Question

Question: To find: The rate of heat flow inward across the cylindrical surface \({y^2} + {z^2} = 6,0 \le x \le 4\).

Short answer

The rate of heat flow inward across the cylindrical surface \({y^2} + {z^2} = 6,0 \le x \le 4\) is \(\underline {1248\pi } \).

Step-by-step solution

Concept of Flux and Formula Used.

A surface integral is a surface integration generalisation of multiple integrals. It can be thought of as the line integral's double integral equivalent. It is possible to integrate a scalar field or a vector field over a surface.

Given:

\(K = 6.5,u(x,y,z) = 2{y^2} + 2{z^2}\)and cylindrical surface\({y^2} + {z^2} = 6,0 \le x \le 4\)

Formula used:

Calculate the Flux.

Let\(x = x\)and\(y = \sqrt 6 \cos \theta \)and\(z = \sqrt 6 \sin \theta \)with limits\(0 \le x \le 4\)and\(0 \le \theta \le 2\pi \).\({\rm{r}}(x,\theta ) = x{\rm{i}} + \sqrt 6 \cos \theta {\rm{j}} + \sqrt 6 \sin \theta {\rm{k}}\)

Find\(\nabla u\).

\(\begin{array}{l}\nabla u = \left( {{\rm{i}}\frac{\partial }{{\partial x}} + {\rm{j}}\frac{\partial }{{\partial y}} + {\rm{k}}\frac{\partial }{{\partial z}}} \right)\left( {2{y^2} + 2{z^2}} \right)\\ = {\rm{i}}\frac{\partial }{{\partial x}}\left( {2{y^2} + 2{z^2}} \right) + {\rm{j}}\frac{\partial }{{\partial y}}\left( {2{y^2} + 2{z^2}} \right) + {\rm{k}}\frac{\partial }{{\partial z}}\left( {2{y^2} + 2{z^2}} \right)\\ = {\rm{i}}(0) + {\rm{j}}(4y) + {\rm{k}}(4z)\\ = 4y{\rm{j}} + 4z{\rm{k}}\end{array}\)

Find\({r_x}\).

Modify equation (2).

\({{\rm{r}}_x} = \frac{{\partial x}}{{\partial x}}{\rm{i}} + \frac{{\partial y}}{{\partial x}}{\rm{j}} + \frac{{\partial z}}{{\partial x}}{\rm{k}}\)

Substitute\(x\)for\(x,\sqrt 6 \cos \theta \)for\(y\)and\(\sqrt 6 \sin \theta \)for\(z\),

\(\begin{array}{l}{{\rm{r}}_x} = \frac{{\partial (x)}}{{\partial x}}{\rm{i}} + \frac{{\partial (\sqrt 6 \cos \theta )}}{{\partial x}}{\rm{j}} + \frac{{\partial (\sqrt 6 \sin \theta )}}{{\partial x}}{\rm{k}}\\ = (1){\rm{i}} + (0){\rm{j}} + (0){\rm{k}}\\ = {\rm{i}}\end{array}\)

Find \({{\rm{r}}_x} \times {{\rm{r}}_\theta }\).

Apply limits and substitute 6.5 for \(K,4\sqrt 6 \cos \theta {\rm{j}} + 4\sqrt 6 \sin \theta {\rm{k}}\) for \(\nabla u\) and \( - \sqrt 6 \cos \theta {\rm{j}} - \sqrt 6 \sin \theta {\rm{k}}\) for \({{\rm{r}}_x} \times {{\rm{r}}_\theta }\).

Simplify the equation.

Thus, the rate of heat flow inward across the cylindrical surface \({y^2} + {z^2} = 6,0 \le x \le 4\) is \(\underline {1248\pi } \).