[유체] Chap 3. Flow equations

Chap 3. Flow equations

3.1 Reynolds (Kinetic) Transport theorem

\[{d Q_{sys} \over dt} = \iiint_{C.V} {\partial \over \partial t} (\rho q)d \forall+ \iint_{C.S} \rho q \vec V_r\cdot \vec n ds\]

• Interpretation : Material derivative can be divided into sum of rate changed by time in body and flux on the surface

Divergence Theorem

\[\iiint_{V} = \big( \nabla \cdot \vec G) d\forall = \iint_{S} \vec G \cdot \vec n ds\] \[\iiint_V \lbrace \vec G \cdot \nabla f + f(\nabla \cdot \vec G)\rbrace d\forall = \iint_S f\vec G\cdot n ds\]

• Quantity of system and q
  • $q = {d Q_{sys} \over dm}$
  • Mass : 1
  • Linear momentum, $q = \vec u$
  • Angular momentum, $q = \vec r \times \vec u$

Mass conservation

\[{d Q_{sys} \over dt} = 0 \text{ (by mass conservation)}\] \[0 = \iiint_{C.V} {\partial \over \partial t}\rho d\forall + \iint_{C.S} \rho \vec V_r \cdot \vec n ds\]

Apply divergence theorem,

\[0 = \iiint_{C.V} [{\partial \rho \over \partial t} + \nabla \cdot (\rho \vec u) ] d\forall\]

Since the value of integral is zero, integrand is also zero :

Continuity equation derived as follows

\[{\partial \rho \over \partial t} + \nabla \cdot (\rho\vec u) = 0\]

if $\rho$ is constant, $\nabla \cdot \vec u = 0$ : incompressible flow.

Linear momentum conservation

\[{d \vec P \over dt} = \iiint_{C.V} {\partial \over \partial t} \rho \vec u d\forall+ \iint _{C.S} \rho \vec u \vec V_{r} \cdot \vec n ds\] \[m{d\vec u \over dt} = \sum F = \iiint _{C.V} \rho[{\partial \vec u \over \partial t} + \vec u \cdot \nabla \vec u ] d\forall\]

Considering force acting on parallelpiped infidecimal volume,

\[m {d\vec u \over dt} = \rho \iiint_{C.V} \lbrace {\partial \vec u \over \partial t} + \vec u \cdot \nabla \vec u\rbrace d\forall = -\iiint_{C.V} [\nabla p -F] d\forall\] \[{\partial \vec u \over \partial t} + \vec u \cdot \nabla \vec u \ \ (\ \ + \ \ \nu \nabla^2 \vec u \ \ )= -{1\over \rho} \nabla p + \vec f\]

Which leads to Euler / Navier-stokes Equation

3.2 Interpretation of flow

$\vec u = (u, v, w), \ \ p$ : 4 unknowns

$\nabla \cdot \vec u= 0, \ {\partial \vec u \over \partial t} + \vec u \cdot \nabla \vec u = -1/\rho \nabla p + \vec f$ : 4 equations by governing eqn.

• Governing equation : Continuity + Euler (or N-S) : a form of pde, boundary condition matters

3.3 Boundary condition

• Kinematic Boundary condition
  • No flux condition (inpermeability) : $\vec u \cdot \vec n = u_n$
    • normal velocity of the fluid is as same as normal velocity of a body
    • note : normal direction is faced inside the body since it is perspective of fluid
  • No slip condition : $\vec u \cdot \vec t = u_t$
    • tangential velocity is same
  • If satisfies no-slip condition, it automatically satisfies no flux condition
• Dynamic Boundary condition
  • Pressure, Force, Shear, Stress …