Please provide details for full credit, no shortcuts... thanks! An incompressibl

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Please provide details for full credit, no shortcuts... thanks!

An incompressible fluid is at rest between two large horizontal plates (as shown in the figure below). The top plate is suddenly given a velocity U at time t = 0 (along the x- or horizontal direction). You may assume constant viscosity. State the initial (IC) and boundary conditions (BCs) on the u velocity component of the flow and state where there are implemented (e.g. BC at y = ...)(cross out extra entries for BCs) BC1: BC2: IC1: IC2: In this flow, the only non-zero velocity component is u and gradients in x and z for u and v are zero, = (u(y,t), 0, 0). Determine the pressure gradient p/ y and argue or show why p/ x = 0. y-momentum: p/ y = The governing equations for u may be simplified to the following form: u/ t = mu/rho 2u/ y2 = v 2u/ y2. Non-dimensionalize the x-momentum equation using a characteristic velocity, U, the length, h, such that: u* = u/U, t* = Ut/h, y* =y/h The new differential equation will be expressed in terms of u*, t*, y* and any dimensionless parameters that you need to identify and define. Re-state the initial (IC) and boundary conditions (BCs) on the u velocity component of the flow based on the non-dimensionalized velocity and coordinates. BC1: BC2: IC1: IC2:

Explanation / Answer

a) IC,

u=0 at t=0 (only one IC)

BCs,

u=0 at y=0 (no-slip)
u=U at y=h (no-slip)

b) Simplify y-momentum equation to obtain dp/dy

(rho)(dv/dt + udv/dx + vdv/dy +wdv/dz)=-dp/dy + (mu)(d^2 v/dx^2 +d^2 v/dy^2 +d^2 v/dz^2)

v=0(given)

So, dp/dy=0

dp/dx=0 because initially the fluid is stagnant and dp/dx=0 and the flow is then driven by the moving plate on the top and not any pressure gradient....So pressure must be constant along x and hence dp/dx=0...

c) u=Uu*

t=ht*/U

y=hy*

Substituting these,

d(Uu*)/d(ht*/U)= (mu/rho)(d^2 (Uu*)/d(hy*)^2)

(U^2/h)(du*/dt*)=(mu/rho)(U/h^2)(d^2 u*/dy*^2)

or, (Uh (rho)/(mu))(du*/dt*)= d^2 u*/dy*^2

Uh(rho)/(mu) = Re (Reynolds number over refence length h)

So,

(Re)(du*/dt*)= d^2 u*/dy*^2

d) IC

u=Uu* and u=0 at t=0

So u*=0 at t=0 (only one IC)

BCs

u=Uu* and u=0 at y=0

so,

u*=0 at y=0

And, u=U at y=h

therefore, u*=1 at y=h

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