Transient Dynamics of a Mercury–Water Differential Manometer with Laminar Viscous Damping
​Authors: Housam Binous and Ahmed Bellagi

The transient response of a mercury–water differential manometer subjected to a sudden pressure difference is studied. The model accounts for mercury inertia, hydrostatic restoring forces arising from density difference, and laminar regime viscous dissipation. The system behaves as a second-order damped oscillator with a natural frequency and a damping factor equal to 6.741 rad/s and 0.0654 (underdamped - typical for mercury–water manometers (oscillatory)), respectively. We plot the displacement of the mercury interface as well as the velocity versus time. A plot of the Reynolds number indicates that the regime is laminar (Re<2000).​​
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(*fluidsproperties*)​​rhoM=13600;(*Mercurydensitykg/m^3*)​​rhoW=1000;(*Waterdensitykg/m^3*)​​muM=1.5*10^-3;(*MercurydynamicviscosityPa.s*)​​
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g=9.81;(*Gravitym/s^2*)
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(*geometry*)​​R=1*10^-3;(*Tuberadius(m)*)​​L=0.4;(*Totalmovingmercurylength(m)*)​​A=PiR^2;(*Cross-sectionalarea*)
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tmax=20;
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(*Pressurestepinput*)​​DeltaP0=500;(*PressuredifferenceinPa*)​​DeltaP[t_]:=DeltaP0UnitStep[t];
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(*GoverningODE---Newton'ssecondlawofmotion*)​​eq=rhoMALh''[t]+(8muMLA/R^2)h'[t]+2(rhoM-rhoW)gAh[t]==ADeltaP[t];​​
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(*Initialconditions*)​​ics={h[0]==0,h'[0]==0};​​
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(*NumericalsolutionusingNDSolve*)​​sol=NDSolve[{eq,ics},h,{t,0,tmax}];​​
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(*Displacementofmercuryinterface:h(t)*)​​Plot[Evaluate[1000h[t]/.sol],{t,0,tmax},FrameLabel->{"t (s)","h (mm)"},PlotLabel->"Transient Response",PlotRange->All,Frame->True,AspectRatio->1,GridLines->Automatic,PlotStyle->Darker@Red]​​
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omegaN=Sqrt[2(rhoM-rhoW)g/(rhoML)];(*Naturalfrequency*)​​damping=8muM/(rhoMR^2);(*dampingratio*)​​zeta=damping/(2omegaN);(*lessthan1---underdamped--typicalformercury–watermanometers(oscillatory)*)​​​​Print["Natural frequency (rad/s): ",omegaN];​​Print["Damping ratio: ",zeta];
Natural frequency (rad/s): 6.74117
Damping ratio: 0.0654451
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(*velocity:h'(t)*)​​Plot[Evaluate[h'[t]/.sol],{t,0,tmax},FrameLabel->{"t (s)","v (m/s)"},PlotLabel->"Velocity",PlotRange->All,Frame->True,AspectRatio->1,GridLines->Automatic,PlotStyle->Darker@Green]​​
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(*Reynoldsnumber*)​​Plot[Evaluate[2RAbs[h'[t]]rhoM/muM/.sol],{t,0,tmax},FrameLabel->{"t (s)","Re"},PlotLabel->"Reynolds number",PlotRange->All,Frame->True,AspectRatio->1,GridLines->Automatic,PlotStyle->Darker@Blue]​​
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