When a manometer is subjected to a time dependent pressure differential, the level of liquid in the legs changes with time according to the equation:

(L/g)*(d^2h/dt^2)+(8uL/ygr^2)*(dh/dt)+h = dp/yg
where L = length of liquid in tube, h is the height differential, dp is the pressure differential, r is the radius of the manometer tube, u is the viscosity and y is the density of the manometer fluid.

Determine and plot the response of the manometer height h using both analytical and numerical solutions to a step change
dp/yg = [0 then t<=0] or[50 then t>0]

To determine and plot the response of the manometer height h to a step change in pressure differential dp/yg, we can solve the given differential equation analytically and numerically.

Analytical Solution:
1. First, let's consider the case when t <= 0. In this case, dp/yg = 0, which means there is no pressure differential (no step change).

2. Solving the differential equation for t <= 0, we have:
(L/g) * (d^2h/dt^2) + (8uL/ygr^2) * (dh/dt) + h = 0

To solve this second-order linear homogeneous differential equation, we can assume a solution of the form h(t) = A * e^(st), where A and s are constants.

3. Substituting this assumed solution into the differential equation, we get:
(L/g) * (s^2 * A * e^(st)) + (8uL/ygr^2) * (s * A * e^(st)) + A * e^(st) = 0

4. Simplifying and factoring out A * e^(st), we get the characteristic equation:
(s^2 * L/g) + (8uL/ygr^2) * s + 1 = 0

5. Solving this quadratic equation for s will give us the roots s1 and s2. The general solution for h(t <= 0) is then a linear combination of the form:
h(t <= 0) = A1 * e^(s1t) + A2 * e^(s2t)

6. Now let's consider the case when t > 0. In this case, dp/yg = 50, which represents a step change.

7. The differential equation for t > 0 becomes:
(L/g) * (d^2h/dt^2) + (8uL/ygr^2) * (dh/dt) + h = 50

To solve this non-homogeneous differential equation, we can assume a particular solution h_p(t) that satisfies the equation. One possible form is a constant h_p(t) = C.

8. Substituting h_p(t) = C into the differential equation, we get:
(L/g) * (0) + (8uL/ygr^2) * (0) + C = 50

This gives us C = 50.

9. The complete solution for t > 0 is then the sum of the particular solution and the general solution for t <= 0:
h(t > 0) = h_p(t) + A1 * e^(s1t) + A2 * e^(s2t)

Numerical Solution:
To obtain the numerical solution, we can use numerical methods such as the Euler's method or the Runge-Kutta method.

1. Define a time step size, dt, for the numerical integration.

2. Initialize the values:
h(0) = 0 (initial condition for t <= 0)
h'(0) = 0 (initial condition for t <= 0)

3. Start a loop for t > 0. At each time step:
- Calculate the values of h(t), and h'(t) using the derivative equation:
(L/g) * (d^2h/dt^2) + (8uL/ygr^2) * (dh/dt) + h = dp/yg
- Update the values for the next time step using numerical integration:
h(t+dt) = h(t) + h'(t) * dt
h'(t+dt) = h'(t) + (dp/yg - (8uL/ygr^2) * h'(t) - h(t)) * dt / (L/g)

4. Repeat the above step until reaching the desired time limit to obtain the numerical solution.

Plotting the h(t):
Now that we have obtained both the analytical and numerical solutions, we can plot the response of the manometer height h as a function of time t. This can be done using various plotting libraries in programming languages such as Python (e.g., matplotlib) or MATLAB.

By comparing the analytical and numerical solutions, we can assess the accuracy of the numerical method and understand the behavior of the manometer height h in response to the step change in pressure differential.

To determine and plot the response of the manometer height (h) to a step change in pressure differential (dp/yg), we will first analyze the analytical solution and then use numerical methods.

Analytical Solution:
The equation provided for the manometer's liquid height change is a second-order linear differential equation. To find the analytical solution, we need to solve this differential equation subject to the given initial conditions. Let's solve this step-by-step:

1. Assume the solution can be expressed as h(t) = A * e^(st) + B * e^(rt), where A and B are constants to be determined, and s and r are the roots of the characteristic equation.

2. Differentiate h(t) twice with respect to time to get:

d^2h/dt^2 = A * s^2 * e^(st) + B * r^2 * e^(rt)

3. Replace the values of d^2h/dt^2 and dh/dt in the given equation:

(L/g) * (A * s^2 * e^(st) + B * r^2 * e^(rt)) + (8uL/ygr^2) * (A * s * e^(st) + B * r * e^(rt)) + A * e^(st) + B * e^(rt) = dp/yg

4. Group similar terms together:

(A * (L/g * s^2 + 8uL/ygr^2 * s + 1) + B * (L/g * r^2 + 8uL/ygr^2 * r + 1)) * e^(st) + (A * (L/g * r^2 + 8uL/ygr^2 * r + 1) + B * (L/g * s^2 + 8uL/ygr^2 * s + 1)) * e^(rt) = dp/yg

5. Since this equation is true for all t, the coefficients of e^(st) and e^(rt) must individually equal zero:

A * (L/g * s^2 + 8uL/ygr^2 * s + 1) + B * (L/g * r^2 + 8uL/ygr^2 * r + 1) = 0 [Equation 1]
A * (L/g * r^2 + 8uL/ygr^2 * r + 1) + B * (L/g * s^2 + 8uL/ygr^2 * s + 1) = 0 [Equation 2]

6. Solve Equations 1 and 2 for A and B in terms of the given variables.

7. Once A and B are determined, substitute them into the equation h(t) = A * e^(st) + B * e^(rt) to obtain the analytical form of the manometer's response to the step change in pressure differential.

Numerical Solution:
To find the numerical solution, we will use a numerical integration method, such as the Euler's method or the Runge-Kutta method, to solve the differential equation numerically. We will discretize the time interval and iteratively solve the equation at each time step. Since the time-dependent pressure differential is defined as [0 then t<=0] or [50 then t>0], we can start with a small time step for t <= 0 (before the step change) and a larger time step for t > 0 (after the step change).

1. Start with an initial condition h(0) = 0.

2. Choose a time step size, Δt, depending on the desired accuracy.

3. For each time interval, update the height using the numerical integration method:

- For t <= 0 (before the step change), use a smaller time step size.
- For t > 0 (after the step change), use a larger time step size.

4. Repeat the calculation until the desired time has been reached.

5. Plot the numerical solution of h(t) as a function of time.

Note: The exact numerical method and implementation details may vary depending on the programming language or software you are using.

By following these steps, you can determine and plot both the analytical and numerical solutions for the manometer height (h) in response to the given step change in pressure differential (dp/yg).