- posses the advantage of being solvable (at least in principle), especially if the fluid in question - as is assumed throughout in the present text - is a newtonian one, exhibiting simple proportionality between the transverse gradient of longitudinal velocity
and the shear stress - Fig.D-5. The frictional force is evaluated by integration along the surface, upon which this stress acts. We shall discuss cases in which this force is balanced by one other type force:

Frictional and pressure forces

are the only forces acting on a fluid partice
Fig.I-3
in the Poiseuille flow (Fig.I-2). We shall treat here separately the plane case of flow through a slit between two parallel walls (Fig.I-3) and the axisymmetric case of flow in a circular pipe (Fig.I-4). The fluid particle, upon which the force balance is derived, need not be an infinitesimal one. In Fig.I-3 the particle is infinitesimally thin (thickness ) strip of finite length and width .. for keeping the problem only one-dimensional, no variations are assumed in the direction of the width and this calls for the width being much larger than height . The balance of forces in Fig.I-3 leads to the differential equation of the longitudinal velocity profile = f ().
Similar derivation of the velocity profile in Fig.I-4 (where the investigated fluid particle is finite-radius cylinder) is followed by another integration step, leading to
Fig.I-4
mass flow rate through the tube. Dividing the result by the cross-sectional area and multiplying it by leads to the mean velocity .. this is the velocity from which fluid kinetic energy as well as Reynolds number in chapter [D] are evaluated. Insertion of the expression for into the Bernoullis' Theorem applied to the constant-cross-section horizontal pipe flow, Fig. D-4, leads to the Hagen-Poiseuille law as the final result.
Note that for the assumptions being applicable, the above derivations require low Reynolds number ( lower than = 2 000 ) and fully developed flow (which means that in front of the pipe or slit there is to be sufficient length of pipe or slit of equal cross section.

Frictional and gravity forces

- is a combination encountered in the case of a thin liquid layer flowing down the vertical wall, as shown in Fig. I-5.
Fig.I-5
From the force balance condition on an elementary cube particle we obtain, similarly as in the previous cases, the requirement of constant second derivative of the longitudinal velocity profile = f (). As a consequence, the double integration yields also here a parabolic velocity profile. To evaluate the two integration constants requires knowledge of two boundary conditions. The condition at the wall is the obvious no-slip condition = 0 at = 0 . More interesting is the other condition on the free surface at = . The last of the investigated cube particles has no neighbouring particle further to its right-hand side, which would act upon it by a friction force. This means that there is = 0. Because there can be neither = 0 nor = , it is the transverse gradient in the expression of Newton's law that must be zero. The second boundary condition, therefore, is / = 0.


Going to another page: click
This is page Nr. I03 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

WWW server administrators: Jiri Kvarda, Zdenek Maruna ...... Contact: webmaster@vc.cvut.cz
Last change : 23.02.1998