Question

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5. (Hints: This derivation is presented in your textbook briefly. I also discussed that in the class. I would like you to provide step-by-step process for this mathematical derivation. You need to use the continuity equation (Eq. 6-21) for the derivation process) Starting from the first law of Thermodynamics for a differential control volume, derive the general governing equation for temperature (6-35) for a 2D flow over flat plate. Using boundary layer assumptions derive Eq. (6-41) from Eq. (6-35). pres Continuity Equation CHAPTE Sed Theし0 ation of mass principle is simply a statement that mass cannot conservatoved during a process and all the mass must be accounted com r destroyed a analysis. In steady flow, the amount of mass within the control 6- dunnig ains constant, and thus the conservation of mass can be expressed as 0or during an analysis. In Rate of mass flow Rate of mass flow out of the control volume into the control volume (6-18) 6-17 ss flow rate is equal to the product of density, average velocity, t ma sivity of mo- oting tha cross-sectional area normal to flow, the rate at which fluid enters the dy diffusivit.ontrol volume from the left surface is puldy 1). The rate at which the fluid aves the control volume from the right surface can be expressed as their molecu er. The eddy wall because lt (6-19) ах e profiles are Repeating this for the y direction and substituting the results into Eq. 6-18, ry layer, but we obtain rge velocity that the wal Differential contr derivation of ma boundary laye they are fluid proper- theitsthelsoknown as the con than Simplifying and dividing by dx-dy 1 gives du av (6-21) diffusivities difusuesThis is the conserva ax theilso known as the continuity equation or mass balance for steady two- con soconervation of mass relation in differential form, which is tthewall tu ensional flow of a fluid with constant density Momentum Equations t are obtained by Ne ore region e equations of motion in the velocity boundary applying Newton's second law of motion to a differen- ent in the boundary layer. Newton's second law is an onrol vole and can be stated as the net force acting The differential forms o Sigression for momental o wh thaentum balance s equal to the mass times the acceleration of the fluid inh ic alsa equal to the net rate of mo- th within th

Answer 1

I hope you are asking about continuity equation .

conservation of mass states that mass cannot be created or destroyed . So the amount of fluid entering a control volume must exit from the control volume . Here we have to assume that the density of fluid does not change .

Now mass flow rate is given by density * area *  velocity   

area = length* width assume a unit width

let the speed be changing in both x and y direction

​​​change in a property P in x direction is given by P + $\partial p/\partial x *$ dx in x direction

similarly change in velocity = $\partial v /\partial x$ * dxin x direction

we equate the mass entering with mass exiting and simplifying we get the continuity equation .

Continuity equation is necessary condition for fluid flow . Even if density changes continuity equation is valid , only we have to make slight change . We account for density changes also in a similar manner as we did for velocity .  

in the right hand side of equation we multiply by changed density instead of density. Change can be calculated by considering density property .

hope it helps comment for more ...