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Math Help - can anyone help me with this fluid mechanics boundary vaule problem dealing with pde

  1. #1
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    Question can anyone help me with this fluid mechanics boundary vaule problem dealing with pde

    Given the second order non linear BVP

    (ƒ')^n = 1 + γ θ …..……………………… 1

    θ" + (λ +n+1/ 2n +1) ƒ θ' - n (2 λ +1/2n+ 1) * ƒ' θ = 0 ………… 2

    Prime in the above eqn’s describe partial differentiation with respect to η

    Boundary conditions are
    ƒ (0) = 0, θ'(0) = -1
    ƒ' (∞) = 0, θ(∞) = 0

    where η is a function of x & y given by
    Similarity variable, η = x ^ (λ-n/2n+1) * y
    Θ is a dimensionless temperature
    ƒ is a dimensionless stream function given by
    Stream function, ψ = x ^ (λ+n+1/2n+1) * ƒ(η)

    how to solve the above system of PD eqn’s
    the above system of equations are related to mixed convection of non newtonian fluids
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  2. #2
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    Ugly. A couple of queries: are you sure you haven't confused your n's with your η's anywhere? Furthermore, if you're only taking the derivative with respect to η anywhere, isn't it an ODE? It might be a system of ODE's, but if the number of independent variables is just 1, then you've got yourself an ODE.

    What have you done so far?
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  3. #3
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    n is the viscosity index
    I haven't confused with n's and η's anywhere
    I think that equations are differentiable with respect to η and η is function of x and y
    So the equations are PDE’s
    Actually these equations are related to boundary layer mixed convection of non Newtonian fluids on vertical plate
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  4. #4
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    Ok. What is/are the function(s) for which you wish to solve? It looks like you could treat Equations 1 and 2 with those boundary conditions as a system of ODE's, even if, underneath, they are pde's. You've got an extremely complex system there. I doubt if I can be a whole lot of help, since it's a nonlinear system of pde's. Although... Have you ever heard of the Inverse Scattering Transform method? It's possible that method might work, if you're looking for an exact solution. If you only need an approximation, I'd hand it off to a computer.
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  5. #5
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    A couple of things.
    First off - your equations are ODEs not PDEs. There only one variable \eta.
    Second. It might be wise to choose a different similarity variable - \eta and n are just to close to tell the difference.
    Third, you got to be careful with your bracket i.e is 2 λ +1/2n+ 1 = 2 \lambda+\dfrac{1}{2n+ 1}.

    You can eliminate \theta giving a third order ODE in f and then reduce to second order but the resulting ODE is still very nonlinear.

    What was the original set of PDEs that you started with? Work on these might have already been done before. Maybe someone here might know of previous work in this area.
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  6. #6
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    Given the second order non linear BVP

    (ƒ')^n = 1 + γ θ …..……………………… 1

    θ" + [(λ +n+1)/( 2n +1)] ƒ θ' - n [(2 λ +1)/(2n+ 1)] * ƒ' θ = 0 ………… 2

    Prime in the above eqn’s describe partial differentiation with respect to η

    Boundary conditions are
    ƒ (0) = 0, θ'(0) = -1
    ƒ' (∞) = 0, θ(∞) = 0

    The above nonlinear coupled system of equations for ƒ and θ have been derived from conservation laws that govern the boundary layer flow on vertical plate in porous medium by introducing similarity variable η and stream function ψ

    where
    γ can be assumed as a Rayleigh number type i.e it represents the relative importance
    of free to forced convection
    η is a function of x & y given by
    Similarity variable, η = x ^ (λ-n/2n+1) * y
    n is the permeability
    ƒ is a dimensionless stream function given by
    Stream function, ψ = x ^ (λ+n+1/2n+1) * ƒ(η)
    θ is a dimensionless temperature given by
    T = x^ [{n(2 λ+1)/2n+1}]* θ(η) and
    λ is a scalar obtained by assuming the surface heat flux Q(X)=x^ λ at y=0 which vary according to power laws

    I would like to solve them for approximations of λ and n
    I know that the above system of equations can be solved by finite difference methods using shooting technique.
    I would like to know how to start the solution and would like to know what type of PDE’s are these i.e. parabolic, elliptic, and hyperbolic since MATLAB can solve elliptic nonlinear PDE’s as far as I know and also I am not familiar with solving PDE's in MATLAB.
    So can any one suggest me the right path.
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