dt–V(t)F(x,t)dt

If V(t) is a material volume containing the same moving fluid particles, then Un= qn and d/dt is the material derivative.

Theorem 2: If V(t) is a material volume

DDtVFdV =VGtdV+sF (qn)dS

This equation is known as the  Kinematic transportation theorem.

Proof of the Reynolds transport theorem

Let us consider a system where N is the extensive property of the system while n is the intensive properties of the system, therefore, the relation between the two can be written as,

b=Bm

Thus, at time t, the relation between properties of system and Control volume is given by,

N[system] t= B[cv]t……………………..(1)

After some time interval at t+Δt, the properties of the system is given by,

B[system][t+t]=B[CV][t+t]+B[out][t+t]–B[in][t+t]……………(2)

Now, the rate of change in property of the system after the time interval of t will be,

Bsystemt= B[system][t+t]–B[system][t]t

Substitute the values form equation [1] and [2],

Bsystemt= B[CV][t+t]–B[out][t+t]–B[in][t+t]–B[cv][t]t

Bsystent=B[cv][t+t]–B[cv][t]t+B[Out][t+t]t–B[in][t+t]t

Taking the limit on both the sides as Δt→0,

t0Bsystent=t0B[cv][t+t]–B[cv][t]t+t0B[Out][t+t]t–t0B[in][t+t]t…………(3)

Where,

1. t0Bsystem t=dBdtsystem
2. t0B[cv]+[t+t]–B[cv][t]t=t0Bcvt=Btcv
3. t0B[out][t+t]t= t0bOutmoutt.. As (B=bm)

=t0bOutOut∀Outt

=t0bOutOutAOutxOutt

=t0bOutOutAOutVOuttt

=t0bOutOutAOutVOut

=bOutOutAOutVOut

t0B[out][t+t]t=bOutOutAOutVOut

1. Similarly, t0B[In][t+t]t=bInInAInVIn

Further putting all the equations together

dBdtsystem=BtCV+{bOutOutAOutVOut}-{bInInAInVIn}

dBdtsystem=BtCV+[BOut–BIn]

Where  [BOut–BIn] is the reflux

BOut={bOutOutAOutVOut}, and

BIn={bInInAInVIn}

Conclusion

The Reynolds Transport theorem is also known as the Leibniz-Reynolds transport theorem which is the generalised form of the Leibniz integral Rule in three dimensions. The theorem states that the sum of the rate of change of property per unit time for a control volume and rate of efflux of the property are equal to the rate of the change of the extensive property of the system with respect to time.