Unger outer surface inFluids 2021, six,7 ofthe path in the top to the bottom. Moreover, due to the modest gap size, it is actually affordable to assume the shear force acting on the outer surface of the plunger is the identical as that of the inner surface of the barrel [23,24]. CGS 12066 dimaleate Autophagy Therefore, these two surface shear forces will balance the total typical force as a result of pressure distinction more than the plunger length, namely, 2Fp = 2R a p, (19)where Fp stands for the viscous shear force acting around the plunger outer surface on account of Poiseuille flow. It’s clear that Equation (19) is constant with Equation (18) and the major term in Equation (eight). In reality, in engineering practice, the dominant term is normally sufficient. It is apparent that using the enable from the physics and mathematics insights [25,26], the simplified rectangular domain is substantially less complicated to deal with than the annulus area and this advantage will probably be additional vital when we go over the relaxation time and the case with eccentricities in Section 3. Similarly, for the Couette flow, around the inner surface on the pump barrel at y = h and also the outer surface on the plunger at y = 0, we’ve got the kinematic situations w(0) = U p and w(h) = 0. Therefore, the velocity profile within the annulus or rather simplified rectangular region might be expressed as U p (h – y) . (20) h Moreover, we are able to easily establish the flow rate Qc via the concentric annulus area with h = as w(y) =hQc =2R a w(y)dy.The flow rate due to the shear motion at y = 0 (outer surface in the plunger) is established as Qc = R a U p h, (21)which matches using the top term in Equation (12). Consequently, the viscous shear force acting on the plunger outer surface inside the direction from the top rated towards the bottom could be calculated as Fc = – 2R a L p w y=y =2L p a U p ,(22)exactly where Fc may be the viscous shear force acting on the plunger outer surface resulting from Couette flow. In comparison with Equation (13), it truly is again confirmed that the top term matches with the simplified expression in (22). Furthermore, in order for us to derive Equation (23) from a full-fledged Navier-Stokes equations, we have to recognize irrespective of whether or not the fluid flow is within the turbulent area too because the transient effects [27,28]. Initial of all, inside the gap which is measured in mills, for common oils, the kinematic viscosity at 100 C is about 5.three cSt or five.three 10-6 m2 /s, about 5 instances that of the water and also the plunger velocity U p is no greater than 40 in/s, as a result the so-called Reynolds quantity Re = U p / is much smaller than 100 let alone the turbulent flow threshold about 2000. While the Reynolds quantity is often a clear indication about the quasi-static nature from the Couette and Poiseuille flows within the Desacetylcefotaxime web narrow annulus area, in an effort to have some guidance with respect towards the choice of the sampling time within the experimental measurements in the pressure along with the displacement inside the sucker rod pump unit, we need to investigate further the inertia effects along with other time dependent troubles. Take into account the general governing equation for the viscous flow inside the annulus region R a r Rb as expressed as w p 1 w = – r , t z r r r (23)Fluids 2021, 6,8 ofwhere the plunger length is L p along with the pressure gradientp p is expressed as – . z Lp Note that the stress distinction p is good when the upper area (major) stress is greater than the reduce region (bottom) stress which is consistent together with the leakage definition. Assuming the plunger velocity is U p , namely, w( R a) = U p , by combining the.
