Question

2. Use the scale analysis to derive momentum equation and energy equation of boundary layer under an external forced convection, and then determine the order pf magnitude of the ratio of thermal boundary layer thickness to hydrodynamic boundary layer thickness

Answer 1

4.4.2 Momentum Equation Momentum equation is based on Newton's second law of motion which states that rate of change of momentum of a fluid element in a particular direction in control volume is equal to net force acting on control volume in that direction Momentum of mass of fluid entering the control volume in x direction is Where m, is the mass of the fluid flowing in x-direction The momentum efflux through the right face in x-direction is Let the mass flowing to the control volume through the bottom face in y direction be m,. Since we are concerned only with momentum in x-direction, therefore the momentum influx from the bottom face in x-direction is And the momentum effux from the top face in x-direction is oy су λι oy oy

m, u Control Volume myu Figure 4.7 Momentum forces acting on control volume in x-direction The resultant change in momentum in x-direction Total momentum of mass leaving the control volumeTotal momentum of mass entering the control volume λι ou au au ουόν охоу We know that (au/ox)+(avloy)-0from continuity equation and therefore the net momentum transfer to the control volume in x-direction becomes: au ou охоу A fluid motion is subjected to several forces which results in variation of fluid velocity and hence momentum. The various forces that influence the motion of fluid are Gravity Force ( due to weight of fluid acting in vertical downward direction, Pressure Force (5, is exerted on the fluid mass due to difference in pressure between two point, Viscous Shear Force (F) due to viscosity, Turbulence Force (Fr) due to random movement of fluid particles between adjacent fluid layers, Surface Tension Force F) acting at the free surface due to cohesive property of fluid and Compressibility Force F) due to elastic property of compressible fluids. From Newton's second law of motion ou au Taking into consideration only viscous shear forces ou ou 4.10) ох ду

Control Volume Figure 4.7: Viscous shear forces acting on control volume in x-direction The shearing force due to viscosity at the lower face of the control volume is λι The shearing force due to viscosity at the upper face of the control volume is dy dx Neglecting the shearing force in y-direction, the net viscous force in the direction of motion is dx dy From equation (4.10) and (4.11) ou ou dx dy ou au Since term(u/p) can be written as kinematic viscosity (v), we obtain This is the force or momentum equation in the laminar boundary layer region in x-direction for a steady, incompressible fluid flow

4.4.3 Energy Equation For a control volume of dimensions (dxxdyx1) in the boundary layer, the quantities of energy transfer across the boundaries of control volume have been shown in Figure 4.8. Conduction heat transfer in x-direction is neglected and conduction is considered only in the y «liveclieon

Viscous heat generation Control Volume -kdx Figure 4.8 Differential control volume for energy conservation Rate of energy entering in the control volume in x-direction (Exl)-mx specific heat × temperature = (ρ u dy)c T Rate of energy leaving the control volume in x-direction (Ex2)-ρ| u +--dx |dy c| T +-jax Neglecting the product of small quantities (Ex2)-ρ c dy uT-u-dx + T-dx Net rate of energy transfer in x direction due to convection OT Similarly net rate of energy convection in the y-direction охоу in Net rate of conduction heat transfer y-direction (4.16) Due to viscosity of flowing fluid, viscous heat generation-μ For steady state conditions, net accumulation of energy in the control volume is zero and thus the algebraic sum of all the energies crossing the boundaries of control volume equals zero. Thus o-T

We know that (au/ox)+(avlay) 0 from continuity equation and therefore the equation (4.18) can be written as: If the heat generation due to viscous effects is neglected, the energy conservation equation (4.19) takes the form; (4.20)

Expression for the convective heat transfer coefficient for laminar flow over a flat plate: In order to derive an expression for convective heat transfer coefficient for laminar flow over a flat plate (that has an unheated starting length xo), let us use cubic velocity and temperature distribu- tions in the integral boundary layer energy equation as follows The cubic velocity profile within the boundary layer is of the form [En. (7.33)] The conditions which are satisfied by the temperature distribution within the boundary layer are: ( Aty

Upon substitution in eqn. (7.82), we get dx 3., ds 20 dx 10 α | 2842 .(7.84) or 10 dx δ 140 μ 140 dr 13ρυ 13 U 2 280 280 vx ux Also ..Refer Art 7.5] 13 pU 3U Putting these values in eqn. (7.84), we obtain 13U dx 13U (7.85) or dx 14 v Using the equality-te(r')-Y , we can write eqn. (7.85) as = 3r-- dr 13α dx 14v .(7.86) The general solution of the equation (which is a linear differential equation of the first order in r3) is given by The value of the constant C can be evaluated from the following boundary conditions 13α.3/4 , and Thus, or 14 v 130 314 -3/4 + 13α 14 v 3/4 or, 1/3 1/3 3/4 13 34 ך1/3 δ1h_ 0.975 I: Pr (Prandtl number)] S 0.975 8(Pr or 1/3 when the plate is heated over the entire length i.e., xo = 0