# Mathematic Modelling

**Question description**

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

## Tutor Answer

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors