The Orr–Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. The solution to the Navier–Stokes equations for a parallel, laminar flow can become unstable if certain conditions on the flow are satisfied, and the Orr–Sommerfeld equation determines precisely what the conditions for hydrodynamic stability are. The equation is named after William McFadden Orr and Arnold Sommerfeld, who derived it at the beginning of the 20th century.
Formulation
The equation is derived by solving a linearized version of the Navier–Stokes equation for the perturbation velocity field
u = ( U ( z ) + u ′ ( x , z , t ) , 0 , w ′ ( x , z , t ) ) {\displaystyle \mathbf {u} =\left(U(z)+u'(x,z,t),0,w'(x,z,t)\right)} , where ( U ( z ) , 0 , 0 ) {\displaystyle (U(z),0,0)} is the unperturbed or basic flow. The perturbation velocity has the wave-like solution u ′ ∝ exp ( i α ( x − c t ) ) {\displaystyle \mathbf {u} '\propto \exp(i\alpha (x-ct))} (real part understood). Using this knowledge, and the streamfunction representation for the flow, the following dimensional form of the Orr–Sommerfeld equation is obtained:
μ i α ρ ( d 2 d z 2 − α 2 ) 2 φ = ( U − c ) ( d 2 d z 2 − α 2 ) φ − U ″ φ {\displaystyle {\frac {\mu }{i\alpha \rho }}\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)^{2}\varphi =(U-c)\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)\varphi -U''\varphi } , where μ {\displaystyle \mu } is the dynamic viscosity of the fluid, ρ {\displaystyle \rho } is its density, and φ {\displaystyle \varphi } is the potential or stream function. In the case of zero viscosity ( μ = 0 {\displaystyle \mu =0} ), the equation reduces to Rayleigh's equation. The equation can be written in non-dimensional form by measuring velocities according to a scale set by some characteristic velocity U 0 {\displaystyle U_{0}} , and by measuring lengths according to channel depth h {\displaystyle h} . Then the equation takes the form
1 i α R e ( d 2 d z 2 − α 2 ) 2 φ = ( U − c ) ( d 2 d z 2 − α 2 ) φ − U ″ φ {\displaystyle {1 \over i\alpha \,Re}\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)^{2}\varphi =(U-c)\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)\varphi -U''\varphi } , where
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