* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Derive from first principles the Poiseuille equation for
Drag (physics) wikipedia , lookup
Magnetorotational instability wikipedia , lookup
Lattice Boltzmann methods wikipedia , lookup
Euler equations (fluid dynamics) wikipedia , lookup
Lift (force) wikipedia , lookup
Magnetohydrodynamics wikipedia , lookup
Stokes wave wikipedia , lookup
Hemodynamics wikipedia , lookup
Wind-turbine aerodynamics wikipedia , lookup
Boundary layer wikipedia , lookup
Coandă effect wikipedia , lookup
Compressible flow wikipedia , lookup
Flow measurement wikipedia , lookup
Airy wave theory wikipedia , lookup
Hydraulic machinery wikipedia , lookup
Fluid thread breakup wikipedia , lookup
Aerodynamics wikipedia , lookup
Computational fluid dynamics wikipedia , lookup
Flow conditioning wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
Reynolds number wikipedia , lookup
Derivation of the Navier–Stokes equations wikipedia , lookup
Derive from first principles the Poiseuille equation for pressure drop generated by the steady flow of a Newtonian fluid through a straight tube of circular cross-section. If the flow is laminar, what is the form of the velocity profile with in the tube? Show that the mean velocity is Half the peak in such circumstances. Consider steady flow of a Newtonian fluid through a straight tube of circular crosssection with a radius R. Consider an element of fluid with a radius r and axial length L. Let the pressure at the end be p1 and p2 as shown in the figure. P1 r L P2 The net force pushing the fluid is FP =p1πr2 - p2πr2 = πr2(p1 – p2) Note that the force inducing the motion of the fluid is the difference or gradient in pressure and does not depend upon the absolute magnitude of the pressure itself. In other words, even if the pressure in the tube is very large, there will be no motion of the fluid if there is no difference in pressure between the two ends and the motion will be in the direction of the positive pressure difference. In an ideal fluid with no viscosity, the fluid will move in bulk. However in viscous fluids, there will be a resistance to the fluid motion as shear force is induced between adjacent layers of fluid. In the figure, the shear force retarding the motion is acting on the surface of the fluid element and given by FT = τ2πrL In fully-developed steady flow, these two forces balance each other and by equating the above two expressions, we obtain πr2(p1 – p2) = τ2πrL and this equation can be reduced to τ = r(p1 – p2)/(2L) The constitutive law for a Newtonian fluid is given by: du dr (1) (2) where u and μ are the velocity and viscosity of the fluid, respectively. In flow through tubes, the velocity magnitude is zero near the wall and increases towards the center of the tube. Thus, the ratio du/dr is negative and a negative sign is arbitrarily introduced. Combining (1) and (2) and rewriting, we obtain: P rdr du 2L with ΔP= p1 – p2 Integrating the above relationship, we obtain: r 2 P u C 4L where C is a constant (3) Since the tube wall does not move, the fluid adjacent to the wall is stationary. Thus the boundary condition can be specified as u = 0 at r = R. Applying the boundary condition to (3), we can obtain C: R 2 P C 4L and the expression for the velocity profile over the tube section will be given by: R 2 r 2 P (4) u 4L Now we need to relate (4) with the flow Q. The flow in an annulus of thickness r is given by: So the flow can be written: R 4 P Q 8L (5) Equation 5 is referred to as the Poiseuille Equation and relates the flow rate to the pressure drop across the tube of radius R and a Newtonian fluid with a viscosity coefficient μ. If the flow is laminar, from equation (4), the velocity profile will be parabolic with the maximum velocity umax occurs at the center of the tube, as shown in the below figure: From equation (4) the peak velocity (r=0) is u max R 2 P 4L The mean velocity can be calculated using equation (5) u mean Q R 4 P 1 R 2 P 2 A 8L 8L R Hence the ratio u mean u max R 2 P 1 8L 2 R P 2 4 L References: 1. http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Examples/eg7_ans.htm 2. http://hyperphysics.phy-astr.gsu.edu/Hbase/pfric.html 3. Transport Phenomena, 3rd ed. by Bird et.al.