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Problem 3 (30 pts). A 2-D flow field between two infinitely large horizontal plates is initially static (V = 0). The fluid constant density of p and constant dynamic viscosity of H. The bottom plate starts moving with a constant velocity of u = U for t 20, and the velocity field shown in the figure evolves with time. a) (15 pts) Simplify the Navier-Stokes equation in the x-direction and only keep the non-trivial terms, justify the elimination of the terms. Nondimensionalize the simplified equation by using appropriate reference quantities. (hint: can use viscous diffusion time to normalize t) b) (15 pts) List all the quantities involved in the simplified equation in a), obtain the PI terms based on Buckingham PI theorem by choosing p,t,y as base quantities.

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Answer 1

Solution:-

Navier - stokes theorem in two dimension

let say x

$\rho g_x-\frac{\partial p}{\partial x}+\mu(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2})=\rho\vec{a_x}$

$a_x=\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}$

Assuming

(1) no body forces in x - direction

(2) steady state

(3)

Fu Fu 2.x2 = 220

so, navier stokes theorem will reduced to

0 น ="อบ

on integrating it we will get

$u=\frac{1}{2\mu}\frac{dp}{dx}y^2+c_1y+c_2$

Boundary conditions

u = 0 at y = 0

so, 0 = 0 + 0 + $c_2$

$c_2$ = 0

so,  $u=\frac{1}{2\mu}\frac{dp}{dx}y^2+c_1y$

and y = h, u = U

so,  $U=\frac{1}{2\mu}\frac{dp}{dx}h^2+c_1h$

$C_1=\frac{U}{h}-\frac{1}{2\mu}\frac{dp}{dx}\cdot h$

hence

$u=\frac{1}{2\mu}\frac{dp}{dx}\cdot y^2+(\frac{U}{h}-\frac{1}{2\mu}\frac{dp}{dx}\cdot h)y$

on re-arranging terms

$\frac{u}{U}=\frac{y}{h}-\frac{h^2}{2\mu U}\frac{df}{dx}\frac{y}{h}\cdot (1-\frac{y}{h})$

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