# Viscous Flow

When liquid flows along a solid surface (see Fig. 4.2) a shearing stress is set up (friction power/surface), which is expressed by

2

R = for a plate (4.34) FIGURE 4.1 Tube flow velocity profile.

2 2

T = feja = for a tube, (4.35)

Where

Vm is mean velocity (velocity is zero at the surface)

F is a dimensionless friction factor

Ј, = 4f; this is the Blasius friction factor

In some literature A or /3 is used instead of Ј; it is essential to use be­cause A is used for thermal conductivity and /3 is used for cubic expansion of air or an angle.

In a tube, the pressure is constant in all radial directions perpendicular to the axis; it varies only in the flow direction x. The power balance of the ele­ment dx, denoting the sectional surface area as A and the periphery as C, gives

-rCdx-&dxA = 0 and = =

Dx dx A I ‘ 2 A

Denoting the hydraulic diameter as dh = 4A/C, we have

Ґ=trf=<4js>

And the friction resistance head corresponding to the pressure difference is

Hf=^- = lTTl — (4-37’l

; Pg sdh2g y

The factor Ј depends on the Reynolds number, which is a dimensionless vari­able that denotes the nature of flow:

Re = , (4.38)

V t] TTd 17

Where

Vm is the mean velocity

D is the characteristic length of a surface; in the case of flow in a tube it is the tube diameter (note that d may be expressed by L in the case of a plate) v is the kinematic viscosity 77 is the dynamic viscosity qm in the mass flow

Viscosity is defined by means of the equation

Where dv/dy is the velocity gradient, t is the shearing stress between two flow layers, and rf is the dynamic viscosity.