Mathematic Modelling

An internal boundary layer is created by a narrow jet of fluid entering a large, stagnant volume of the same fluid as shown in the following figure. Both velocity components are important here, so the governing equation is written in terms of the stream function ψ(x, y). ∂ψ/∂y*∂^2ψ/∂x∂y-∂ψ/∂x*∂^2ψ/∂y^2=v*∂^3ψ/∂y^3 The absence of a fixed length scale suggests that a similarity solution is possible. Introducing a scale factor to account for the slowing of the jet as it moves away from the wall, we assume a similarity solution of the form ψ(x, y)=x^p*f(η), η=y/g(x) where the constant p and the functions f(η) and g(x) must be determined. (a) With the above similarity variable, show that the governing equation becomes f““+C1ff“+C2(f`)^2=0 where C1=px^(p-1)*g/v and C2={x^p*g`-px^(p-1)*g}/v (b) Let C1 = 2. Show that g=2vx^1-p/p and C2=2(1-2p)/p (c) Let p = 1/3. Show that the governing equation may then be written as f“`+2{ff“+(f`)^2}=0

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