Hello all,

Pressure gradient for a purely oscillatory flow is considered in my study, so after applying the Crank-Nicolson method to solve the equation, should I adopt an initial condition for pressure gradient?

The equation below represents the dimensionless form of the momentum equation:

Re (∂U/∂t) = - ∂p/∂x + 1/((1+λ_1)) (∂^2 U)/〖∂y〗^2 - (Ha + 1/(Da(1+λ_1))) U ...(1)

Boundary conditions:

U = 0 on y = h_1

U = 0 on y = h_2

where h_1 and h_2 are channel wall equations

I have considered an oscillatory flow, so the pressure gradient is taken as follows:

- ∂p/∂x = λ e^iωt ....(2)

where 'ω' represents the frequency of oscillations and 't' is time (t=0.1).

I have used the crank-nicolson method to solve this problem

After discretizing eq (1) :

Re ((U_(i,j+1) - U_(i,j))/Δt) = λ e^iωt + ((U_(i-1,j) - 2 U_(i,j) + U_(i+1,j) + U_(i-1,j+1) - 2 U_(i,j+1) + U_(i+1,j+1))/(2〖(Δy)〗^2 )) (1/((1+λ_1)))

- (Ha + 1/(Da(1+λ_1)))((U_(i,j+1) + U_(i,j))/2)

Note: I have substituted eq (2) directly into the discretizing step.

The corresponding initial and boundary conditions are formulated as

follows:

U_(i,j) = 0

U_(h_1,j) = 0

U_(h_2,j) = 0

Is a direct substitution of the pressure gradient assumption sufficient or should I adopt an initial condition?

How can I claim it with a valid reason or explanation?