Question

Consider the case of a Newtonian fluid undergoing laminar, pressure-driven flow between two parallel, infinite flat plates separated by a distance B (Figure). The bottom plate is stationary and the top plate moves at a constant velocity Vup. For a constant dynamic pressure gradient, AP/AX, P-p-g r, we wish to calculate the resulting velocity profile. 9--(%) + mai Differentiation equation: B.C.v. (y=0) -0,vxly - B) - Vu Figure 1.10 Pressure-driven flow between two infinite, parallel, flat plates. (i) () Use the finite difference method to write a set of equations that describe the velocity between two plates. Make sure to include the boundary conditions. Write a MATLAB function that solves the system of equations in part(). Make sure to calculate v, for at least 100 points within the slab. Generate a plot of v, vs. y for Vup - le m/s, (Ap/Ax) - 1, -1, B 5e m. Hint: use spdiags to create linear banded matrix

Answer 1

i) Let,

α = 1 ΔΡ μ Δr

Then,

D = zhp

At the m^th node, this second derivative can be expressed as,

d2a_al+ + - - 21 2(رود)

where,

Ay =

n is the number of elements.

Hence,

M,++MI,-- 21, ܐܐ 2(wܠ)

un = 0.5(u 2+1 +um-1 - a(Ay))

With,

0 = 'n
and
un+1 = Vur

ii) The code is as follows.

% Assign the given parameters to variables.
v_up = 0.01;
dp_by_dx = 1; mu = 1;
a = dp_by_dx/mu;
B = 0.005;

n = 100;
delta_y = B/n;
y = 0:delta_y:B;

% Initialse the velocity vector, along with Boundary Conditions.
u = v_up*ones(n+1,1);
u(1) = 0;

% Calculate the velocity vector.
u_old = u;
u = fdiff(u,n,delta_y,a);

while norm(u - u_old) > 0.000001*v_up
    u_old = u;
    u = fdiff(u,n,delta_y,a);
end

% Plot of solution from finite difference.
plot(y,u)
hold on
% Plot of analytical solution.
plot(y, y.^2+2*y)
hold off

% Function to carry out iterative finite difference operations.
function [u] = fdiff(u,n,delta_y,a)
    for i = 2:n
        u(i) = 0.5*( u(i+1) + u(i-1) - a*delta_y^2);
    end
end

The output is as follows.

0.012 0.01 0.008 0.006 0.004 0.002 OLL O 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 *10-3