Math, asked by rohangarg25, 1 year ago

derivation of Kelvin's minimum energy theorem.​

Answers

Answered by Anonymous
4

Answer:

explanation:

In fluid mechanics, Kelvin's minimum energy theorem (named after William Thomson, 1st Baron Kelvin who published it in 1849) states that the steady irrotational motion of an incompressible fluid occupying a simply connected region has less kinetic energy than any other motion with the same normal component of velocity ...


khalid0786: yes or no
khalid0786: answer in yes or no do u want to follow me
khalid0786: ok
khalid0786: hii
Answered by anildeshmukh
1

Answer:

Step-by-step explanation:

In fluid mechanics, Kelvin's minimum energy theorem (named after William Thomson, 1st Baron Kelvin who published it in 1849[1]) states that the steady irrotational motion of an incompressible fluid occupying a simply connected region has less kinetic energy than any other motion with the same normal component of velocity at the boundary (and, if the domain extends to infinity, with zero value values there)[2][3][4][5].

Mathematical Proof

Let {\displaystyle \mathbf {u} } \mathbf {u}  be the velocity field of an incompressible irrotational fluid and {\displaystyle \mathbf {u_{1}} } {\displaystyle \mathbf {u_{1}} } be that of any other incompressible fluid motion with same normal component velocity {\displaystyle \mathbf {u} \cdot \mathbf {n} =\mathbf {u_{1}} \cdot \mathbf {n} } {\displaystyle \mathbf {u} \cdot \mathbf {n} =\mathbf {u_{1}} \cdot \mathbf {n} } at the boundary of the domain, where {\displaystyle \mathbf {n} } \mathbf {n}  is the unit vector of the bounding surface (and, if the domain extends to infinity, {\displaystyle \mathbf {u} \cdot \mathbf {n} =\mathbf {u_{1}} \cdot \mathbf {n} =0} {\displaystyle \mathbf {u} \cdot \mathbf {n} =\mathbf {u_{1}} \cdot \mathbf {n} =0} there). Then the difference between the kinetic energy is given by

{\displaystyle T_{1}-T={\frac {1}{2}}\rho \int (\mathbf {u} _{1}^{2}-\mathbf {u} ^{2})\ dV} {\displaystyle T_{1}-T={\frac {1}{2}}\rho \int (\mathbf {u} _{1}^{2}-\mathbf {u} ^{2})\ dV}

can be rearranged to give

{\displaystyle T_{1}-T={\frac {1}{2}}\rho \int (\mathbf {u} _{1}-\mathbf {u} )^{2}\ dV+\rho \int (\mathbf {u} _{1}-\mathbf {u} )\cdot \mathbf {u} \ dV.} {\displaystyle T_{1}-T={\frac {1}{2}}\rho \int (\mathbf {u} _{1}-\mathbf {u} )^{2}\ dV+\rho \int (\mathbf {u} _{1}-\mathbf {u} )\cdot \mathbf {u} \ dV.}

Since {\displaystyle \mathbf {u} } \mathbf {u}  is irrotational and the domain is simply-connected, a single-valued velocity potential exists, i.e., {\displaystyle \mathbf {u} =\nabla \phi } {\displaystyle \mathbf {u} =\nabla \phi }. Using this, the second integral in the above equation can be written as

{\displaystyle \int (\mathbf {u} _{1}-\mathbf {u} )\cdot \nabla \phi \ dV=\int \nabla \cdot [(\mathbf {u} _{1}-\mathbf {u} )\phi ]\ dV-\int \phi \nabla \cdot (\mathbf {u} _{1}-\mathbf {u} )\ dV.} {\displaystyle \int (\mathbf {u} _{1}-\mathbf {u} )\cdot \nabla \phi \ dV=\int \nabla \cdot [(\mathbf {u} _{1}-\mathbf {u} )\phi ]\ dV-\int \phi \nabla \cdot (\mathbf {u} _{1}-\mathbf {u} )\ dV.}

The second integral is identically zero for steady incompressible fluid, i.e., {\displaystyle \nabla \cdot \mathbf {u} =\nabla \cdot \mathbf {u} _{1}=0} {\displaystyle \nabla \cdot \mathbf {u} =\nabla \cdot \mathbf {u} _{1}=0}. Applying the Gauss theorem for the first integral we find

{\displaystyle \int (\mathbf {u} _{1}-\mathbf {u} )\cdot \nabla \phi \ dV=\int \phi (\mathbf {u} _{1}-\mathbf {u} )\cdot \mathbf {n} \ dA} {\displaystyle \int (\mathbf {u} _{1}-\mathbf {u} )\cdot \nabla \phi \ dV=\int \phi (\mathbf {u} _{1}-\mathbf {u} )\cdot \mathbf {n} \ dA}

where the surface integral is zero since normal component of velocities are equal there. Thus, one concludes

{\displaystyle T_{1}-T={\frac {1}{2}}\rho \int (\mathbf {u} _{1}-\mathbf {u} )^{2}\ dV\geq 0} {\displaystyle T_{1}-T={\frac {1}{2}}\rho \int (\mathbf {u} _{1}-\mathbf {u} )^{2}\ dV\geq 0}

or in other words, {\displaystyle T_{1}\geq T} {\displaystyle T_{1}\geq T}, where the equality holds only if {\displaystyle \mathbf {u} _{1}=\mathbf {u} } {\displaystyle \mathbf {u} _{1}=\mathbf {u} }, thereby proving the theorem.


dipi84: ;-)
anildeshmukh: ACTUALLY ITS GRAPHICS SO
anildeshmukh: sorry
dipi84: Why sorry
anildeshmukh: that u can not read
dipi84: I can read
anildeshmukh: ;)
anildeshmukh: ;)
dipi84: hii
Similar questions